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# TEXTBOOK SOLUTION MANUAL Economics 103: Econometrics

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Instructor s Manual For Principles of Econometrics Fourth Edition Instructor s Manual For Principles of Econometrics Fourth Edition WILLIAM E GRIFFITHS University of Melbourne R CARTER HILL Louisiana State University GUAY C LIM University of Melbourne SIMON YUNHO CHO University of Melbourne SIMONE SIYIN WONG University of Melbourne JOHN WILEY amp SONS INC New York Chichester Weinheim Brisbane Singapore Toronto PREFACE This Instructor s Manual contains solutions to the Exercises in the Probability Primer Chapters 2 16 and Appendices A B and C in Principles of Econometrics 4th edition by R Carter Hill William E Grif ths and Guay C Lim John Wiley amp Sons 2011 There are several other resources available for both students and instructors Full details can be found on the Web page httpprinciplesofeconometricscomp0e4p0e4htm These resources include 0 Answers to Selected Exercises These answers are available to both students and instructors They are shortened versions of the solutions in this Manual for exercises that are marked in POE4 with a 0 Supplementary computer handbooks designed for students to learn software at the same time as they are using Principles of Econometrics to learn econometrics These handbooks are available for the following software packages gt EViews Stata GRETL Excel SAS VVVV 0 Data les for all text examples and exercises The following types of les are available V Data de nition les def are text les containing variable names de nitions and summary statistics Text les dat containing only data Variable names are in def les EViews work les wf1 compatible with EViews Versions 6 or 7 Stata data sets dta readable using Stata Version 9 or later Excel spreadheets xlsx for Excel 2007 or 2010 GRETL data sets gdt gt SAS data sets sas7bdat compatible with SAS Version 7 or later VVVVV We welcome any comments on this manual Please feel free to contact us if you discover errors or have suggestions for improvements William E Grif ths Simon Yunho Cho wegrifunimeZb edu cm yunhocunimeZb edu an R Carter Hill Simone SiYin Wong eohillZsu edu simone sy wgmaiZ com Guay C Lim g ZimunimeZb eduou October 1 2011 CONTENTS Solutions to Exercises in Probability Primer 1 Chapter 2 The Simple Linear Regression Model 21 Chapter 3 Interval Estimation and Hypothesis Testing 54 Chapter 4 Prediction Goodness of Fit and Modeling Issues 97 Chapter 5 The Multiple Regression Model 132 Chapter 6 Further Inference in the Multiple Regression Model 178 Chapter 7 Using Indicator Variables 225 Chapter 8 Heteroskedasticity 271 Chapter 9 Regression with Time Series Data Stationary Variables 308 Chapter 10 Random Regressors and Moment Based Estimation 360 Chapter 11 Simultaneous Equations Models 387 Chapter 12 Regression with Time Series Data NonStationary Variables 424 Chapter 13 Vector Error Correction and Vector Autoregressive Models 448 Chapter 14 TimeVarying Volatility and ARCH Models 472 Chapter 15 Panel Data Models 489 Chapter 16 Qualitative and Limited Dependent Variable Models 527 Appendix A Mathematical Tools 5 7 6 Appendix B Probability Concepts 586 Appendix C Review of Statistical Inference 604 PROBABILITY PRIMER Exercise Solutions EXERCISE P1 a X is a random variable because attendance is not known prior to the outdoor concert Before the concert attendance is uncertain because the Weather is uncertain b Expected attendance is given by EX Zxfx 500gtlt 02 1000gtlt 06 2000gtlt 02 1100 c Expected pro t is given by E0 E5X 2000 5EX 2000 5 gtlt1100 2000 3500 d The variance of pro t is given by Probability Primer Exercise Solutions Principles of Econometrics 4e 2 varY var5X 2000 52 varX 25 x 240000 6000000 Probability Primer Exercise Solutions Principles of Econometrics 4e EXERCISE P2 a The completed table is Y f xy 0 1 f X 10 018 000 018 x 0 000 030 030 10 007 045 052 fy 025 075 b EXZxfx lOgtltOl80gtltO3lOgtltO5234 You should take the bet because the expected value of your Winnings is positive c The probability distribution of your Winnings if you know she did not study is fxy1E for x 1001O 39 fY1 It is given in the following table X fx1fy1 fxy1 10 000075 00 0 030075 04 10 045075 06 d Given that she did not study your expected Winnings are EXY1Zxfxy1 10gtlt00OgtltO410gtltO66 Probability Primer Exercise Solutions Principles of Econometrics 4e EXERCISE P3 Assume that total sales X are measured in millions of dollars Then X N 25032 and PXgt3PZgt 33925 03 PZ gt 16667 1 PZ lt16667 1 09522 00478 14 fx 12 10 08 06 04 00478 02 L I I I I I I I I I I Probability Primer Exercise Solutions Principles of Econometrics 4e 5 EXERCISE P4 Extending the table to include the marginal distributions for political af liation PA and CITY yields Political Affiliation PA R I D f CITY Southern 024 004 012 04 CITY Northern 018 012 030 06 fPA 042 016 042 R 1 a PR CITY Northern f N0rthem O 8 0 3 fCTYNorthern 06 i b Political affiliation and region of residence are not independent because for example fR Northern 018 at fPA R X fcm Northern 042 X 06 0252 C EUDARXfPAR1XfPADXfPAD 0gtlt0422gtlt0l65gtlt042 242 d EX E2PA 2P12 2EPA 2EP12 Where EP12 R2 x fpAR I2 x fp1I D2 x MD 02 gtlt04222 gtlt0l652 gtlt042 1114 Thus EX 2EPA 2EP12 2x 242 2gtltlll4 2712 Probability Primer Exercise Solutions Principles of Econometrics 4e 6 EXERCISE P5 3 b The probability that the NFC Wins the 12th ip given they have WOI1 the previous 11 ips is 05 Each ip is independent so the probability of Winning any ip is 05 irrespective of the outcomes of previous ips Because the outcomes of previous ips are independent and independent of the outcomes of future ips the probability that the NFC will Win the next two consecutive ips is 05 multiplied by 05 That is 052 025 Go Saints Probability Primer Exercise Solutions Principles of Econometrics 46 EXERCISE P6 a ESALES E4071O 43 OPRICE 40710 43OEPRICE 40710 430 x 75 8460 b VarSALES Var4O71O 43OPRCE 4302 VarPRICE 4302 X 25 4622 500 C PSALES gt 6300 PZ gt 6300 39 8460 44622500 PZ gt 100465 PZ lt100465 08425 ooo2o fSALES 00016 00012 00003 08425 00001 I I I I I I I I I I I I I I 1000 3000 5000 17000 9000 11000 13000 15000 L4lL Probability Primer Exercise Solutions Principles of Econometrics 4e 8 EXERCISE P7 After including the marginal probability distributions for both C and B the table becomes 3 b C d 6 f B 0 1 2 f C 0 005 005 005 015 C 1 005 020 015 040 2 005 025 015 045 f 19 015 050 035 The marginal probability distribution for C is given in the last column of the above table ECZcfcOgtlt0l5lgtltO402gtltO45l3 varC Zc2fc EC2 02 x 015 12 x 040 22 x 045 132 051 For the two companies advertising strategies to be independent the condition f 6319 fc CfB 19 must hold for all c and b We find that f00 005 75 fCOfBO O15gtlt 015 00225 Thus the two companies advertising strategies are not independent Values for A are given by the equation A 50001000B Its probability distribution is obtained by matching values obtained from this equation with corresponding probabilities for B A f a 5000 015 6000 050 7000 035 Since the relationship between A and B is an exact linear one they are perfectly correlated The correlation between them is 1 Probability Primer Exercise Solutions Principles of Econometrics 4e 9 EXERCISE P8 3 X f X 1 16 2 16 3 16 4 16 5 16 6 16 1 l l l b PX4g PX4orX5gg 6 EXxfx1gtlt2gtlt3gtlt4gtlt5gtlt6gtlt35 The result EX 35 means that if a die is rolled a very large number of times the average of all the values shown will be 35 it will approach 35 as the number of rolls increases 2 2 2 1 2 1 2 1 2 1 2 1 2 1 d EX x fx 1gtltg2 gtltg3 gtltg4 gtltg5 gtltg6 gtltE 1516667 6 varX EX2 EX2 1516667 352 291667 f The results for this part will depend on the rolls obtained by the student Let n denote the average value after n rolls The values obtained by one of us and their averages are 20 values of X 2l534l55242 2 4 2 4 43263 X5 3000 X10 3200 X20 3200 These values are relatively close to the mean of 35 and are expected to become closer as the number of rolls increases Probability Primer Exercise Solutions Principles of Econometrics 4e 10 EXERCISE P9 3 fx23 29x The area under the curve is equal to one Recalling that the formula for the area of a triangle is halfthe base multiplied by the height it is given by gtlt 3 gtlt 1 b When x 1 2 f x 5 9 The probability is given by the area under the triangle between 0 and 12 This can be calculated as 1 P1 2 lt X lt 3 The latter probability is P12ltXlt3lhhlx x 069444 2 2 2 9 36 Therefore P0ltXlt121 P12ltXlt31 o30555 36 36 c To compute this probability We can subtract the area under the triangle between 34 to 3 from the area under the triangle from 14 to 3 Doing so yields P14ltXlt34PGltXlt3 PltXlt3 gtlt2gtltfl gtlt2lgtltf 4 4 4 1 11 11 1 9 1 gtlt gtlt gtlt gtlt 2 4 18 2 4 2 13 144 16 5 O27778 18 Probability Primer Exercise Solutions Principles of Econometrics 4e 11 EXERCISE P10 8 b C EZE jEXEYMuM Assuming X and Y are independent 2 X 17 m VarX VarY o2 62 67 VarZ Var Assuming that coVXY 0562 XY VarZ Var J VarX VarY 2eoVXY 2o22gtltO5o2 Probability Primer Exercise Solutions Principles of Econometrics 4e 12 EXERCISE P11 Let X denote the length of life of a personal computer selected at random The fraction of computers that fail Within a given time interval is equal to the probability that X lies in that interval 134 PXlt1P Zlt PZlt 18974 00289 ltagt lt gt M lt gt b PX24PZgt 43394jPZgtO4743O3176 16 c PX22PZgt jPZgt 11068O8658 16 2534 434 d P25ltXlt4 P Zltj P O7115ltZltO4743 PZltO4743 PZlt O7115 06824 02384 O444 e We Want X0 such that PXltX0OO5 Now PZlt1645005 and thus a suitable X0 is such that X0 34 x164 1645 Solving for X0 yields X0 34 1645x16 1319 which is approximately 16 months Probability Primer Exercise Solutions Principles of Econometrics 4e EXERCISE P12 3 C 6 he probability function of X is shown below 40 u nuuuuuuu d w oow oow oow oow oow oow oow oow oow oow ooooooou oow oow oow oow oow oow oow oow oow oow oow oow 4 4 4 r 4 39vv vlt wvvv vvvv an an vvvv wvvvvv 2 gt an vv 4 39 V 2 V A he probability that on a given onday either 2 or 3 or 4 students will be absent is foc f2 f3 f4 026 034 022 082 x22 he probability that on a given Monday more than 3 students are absent is ifoc f4 f5 f6 f7 O22008004001O35 x4 7 EX Zxfx Ogtlt0021gtlt0032gtltO263gtltO344gtltO22 x0 5xOD86xOD47xOD1 3l6 ased on information over many Mondays the average number of students absent on ondays is 316 varXE 2 EX2 7 X2 Zx2fx 02 gtltOO212 gtltOO322 gtltO2632 gtltO34 x20 2gtltO2252gtltOO862gtltOO472gtltOO1 1158 varX 62 1158 3162 15944 G 52 12627 YE7X37EX37gtlt31632512 varY var7X 3 72 varX 49gtlt 15944 781256 Probability Primer Exercise Solutions Principles of Econometrics 4e 14 EXERCISE P13 LetX be the annual return from the mutual fund Then X N 005 0042 a PX lt0 PZ lt OO5PZ lt 125 01056 b PX gt 015 PZ gt PZ gt 25 00062 c Let Y be the return from the alternative portfolio Then Y N 007 0072 0 007 007 PYlt0 PZ lt PZlt 101587 PX gt 015 PZ gt j PZ gt 11429 01265 The calculations show that the probability of a negative return has increased from 1056 to 1587 While the probability of a return greater than 15 has increased from 062 to 1265 Whether fund managers should or should not change their portfolios depends on their risk preferences Probability Primer Exercise Solutions Principles of Econometrics 4e 15 EXERCISE P14 Expressing the returns in terms of percentages We have R A 4 82 and RB 8 122 a EU EO25RA O75RB O25ERA O75ERB O25gtlt4O75gtlt8 7 b VarP 52 VarO25RA 07512 0252 VarRA 0752 VarRB 2 X O25gtlt O75gtlt coVRARB Now 1 COVRARB p VarRAVarRB Hence C0VRARB GAGE 8x12 96 VarP 0252 X 82 0752 X 122 2 X 025 X 075 X 96 121 cs 12111 c When 2 cov RARB p 0395 VarRAVarRB coVRARB O5gtlt6AoB O5gtlt8gtlt12 48 VarP 0252 X 82 0752 X122 2 x 025 X 075 X 48 103 GP x103 1015 d When p O COV R ARB O and the Variance and standard deviation of the portfolio are VarP 0252 x 82 0752 X122 85 cs E 922 Probability Primer Exercise Solutions Principles of Econometrics 4e 16 EXERCISE P15 81 b 0 C1 6 D LM xlx1x2729 I 2ixl4x1x2x3x4i724 715 i1 WME I1 xi Tx1 Tx2 Tx3 x4 7 7 152 154 15 7 15550525 850 xi T2 x1 T2 x2 c2 x3 T2 x4 T2 iquot44gt D1 7 152 2 152 4 152 7 152 552 052 252 852 109 4 1 I J2I4Zy1y2y3y4Z5231255 Z4xl Tyl 7 7 15gtlt5 552 15gtlt2 554 15gtlt3 55 7 15 gtlt12 55 55 X O5 05 X 35 25 X 25 85 x 65 275 175 625 5525 66 4 4gtlt gtlt l1x y X yx1y1x2y2x3y3x4y4 4xy 4 2 2 2 2 2 Zxl24Xf2 x1x2x3x4 4x i1 7gtlt52gtlt24gtlt3 7gtlt12 4gtlt15gtlt55 4941649 4gtlt225 66 O6055 109 Probability Primer Exercise Solutions Principles of Econometrics 4e EXERCISE P16 4 a x1x2x3x4Zxl i1 3 b x2 x3 Zxl i2 4 C X11 X22 X3373 35434 Zxiyi i1 4 d X1373 35234 35335 x4y6 Zxiyi2 i1 2 2 4 2 6 35333 x4y4 Z Zxiyi i3 lt4 xl ylx2 y2x3y3 xj yj Probability Primer Exercise Solutions Principles of Econometrics 4e 18 EXERCISE P17 a Z4abxl abxlabx2abx3abx4 4abx1x2 x3 x4 3 b Ziz 122232 14914 i1 3 c Zx22x2O22gtltO212 2gtlt12222gtlt22322gtlt32 x0 25101734 d ifx2f22f32f42 f4f5f6 6 iofbcay f0yf1yf2y 2 f 3 1x2yix2gtlt1x2gtlt2i2x6 2gtlt262gtlt362gtlt4610121436 Probability Primer Exercise Solutions Principles of Econometrics 4e 19 EXERCISE P18 a b 0 CI 6 4 fZx4x1x2x3x44135343 i1 II 5 gt t I U l 5 U I U l 5 U I U l 5 U I U l II 00 4 Zxf 4 2 xfx x x 472 19259 4gtlt32 8 i1 H H H 1 2 xl2 2xn Zxl nx2 39 71 i1 2302 2m72 mc2 Znllxf m72 1 i1 5 3lt I gtlt 1 II a 1M 3lt l I l 1 3lt an E an F Probability Primer Exercise Solutions Principles of Econometrics 4e EXERCISE P19 Znxxi fyz39 J x139yz39 xz397 yz39 W i1 13921 i1 13921 i1 1 1 Zxlyl yn Zxij xn Zyijnxy i1 7121 1721 Znxlyl 2mTy1DE i1 2 352quot239 quotW 13921 20 CHAPTER 2 Exercise Solutions 21 Chapter 2 Exercise Solutions Principles of Econometrics 4e 22 EXERCISE 21 3 x y x x 72 J7 x cy 7 0 6 2 4 36 72 1 2 1 1 04 04 2 3 0 0 06 0 3 1 1 1 14 14 4 0 2 4 24 48 Zxi 1139 Zxif xz39C2 Z0 2 Zx 3C 2 10 12 0 10 0 13 972 y24 b I Zx 7yZy 513 2ltxxgt 10 0 C1 92 is the estimated slope of the fitted line b17 b22 c24 13gtlt25 bl is the estimated Value of E y when x O it is the intercept of the fitted line 5 Zxf O212223242 30 i1 5 Zxiyi Ogtlt61gtlt22gtlt33gtlt14gtltO11 i1 ix Nf2 3O 5gtlt22 1O5xl 72 i1 i1 Zslxiyl Nc711 5gtlt2gtlt24 135xl 7yl y i1 i1 xi yi 31 i P i xi i 0 6 5 1 1 0 1 2 37 17 289 17 2 3 24 06 036 12 3 1 11 01 001 03 4 0 02 02 004 08 Zxi Zi Z133239 Z i Zxi i 10 12 12 0 43 0 Chapter 2 Exercise Solutions Principles 0fEc0n0metrics 4e 23 Exercise 21 continued 6 D g h i i Figure xr21 Observations and fitted line I I I I I I I I I I 0 1 2 3 4 X 0 y 0 Fitted values See gure above The tted line passes through the point of the means C 2 E 24 Given bl 5 b 13 and 7 bl bf We have 7 24 bl bc 5 132 24 Z N12524y A2 62 Z61 4393 14333 62 14333 O14333 222 TCThapter 2 Exercise Solutions Principles of Econometrics 4e a MEXERCISE 22 180 H yx2000 lt X H yx2000 lt 215 Hyx2000 2 2 2 Gyx2000 Gyx2000 Gyx2000 P180 200 Z 215200 100 100 P 2ltZ lt15 a P180ltXlt215P 09104 Figure xr22a X 190 b P X gt190 P l yx2000 gt 2H yx2000 Gyx2000 6yx2000 PZgt190 200j M 1 PZs 1 08413 Figure xr22b 24 Chapter 2 Exercise Solutions Principles of Econometrics 4e 25 Exercise 22 continued 180 H39yx2000 lt X H39yx2000 lt 215 H39yx2000 2 2 2 Gyx2000 Gyx2000 Gyx2000 P18O 2OO Z 215200 M 1 P 22222 lt Z lt 16666 6 P180ltXlt215P 09391 X H39yx2000 190 H39yx2000 d PXgt190P gt 5ix2000 Gix2000 190 200 P Z gt T K x5 J 1 PZS 11111 08667 Chapter 2 Exercise Solutions Principles of Econometrics 4e 26 EXERCISE 23 a The observations on y and x and the estimated leastsquares line are graphed in part b The line drawn for part a will depend on each student s subjective choice about the position of the line For this reason it has been omitted b Preliminary calculations yield Zx21 Zy33 Zx fy 7 265 Zx 72175 y55 9735 The least squares estimates are bZxfy 26395 1514286 ZxCf 175 19 7 b27 55 1514286gtlt35 108 Figure xr23 Observations and fitted line 10 X 0 y O Fitted values c 372IN33655 ZxN21635 The predicted Value for y at x f is j19b22T108 1514286gtlt3555 We observe that 3 bl 9237 7 That is the predicted Value at the sample mean f is the sample mean of the dependent Variable This implies that the leastsquares estimated line passes through the point cf This point is at the intersection of the two dashed lines plotted on the graph in part b Chapter 2 Exercise Solutions Principles 0fEc0n0metrics 4e 27 Exercise 23 Continued d The Values of the least squares residuals computed from l yl 3 yl bl b2x are xl yz z 1 10 0714286 2 8 0228571 3 5 1257143 4 5 0257143 5 2 1228571 6 3 1285714 Their sum is Z l 0 e Zxl l 1gtltO7142862gtlt02285713gtlt12571434gtlt0257143 5gtlt 12285716gtlt1285714 Chapter 2 Exercise Solutions Principles of Econometrics 4e 28 EXERCISE 24 a If 31 O the simple linear regression model becomes y 139 B239 3 b Graphically setting 3120 implies the mean of the simple linear regression model E yl 52x passes through the origin 0 O c To save on subscript notation We set 52 B The sum of squares function becomes N N d 0 Bx 2 p Z xiyz B2392 4 z zxiyi B22X12 i1 i1 352 2gtlt176B91B2 352 35239132 40 35 30 25 SUM SQ 20 16 18 2390 2392 24 B ETA Figure xr24a Sum of squares for B2 The minimum of this function is approximately 12 and occurs at approximately 32 195 The significance of this value is that it is the leastsquares estimate d To nd the value of B that minimizes S We obtain E 22 wt 23296 03913 Setting this derivative equal to zero We have xi Thus the leastsquares estimate is 92 19341 91 which agrees with the approximate value of 195 that We obtained geometrically Chapter 2 Exercise Solutions Principles of Econometrics 4e 29 Exercise 24 Continued 6 12 10 8 7333 6 4 2 0 O 1 2 3 4 5 6 X1 Figure xr24b Fitted regression line and mean The tted regression line is plotted in Figure Xr24 b Note that the point 96 does not lie on the tted line in this instance f The least squares residuals obtained from l y bzxl are 120659 221319 3 11978 4 07363 s O6703 6 06044 Their sum is Z 33846 Note this Value is not equal to zero as it was for 51 at 0 g Zxi l20659gtlt121319gtlt211978gtlt3 07363 x 4 06703 x 5 06044gtlt 6 0 Chapter 2 Exercise Solutions Principles of Econometrics 4e EXERCISE 25 3 equations 30 The eonsultant s report implies that the least squares estimates satisfy the following two 19 5000 10000 19 7500 12000 Solving these two equations yields 250b2000 gt 0 2000 8 b6000 250 Therefore the estimated regression used by the consultant is SYLFS 6000 8x ADVERT sales 10000 12000 14000 I I 8000 I Figure xr25 Regression line 6000 I I I 800 1000 Chapter 2 Exercise Solutions Principles of Econometrics 4e 31 EXERCISE 26 a The intercept estimate bl 240 is an estimate of the number of sodas sold when the temperature is 0 degrees Fahrenheit A common problem when interpreting the estimated intercept is that We often do not have any data points near x 0 If We have no observations in the region Where temperature is 0 then the estimated relationship may not be a good approximation to reality in that region Clearly it is impossible to sell 240 sodas and so this estimate should not be accepted as a sensible one The slope estimate 92 8 is an estimate of the increase in sodas sold when temperature increases by 1 Fahrenheit degree This estimate does make sense One would expect the number of sodas sold to increase as temperature increases b If temperature is 80 F the predicted number of sodas sold is f2 2408gtlt80400 c If no sodas are sold y 0 and 0 2408x or x30 Thus she predicts no sodas will be sold below 30 F d A graph of the estimated regression line Figure xr26 Regression line 600 I 400 I 200 I O I I I I I I I O 20 40 60 80 100 200 Chapter 2 Exercise Solutions Principles of Econometrics 4e EXERCISE 27 8 b C d 9 0 Since A2 62 261 204672 N 2 it follows that 26 204672 N 2 204672gtlt 49 10029 The standard error for 22 is seb2 vEb 000098 0031305 Also A2 Zx39 f2 Varb2 Thus A2 204672 2 20 ZxiC2 cs 885 Va 12 000098 32 The Value 22 018 suggests that a 1 increase in the percentage of males 18 years or older who are high school graduates will lead to an increase of 180 in the mean income of males who are 18 years or older 15 7 b2T15187 018gtlt69139 2742 Since Zxl 72 0 Q Nf2 We have Zxf Zx 572 N272 20885 51gtlt 691392 245879 For Arkansas l yl f1 yl 191 b2xl 12274 2742 018gtlt 583 0962 Chapter 2 Exercise Solutions Principles of Econometrics 4e 33 EXERCISE 28 a The EZ estimator can be Written as Where 2 Thus lag is a linear estimator b Taking expectations yields EbEZE x21x1Ey2 21Ey1 x1 B1B2x2 B1B2x1 Thus ZDEZ is an unbiased estimator C The Variance is given by VarbEZ VarZ kiyl Zkf Varel 62 Zkf 2 2 d If e NOo2 then 19 N3i X2 951 Chapter 2 Exercise Solutions Principles of Econometrics 4e 34 Exercise 28 continued e To convince EZ Stuff that varb2 lt varbEZ We need to show that 262 gt G2 or that 2061 JC2 gt 2 X1 2 X2 12 20939 quot z 2 Consider X2 x12 962 Cx1 C 2 2 2 Thus We need to show that 2 1xl 72 gtx2 c2x1 72 2x2 cx1 c or that or that x 7 x 72 2lZN3x 2 gt 0 This last inequality clearly holds Thus bEZ is not as good as the least squares estimator Rather than prove the result directly as We have done above We could also refer Professor EZ Stuff to the Gauss Markov theorem Chapter 2 Exercise Solutions Principles of Econometrics 4e 35 EXERCISE 29 8 b Plots of the occupancy rates for the motel and its competitors for the 25month period are given in the following gure Figure xr29a Occupancy Rates 100 90 80 70 60 50 40 30 I I I I I I I I I I I I month 1march 2003 25march 2005 o percentage motel occupancy percentage competitors occupancy The repair period comprises those months between the two Vertical lines The graphical evidence suggests that the damaged motel had the higher occupancy rate before and after the repair period During the repair period the damaged motel and the competitors had similar occupancy rates A plot of MOT ELPC T against C OMPPC T yields Figure xr29b Observations on occupancy 100 0 0 90 gt 8 o o lt5 0 1 80 E 9 0 9 5 70 39 E 1 83 o 39 Q 60 D Q 50 O 40 I I I I I 40 50 60 70 80 percentage competitors occupancy There appears to be a positive relationship the two Variables Such a relationship may exist as both the damaged motel and the competitors face the same demand for motel rooms That is competitor occupancy rates re ect overall demand in the market for motel rooms Chapter 2 Exercise Solutions Principles of Econometrics 4e 36 Exercise 29 continued 0 d 6 D The estimated regression is A 2140 08646 x COMPPCT The competitors occupancy rates are positively related to motel occupancy rates as expected The regression indicates that for a one percentage point increase in competitor occupancy rate the damaged motel s occupancy rate is expected to increase by 08646 percentage points 30 Repair period 20 10 0 1 39 E o 39 39 U 0 0 9 39 0 0 10 39 0 2o 0 3930 I I I I I I 0 4 8 12 16 20 24 28 month 1march 2003 25march 2005 Figure xr29d Plot of residuals against time The residuals during the occupancy period are those between the two vertical lines All except one are negative indicating that the model has overpredicted the motel s occupancy rate during the repair period We would expect the slope coef cient of a linear regression of MOT ELPCT on RELPRICE to be negative as the higher the relative price of the damaged motel s rooms the lower the demand will be for those rooms holding other factors constant The estimated regression is A 16666 12212 x RELPRICE The sign of the estimated slope is negative as expected The linear regression with an indicator variable is MOTEL PCT 81 BZREPAIR e From this equation we have that 3 3 ifREPAIR 1 EMOTEL PCT 3 BZREPAIR 3 ifREPAIR 0 Chapter 2 Exercise Solutions Principles of Econometrics 4e 37 Exercise 29f continued 8 h The expected occupancy rate for the damaged motel is 51 32 during the repair period it is 31 outside of the repair period Thus 32 is the difference between the expected occupancy rates for the damaged motel during the repair and nonrepair periods The estimated regression is A 793500 132357 gtlt REPAIR In the nonrepair period the damaged motel had an estimated occupancy rate of 7935 During the repair period the estimated occupancy rate was 79351324 6611 Thus it appears the motel did suffer a loss of occupancy and pro ts during the repair period From the earlier regression we have 0 6 7935 1 6 6 7935 1324 6611 For competitors the estimated regression is 624889 08825 x REPAIR Thus 0 6 6249 W1 191 6 6249 088 6337 During the nonrepair period the difference between the average occupancies was MOTEL0 COMP0 7935 6249 1686 During the repair period it was MOTEL1 COMP1 6611 6337 274 This comparison supports the motel s claim for lost pro ts during the repair period When there were no repairs their occupancy rate was 1686 higher than that of their competitors during the repairs it was only 274 higher The estimated regression is MOTELPCT COMPPCT 168611 141183gtlt REPAIR The intercept estimate in this equation 1686 is equal to the difference in average occupancies during the nonrepair period M0TEL6 COMP6 The sum of the two coef cient estimates 1686 1412274 is equal to the difference in average occupancies during the repair period MOT EL1 COMP1 This relationship exists because averaging the difference between two series is the same as taking the difference between the averages of the two series Chapter 2 Exercise Solutions Principles of Econometrics 4e EXERCISE 210 38 a The model is a simple regression model because it can be Written as y 31B2xe Where yr rf xrm rf B1209 and B2 BJ b General General Exxon Firm Microsoft Electric Motors IBM Disney Mobil 2 3 13189 08993 12614 11882 08978 04140 The stocks Microsoft General Motors and IBM are aggressive with Microsoft being the most aggressive with a beta value of I92 13189 General Electric Disney and Exxon Mobil are defensive with ExxonMobil being the most defensive with a beta value of 192 04140 C General General Exxon Firm Microsoft Electric Motors IBM Disney Mobil bl 59 00061 00012 00116 00059 00011 00079 All estimates of the 0c are close to zero and are therefore consistent with finance theory The tted regression line and data scatter for Microsoft are plotted in Figure xr210 MSFTRISKF REE M KTRISKFREE Fig xr210 Scatter plot of Microsoft and market rate d The estimates for B given 06 0 are as follows General General Exxon Firm Microsoft Electric Motors IBM Disney Mobil A3 13185 08993 12622 11878 08979 04134 The restriction on 0 has led to small changes in the 0 it has not changed the aggressive or defensive nature of the stock Chapter 2 Exercise Solutions Principles of Econometrics 4e 39 EXERCISE 211 3 1000000 39 300000 E 0 2 3 sooooo E 4oamo 39 E Q Q U C 200000 0 0 I I I I I 0 M I r2iim W mm Figure xr211a Price against square feet for houses of traditional style b The estimated equation for traditional style houses is W 28408 73772 SQFT The slope of 73772 suggests that expected house price increases by approximately 7377 for each additional square foot of house size The intercept term is 28408 which would be interpreted as the dollar price of a traditional house of zero square feet Once again this estimate should not be accepted as a serious one A negative Value is meaningless and there is no data in the region of zero square feet Figure xr211b Observations and fitted line 1oooooo I 0 I I sooooo I I 0 l C 600000 I o i I 4ooooo I I I 200000 i jj i 0 I I I I 0 2000 4000 6000 8000 total square feet 0 sale price dollars Fitted values Chapter 2 Exercise Solutions Principles of Econometrics 4e 40 Exercise 211 continued c The estimated equation for traditional style houses is I 68710 0012063 SQFT2 The marginal effect on price of an additional square foot is m slope p For a home with 2000 square feet of living space the marginal effect is 6 PRICE Q 200120632000 4825 dSQFT That is an additional square foot of living space for a traditional home of 2000 square feet is expected to increase its price by 4825 To obtain the elasticity We first need to compute an estimate of the expected price when SQF T 2000 I 68710 0012063220002 116963 Then the elasticity of price with respect to living space for a traditional home with 2000 square feet of living space is A d I slopegtlt SQFT gtlt SQFT 2001206322000 2000 0825 PRICE dSQFT PRICE 1 16963 That is for a 2000 square foot house We estimate that a 1 increase in house size will increase price by 0825 Figure xr211c Observations and quadratic fitted line 1000000 0 800000 600000 400000 200000 0 I I I I 0 2000 6000 8000 4000 total square feet 0 sale price dollars Fitted values tangent Chapter 2 Exercise Solutions Principles of Econometrics 4e 41 Exercise 211 continued d Residual plots Figure xr211d Residuals from linear relation 600000 400000 g 0 3 200000 c T9 0 0 8 2 39 39 m cuDquoto 0 0 J o 0 339 0 O 0 39o0 39 0 39 0 200000 0 I I I I I 0 2000 4000 6000 8000 total square feet Figure xr211d Residuals from quadratic relation 400000 0 200000 39 o 0 0 0 O 390 39 0 0 Residuals O I 3 o O o t 0 O 39 039 quot0 o o 0 200000 0 400000 I I I I I 0 2000 4000 6000 8000 total square feet The magnitude of the residuals tends to increase as housing size increases suggesting that SR3 Vare x 62 homoskedasticity could be violated The larger residuals for larger houses imply the spread or Variance of the errors is larger as SQF T increases Or in other words there is not a constant Variance of the error term for all house sizes e SSE of linear model b SSE 26 137 X1012 SSE of quadratic model c SSE 26 123 X1012 The quadratic model has a lower SSE A lower SSE or sum of squared residuals indicates a lower Value for the squared distance between a regression line and data points indicating a line that better ts the data Chapter 2 Exercise Solutions Principles of Econometrics 4e 42 Exercise 211 continued f The estimated equation for traditional style houses is miiR713 1079894 0000413235SQFT The tted line with a tangent line included is Figure xr21 1f Observations and loglinear fitted line 1oooooo 0 800000 600000 400000 200000 0 I I I I 0 2000 6000 8000 4000 total square feet 0 sale price dollars pricel tangentl g The SSE from the loglinear model is based on how Well the model ts lnPRICE Since the log scale is compressed the SSE from this speci cation is not comparable to the SSE from the models with PRICE as the dependent Variable One Way to correct this problem is to obtain the predicted Values from the loglinear model then take the antilogarithm to make predictions in terms of PRICE Then a residual can be computed as PRICE eXpl Using this approach the SSE from loglinear model is 131gtlt1O12 This is smaller than the SSE from the tted linear relationship but not as small as the SSE from the tted quadratic model Chapter 2 Exercise Solutions Principles of Econometrics 4e 43 EXERCISE 212 a The scatter plot in the gure below shows a positive relationship between selling price and house size 300000 I E 39 4E 000000 3 I c o g 39 1 o 0 395 400000 o 4 39 9 n 3 9 I 939c I g to I quot E 200000 I u zulli 0 I I I I I m llii 1113 area hundred olf square eeio 50 Figure xr212a Scatter plot of selling price and living area b The estimated equation for all houses in the sample is 51715175 30069 91817LIlAREA The coef cient 91817 suggests that selling price increases by approximately 9182 for each additional 100 square foot in living area The intercept if taken literally suggests a house with zero square feet would cost 30069 a meaningless value The model should not be accepted as a serious one in the region of zero square feet Figure xr212b Observations and fitted line 800000 600000 400000 200000 10 20 30 40 50 living area hundreds of square feet 0 selling price of home dollars Fitted values Chapter 2 Exercise Solutions Principles of Econometrics 4e 44 Exercise 212 continued 0 d 6 The estimated quadratic equation for all houses in the sample is 31715173 57728 212611LIIAREA2 The marginal effect of an additional 100 square feet is A dt z s slope T 2212611LIlAREA dLIIAREA For a home with 1500 square feet of living space the marginal effect is 221261115 637833 dLIVAREA That is adding 100 square feet of living space to a house of 1500 square feet is estimated to increase its expected price by approximately 6378 Figure xr212d Linear and quadratic fitted lines 800000 600000 400000 200000 30 living area hundreds of square feet 0 selling price of home dollars Fitted values O Fitted values The quadratic model appears to fit the data better it is better at capturing the proportionally higher prices for large houses SSE of linear model b SSE 2512 223 X1012 SSE of quadratic model c SSE 26 203 X1012 The SSE of the quadratic model is smaller indicating that it is a better fit The estimated equation for houses that are on large lots in the sample is 317 113279 19383LIVIREA2 The estimated equation for houses that are on small lots in the sample is ETYIEIFE 62172 18686LIVARE12 Chapter 2 Exercise Solutions Principles of Econometrics 4e 45 Exercise 212e continued f The intercept can be interpreted as the expected price of the land the selling price for a house with no living area The coefficient of LIVAREA has to be interpreted in the context of the marginal effect of an extra 100 square feet of living area which is 2B2LIIAREA Thus we estimate that the mean price of large lots is 113279 and the mean price of small lots is 62172 The marginal effect of living area on price is 38766 gtlt LIVAREA for houses on large lots and 37372 gtlt LIIAREA for houses on small lots The following figure contains the scatter diagram of PRICE and AGE as well as the estimated equation which is E 137404 62716AGE Figure xr212f sprice vs age regression line sooooo 600000 400000 200000 40 age of home at time of sale years 0 selling price of home dollars Fitted values i We estimate that the expected selling price is 627 less for each additional year of age The estimated intercept if taken literally suggests a house with zero age ie a new house would cost 137404 The model residuals plotted below show an asymmetric pattern with some very large positive values For these observations the linear fitted model under predicts the selling price Figure xr212f residuals from linear model eooooo o 400000 200000 Residuals o O O 200000 I I 0 20 40 60 80 I00 age of home at time of sale years Chapter 2 Exercise Solutions Principles of Econometrics 4e 46 Exercise 212f continued g The following gure contains the scatter diagram of lnPRICE and AGE as well as the estimated equation which is 1 11746 000476AGE Figure xr212f logsprice vs age regression line 40 60 age of home at time of sale years O lsprice Fitted val ues i In this estimated model each extra year of age reduces the selling price by 048 To nd an interpretation from the intercept we set AGE 0 and nd an estimate of the price of a new home as expl expll74597 126244 The following residuals from the tted regression of lnSPRICE on AGE show much less of problem with underprediction the residuals are distributed more symmetrically around zero Thus based on the plots and Visual t of the estimated regression lines the loglinear model is preferred Figure xr212f transformed residuals from loglinear model 2 Residuals O 0 Q 0 I I 40 80 100 age of home at time of sale years The estimated equation for all houses is EEEFE 115220133797LGELOT The estimated expected selling price for a house on a large lot LGELOT 1 is 115220133797 249017 The estimated expected selling price for a house not on a large lot LGELOT 0 is 115220 Chapter 2 Exercise Solutions Principles of Econometrics 4e 47 EXERCISE 213 8 b C d The estimated equation using a sample of small and regular classes is i 918043 13899SMALL Comparing a sample of small and regular classes We nd students in regular classes achieve an average total score of 9180 While students in small classes achieve an average of 9180 139 9319 This is a 150 increase This result suggests that small classes have a positive impact on learning as measured by higher totals of all achievement test scores The estimated equations using a sample of small and regular classes are R 434733 5819SMALL A 483310 8080SMALL Students in regular classes achieve an average reading score of 4347 While students in small classes achieve an average of 43473 582 4406 This is a 134 increase In math students in regular classes achieve an average score of 48331 While students in small classes achieve an average of 48331 808 4914 This is a 167 increase These results suggests that small class sizes also have a positive impact on learning math and reading The estimated equation using a sample of regular classes and regular classes with a full time teacher aide is i 918043 0314IIDE Students in regular classes Without a teacher aide achieve an average total score of 9180 While students in regular classes with a teacher aide achieve an average total score of 91804 031 9184 These results suggest that having a fulltime teacher aide has little impact on learning outcomes as measured by totals of all achievement test scores The estimated equations using a sample of regular classes and regular classes with a full time teacher aide are R 434733 0705IIDE A 483310 0391IIDE The effect of having a teacher aide on learning as measured by reading and math scores is negligible This result does not differ from the case using total scores Chapter 2 Exercise Solutions Principles of Econometrics 4e 48 EXERCISE 214 a xr214 Vote versus Growth o 39 LO Q 0 Q 0 go 9 gtLD E 0 0 E O 0 UV C 0 O V I I I I I I 15 10 10 5 0 Growth rate before election There appears to be a positive association between VOTE and GROWTH b The estimated equation for 1916 to 2008 is I 50848 088595GROWTH The coef cient 088595 suggests that for a 1 percentage point increase in the growth rate of GDP in the 3 quarters before the election there is an estimated increase in the share of Votes of the incumbent party of 0885 95 percentage points We estimate based on the fitted regression intercept that that the incumbent party s expected Vote is 50848 when the growth rate in GDP is zero This suggests that when there is no real GDP growth the incumbent party will still maintain the majority Vote A graph of the fitted line and data is shown in the following figure xr214 Vote versus Growth with fitted regression 6 60 I 50 Incumbent vote 30 40 I I I I 15 10 5 O 10 Growth rate before election 0 Incumbent share of the two party presidential vote Fitted values c The estimated equation for 1916 2004 is I 51053 0877982GROWTH The actual 2008 Value for growth is 0220 Putting this into the estimated equation we obtain the predicted Vote share for the incumbent party Chapter 2 Exercise Solutions Principles of Econometrics 4e 49 Exercise 214c continued d 2008 51053 0877982GR0WTH 51053 08779820220 51246 This suggests that the incumbent party will maintain the majority Vote in 2008 However the actual Vote share for the incumbent party for 2008 was 4660 which is a long way short of the prediction the incumbent party did not maintain the majority Vote The gure below shows a plot of VOTE against INFLATION There appears to be a negative association between the two Variables xr2 14 Vote versus Inflation 60 I Incumbent vote 40 50 I O O 0 30 I 4 Inflation rate before election The estimated equation plotted in the gure below is I 53408 0444312INFLATION We estimate that a 1 percentage point increase in in ation during the incumbent party s rst 15 quarters reduces the share of incumbent party s Vote by 0444 percentage points The estimated intercept suggests that when in ation is at 0 for that party s rst 15 quarters the expected share of Votes won by the incumbent party is 534 the incumbent party is predicted to maintain the majority Vote when in ation during its rst 15 quarters is at 0 xr2 14 Vote versus Inflation o 39 39 60 0 O Q g 0 O O C Incumbent vote 30 40 I I I I O 4 8 Inflation rate before election 0 Incumbent share of the twoparty presidential vote Fitted values Chapter 2 Exercise Solutions Principles of Econometrics 4e 50 EXERCISE 215 8 b 350 300 6 25 Series EDUC 200 Sampe11000 3 150 Observations 1000 II 100 50 Z I Mean 1379900 Maximum 2100000 1 2 3 4 5 6 7 8 9 10 11 12 13 1415 16 17 18 19 20 21 Minimum 0000000 years of education Sid Dev 2711079 Figure xr215a Histogram and statistics for EDUC Most people had 12 years of education implying that they nished their education at the end of high school There are a few observations at less than 12 representing those who did not complete high school The spike at 16 years describes those who completed a 4 year college degree while those at 18 and 21 years represent a master s degree and further education such as a PhD respectively Spikes at 13 and 14 years are people who had one or two years at college 140 Series WAGE 120 Sample 1 J Observations 1000 100 Mean 2061566 Median 1730000 80 Maximum 7639000 Minimum 1970000 60 Std Dev 1283472 Skewness 1 583909 40 Kurtosis 5921362 20 Jarq ueBera 7737260 Probability 0000000 0 I I I I II Ii I II II II I II 39I II I I39 F 0 10 20 30 40 50 60 70 Figure xr215a Histogram and statistics for WAGE The observations for WAGE are skewed to the right indicating that most of the observations lie between the hourly wages of 5 to 40 and that there is a smaller proportion of observations with an hourly wage greater than 40 Half of the sample earns an hourly wage of more than 1730 dollars per hour with the average being 2062 dollars per hour The maximum earned in this sample is 7639 dollars per hour and the least earned in this sample is 197 dollars per hour The estimated equation is ME 67103 19803EDUC The coefficient 19803 represents the estimated increase in the expected hourly wage rate for an extra year of education The coef cient 67103 represents the estimated wage rate of a worker with no years of education It should not be considered meaningful as it is not possible to have a negative hourly wage rate Chapter 2 Exercise Solutions Principles of Econometrics 4e 51 Exercise 215 continued 0 d 6 The residuals are plotted against education in Figure xr215c There is a pattern evident as ED UC increases the magnitude of the residuals also increases suggesting that the error Variance is larger for larger Values of EDUC a Violation of assumption SR3 If the assumptions SR1SR5 hold there should not be any patterns evident in the residuals 60 50 40 1 0 0 30 RESID 0 O0 TOQOKDOZIDO XX 0 00 O O 0O O O 0 O 3930 000 Figure xr215c Residuals against education The estimated equations are If female ME 141681 23575EDUC If male VVACE 30544 18753EDUC If black WACE 150859 24491EDUC Ifwhite WA 65507 19919EDUC The White equation is obtained from those Workers who are neither black nor Asian From the results We can see that an extra year of education increases the Wage rate of a black Worker more than it does for a White Worker And an extra year of education increases the Wage rate of a female Worker more than it does for a male Worker The estimated quadratic equation is ME 608283 OO73489EDUC2 The marginal effect is therefore dV dEDUC mm slope 2 0073489 EDUC For a person with 12 years of education the estimated marginal effect of an additional year of education on expected Wage is dV F dEDUC mm Slope 2007348912 17637 That is an additional year of education for a person with 12 years of education is expected to increase Wage by 176 Chapter 2 Exercise Solutions Principles of Econometrics 4e 52 Exercise 215e continued For a person with 14 years of education the marginal effect of an additional year of education is A cz IWETE slope An additional year of education for a person with 14 years of education is expected to increase Wage by 206 2007348914 20577 The linear model in b suggested that an additional year of education is expected to increase Wage by 198 regardless of the number of years of education attained That is the rate of change is constant The quadratic model suggests that the effect of an additional year of education on Wage increases with the level of education already attained f 80 70 quadratic equation LU 3 Iinear equation E Figure xr215f Quadratic and linear equations for wage on education The quadratic model appears to fit the data slightly better than the linear equation g The histogram of ln WA GE in the gure below is more symmetrical and bellshaped than the histogram of WA GE given in part a L It airiN K K 4 5 1 2 3 lwage Figure xr215g Histogram for lnWAGE Chapter 2 Exercise Solutions Principles of Econometrics 4e 53 Exercise 215 continued The estimated loglinear model is l 160944 0090408ED UC We estimate that each additional year of education increases expected Wage by approximately 904 The estimated marginal effect of education on WAGE is dWAGE dEDUC This marginal effect depends on the Wage rate For Workers with 12 and 14 years of education We predict the Wage rates to be 32 XWAGE WAc expl60944 0090408 X 12 14796 EDUC12 675 expl 60944 0090408 x 14 17728 EDUC14 Evaluating the marginal effects at these Values We have 13377 EDUC 12 LWAGE 122 x WAGE dEDUC 16028 EDUC 14 For the linear relationship the marginal effect of education was estimated to be 198 For the quadratic relationship the corresponding marginal effect estimates are 176 and 206 The marginal effects from the loglinear model are lower A comparison of the tted lines for the linear and loglinear model appears in the figure below 80 9 n as 39 o 8 I o 39 I 39 39 3 0 o t 0 8 n 39 I z 39 39393939 0 ff quotT 2 n p 39 J 439139339393 I 3 D O n 1 15 20 0 5 yeas opechcatim n3linear Figure xr1215h Observations with linear and loglinear tted lines CHAPTER 3 Exercise Solutions 54 Chapter 3 Exercise Solutions Principles of Econometrics 4e 55 EXERCISE 31 8 b C The required interval estimator is b1itcseb1 When 191283416 tc t 2024 097538 and seb1 43410 We get the interval estimate 83416 i 2024 x 43410 446 17130 We estimate that 31 lies between 446 and 17130 In repeated samples 95 of similarly constructed intervals would contain the true B1 To test H0 13 0 against H1 13 0 We compute the tvalue f b1 B1 1 seb1 834160 2192 43410 Since the t 192 value does not exceed the 5 critical value tc t 2024 We do 097538 not reject H 0 The data does not reject the zerointercept hypothesis The pvalue 00622 represents the sum of the areas under the t distribution to the left of t 192 and to the right of t 192 Since the t distribution is symmetric each of the tail areas that make up the pvalue are p 2 00622 2 00311 The level of signi cance 0c is given by the sum of the areas under the PDF for t gt tc so the area under the curve for tgt Z6 is oc 2 025 and likewise for tlt tc Therefore not rejecting the null hypothesis because oc 2 lt p 2 oroc lt p is the same as not rejecting the null hypothesis because tc lt t lt to From Figure Xr31c We can see that having a pvalue gt 005 is equivalent to having tc ltt lt tc 1 n3 PDF 02 11 quot3939 32 P3939392t I29 I 2 C an 3 2 1 u 1 2 3 Figure xr31c Critical and observed tvalues Chapter 3 Exercise Solutions Principles of Ec0n0meIrics 4e 56 Exercise 31 continued d 6 0 Testing H0 2131 0 against H1213 gt0 uses the same Ivalue as in part b I 192 Because it is a onetailed test the critical value is chosen such that there is a probability of 005 in the right tail That is I6 I09538 1686 Since I 192 gt I6 169 H0 is rejected the alternative is accepted and We conclude that the intercept is positive In this case pvalue PI gt 192 00311 We see from Figure Xr31d that having the pvalue lt 005 is equivalent to having I gt 169 04 Rejection aegion 03 PDF O2 01 192 00 1 n I 1 I 3 2 1 O 1 t 2 3 Figure xr31d Rejection region and observed tvalue The term quotlevel of signi cancequot is used to describe the probability of rejecting a true null hypothesis when carrying out a hypothesis test The term quotlevel of con dencequot refers to the probability of an interval estimator yielding an interval that includes the true parameter When carrying out a twotailed test of the form H0 Bk c versus H1 Bk 7 c nonrejection of H0 implies c lies Within the con dence interval and vice versa providing the level of signi cance is equal to one minus the level of con dence False The test in d uses the level of signi cance 5 which is the probability of a Type I error That is in repeated samples We have a 5 chance of rejecting the null hypothesis when it is true The 5 signi cance is a probability statement about a procedure not a probability statement about B1 It is careless and dangerous to equate 5 level of signi cance with 95 con dence which relates to interval estimation procedures not hypothesis tests Chapter 3 Exercise Solutions Principles of Econometrics 4e 57 EXERCISE 32 3 b C d The coef cient of EXPER indicates that on average a technical artist39s quality rating goes up by 0076 for every additional year of experience 39 36 37 36 35 RAWNG 34 33 32 31 EXPER Figure xr32a Estimated regression function Using the value tc t 2074 the 95 confidence interval for 32 is given by 097522 19 tcseb2 0076 2074 x 0044 0015 0167 We are 95 confident that the procedure We have used for constructing a confidence interval will yield an interval that includes the true parameter B2 To test H0 13 0 against H1 13 7 0 We use the test statistic I Z92seb2 00760044 1727 The I critical value for a two tail test with N 2 22 degrees of freedom is 2074 Since 2074 lt 1727 lt 2074 We fail to reject the null hypothesis To test H 0 13 0 against H1 13 gt 0 We use the Ivalue from part c namely t1727 09522 1717 Since 1727 gt1717 We reject H0 and conclude that 32 is positive Experience has a positive effect on quality rating but the righttail critical value tc t Chapter 3 Exercise Solutions Principles of Econometrics 4e 58 Exercise 32 continued e The pvalue of 00982 is given as the sum of the areas under the tdistribution to the left of 1727 and to the right of 1727 We do not reject H0 because for on 005 pvalue gt 005 We can reject or fail to reject the null hypothesis just based on an inspection of the pvalue Having the pvalue gt on is equivalent to having it lt tc 2074 04 034 PM 02 014 pm pg D39D V 39 Y Y T V Y Y 3 2 1 0 1 2 3 r r T Figure xr32e pvalue diagram Chapter 3 Exercise Solutions Principles of Econometrics 4e EXERCISE 33 8 b C d 6 Hypotheses Calculated IValue Critical IValue Decision Hypotheses Calculated tValue Critical IValue Decision Hypotheses Calculated IValue Critical IValue Decision Hypotheses Calculated IValue Critical IValue Decision TDIST22 NJ H11 13 0 against H1 13 72 0 t 03100082 378 it Jrt10199511 i2819 Reject H11 because t 378 gt tc 2819 H0 13 0 against H1 13 gt0 t 03100082 378 tc t09922 2508 Reject H11 because t 378 gt tc 2508 H1 13 0 against H1 13 lt0 t 03100082 378 tc t00522 1717 Do not reject H11 because t 378 gt to 1717 tcritical 3 39 1 71 7 gt alpha 05 Q 5 0 I I I 6 4 2 0 2 4 5 X Figure xr33 One tail rejection region H11 13 05 against H1 13 7 05 I 0310 050082 232 it Jrt10197511 i2074 Reject H11 because t 232 lt tc 2074 A 99 interval estimate of the slope is given by 92 i tcseb2 0310 i 2819 x 0082 0079 0541 59 We estimate B2 to lie between 0079 and 0541 using a procedure that Works 99 of the time in repeated samples Chapter 3 Exercise Solutions Principles of Econometrics 4e EXERCISE 34 a b1tgtlt seb1 1257 x 2174 2733 24 20 16 E 12 2 8 4 O I I I I I I I I I 0 10 20 30 40 50 60 70 80 90 100 PMHS Figure xr34a Estimated regression function b seb2 92 t 01805754 00313 c pvalue 2 X 1 Pt lt 1257 2 X 1 08926 02147 0 03 PM 02 125 3925 01 pI2 on Figure xr34c pvalue diagram 60 d The estimated slope 92 018 indicates that a 1 increase in males 18 and older who are high school graduates increases average income of those males by 180 The positive sign is as expected more education should lead to higher salaries 6 Using tc t099549 6 itcseb2 0180 268 x 00313 0096 0264 268 a 99 con dence interval for the slope is given by Chapter 3 Exercise Solutions Principles of Econometrics 4e 61 Exercise 34 continued D For testing H0 2132 02 against H1 2132 7 02 We calculate t 0180 O2 00313 0639 The critical values for a twotailed test with a 5 significance level and 49 degrees of freedom are itc i201 Since I 0634 lies in the interval 201 201 We do not reject H0 The null hypothesis suggests that a 1 increase in males 18 or older who are high school graduates leads to an increase in average income for those males of 200 Nonrejection of H 0 means that this claim is compatible with the sample of data Chapter 3 Exercise Solutions Principles of Econometrics 4e 62 EXERCISE 35 3 b C The linear relationship between life insurance and income is estimated as WS 68550 38802INCOME se 7383501121 0 z tinsf13iJ1dJ15n1 n WCOME Figure xr35 Fitted regression line and mean The relationship in part a indicates that as income increases the amount of life insurance increases as is expected If taken literally the value of bl 68550 implies that if a family has no income then they would purchase 6855 Worth of insurance However given the lack of data in the region Where INCOAE 0 this value is not reliable i If income increases by 1000 then an estimate of the resulting change in the amount of life insurance is 388020 ii The standard error of 92 is 01121 To test a hypothesis about 32 the test statistic is g 2 B2 N UH seb2 An interval estimator for 2 is b t se 19 b t se 19 Where I6 is the critical 2 c 2 2 c 2 value for twith N 2 degrees of freedom at the on level of significance To test the claim the relevant hypotheses are H0 32 5 versus H1 32 7 5 The alternative 32 5 has been chosen because before We sample We have no reason to suspect 32 gt 5 or 32 lt 5 The test statistic is that given in part b ii with 32 set equal to 5 The rejection region 18 degrees of freedom is I gt 2101 The value of the test statistic is t Z 5 38802 5 seb2 01121 As I 999lt 2101 we reject the null hypothesis and conclude that the estimated relationship does not support the claim Chapter 3 Exercise Solutions Principles of Econometrics 4e 63 Exercise 35 continued d 6 To test the hypothesis that the slope of the relationship is one We proceed as We did in part c using 1 instead of 5 Thus our hypotheses are H0 B2 1 versus H1 32 7 1 The rejection region is I gt 2101 The value of the test statistic is t 38802 1 01 121 257 Since t 257 gt tc 2101 We reject the null hypothesis We conclude that the amount of life insurance does not increase at the same rate as income increases Life insurance companies are interested in household characteristics that in uence the amount of life insurance cover that is purchased by different households One likely important determinant of life insurance cover is household income To see if income is important and to quantify its effect on insurance We set up the model INSURANCE 13 BZINCOMEI el Where INSURANCE is life insurance cover by the ith household INCOME is household income 31 and B2 are unknown parameters that describe the relationship and e is a random uncorrelated error that is assumed to have zero mean and constant variance 62 To estimate our hypothesized relationship We take a random sample of 20 households collect observations on INSURANCE and INCOJWE and apply the leastsquares estimation procedure The estimated equation with standard errors in parentheses is Uvs c 68550 38802INCOME se 7383501121 The point estimate for the response of lifeinsurance coverage to an income increase of 1000 the slope is 3880 and a 95 interval estimate for this quantity is 3645 4116 This interval is a relatively narrow one suggesting We have reliable information about the response The intercept estimate is not signi cantly different from zero but this fact by itself is not a matter for concern as mentioned in part b We do not give this value a direct economic interpretation The estimated equation could be used to assess likely requests for life insurance and What changes may occur as a result of income changes Chapter 3 Exercise Solutions Principles of Econometrics 4e 64 EXERCISE 36 a The estimated model is A 2140 O8646COMPPCT se 1291 02027 The null and alternative hypotheses are H0B2 0 H1B2gt0 The test statistic and its distribution assuming the null hypothesis is true at the point B2 0 are t Z A Q23 seb2 At a 1 signi cance level We reject H0 when t gt t09923 2500 The calculated Value of the tstatistic is 1 086464 4 t 265 seb2 020271 Since 4265 gt 2500 We reject H0 and conclude that when the competitors occupancy rate is high the motel s occupancy rate is also high and Vice Versa One would expect both occupancy rates to be high in periods of high demand and low in periods of low demand The pValue is 0000145 aI uIated value 110 N I Criti cal 39v39i3iLle Figure xr36a Rejection region and pvalue Chapter 3 Exercise Solutions Principles of Econometrics 4e 65 Exercise 36 continued b The model is MOTEL PCT 13 BZRELPRICE e The null and alternative hypotheses are H0B20 H1B2lt0 The test statistic and its distribution assuming the null hypothesis is true are A Q23 seb2 At a 1 signi cance level We reject H0 when t ltt 2500 00123 2 The estimated regression is A 166656 12212RELPRICE se 4357 5835 The calculated value of the tstatistic is t 92 122122O93 seb2 5835 Since 2093gt 2500 we do not reject H0 at a 1 signi cance level There is insuf cient evidence to conclude that there is an inverse relationship between MOT ELPCT and RELPRICE This result is a surprising one From demand theory We would expect the occupancy rate to be negatively related to relative price However at a 5 signi cance level We would have rejected H 0 the data are not suf ciently informative to do so at a 1 level The pvalue is 00238 4 pvalue is area under the curve to the left of the auIated value 3 xaI ulated value 2 1 I I I I I I I 5 4 1 O 1 2 3 4 5 rejection region t Figure xr36b Rejection region and pvalue Chapter 3 Exercise Solutions Principles of Econometrics 4e 66 Exercise 36 continued 0 C1 The model is MOTEL PCT 51 BZREPAIR e The expected occupancy rate in the repair and nonrepair periods is 51 REPAIR 0 EMOTEL PCT 51 5REPA1R 51 5 REPAIR 1 The null and alternative hypotheses are H11522O H152ltO We wish to show that the motel occupancy rate is less during the repair period which implies that 82 lt 0 If we are able to reject the null hypothesis that the difference in occupancy rates between the repair and nonrepair period is zero or positive we will then conclude beyond reasonable doubt that this difference is negative and that the motel suffered a loss in occupancy during the repair period The test statistic and its distribution assuming H11 is true at the point 52 O are 5 4 S lt82 z 1 where 32 is the least squares estimator of 82 We reject H11 when I ltt 1714 00523 2 The estimated regression model is A 7935 132357REPAIR se 3154 59606 The calculated value of the tstatistic is 53 2 132357 2 2221 3452 59606 Since 2221 lt 1714 we reject H0 at a 5 significance level The data suggest that the motel s occupancy rate is signi cantly lower during the repair period 1 A 95 interval estimate for 52 is given by 00gt 1 z11111se5 132357 20687 x 59606 2557 091 With 95 confidence we estimate that the effect of the repair period is to reduce the motel s occupancy rate by a percentage between 091 and 2557 Our confidence is in the procedure 95 of intervals constructed in this way with new samples of data would yield an interval that contains 52 The effect of repairs on the occupancy rate has not been estimated precisely Our interval suggests it could be anywhere from almost no effect to a 25 effect Chapter 3 Exercise Solutions Principles of Econometrics 4e 67 Exercise 36 continued 6 f The model is MOTEL PCT COMPPCT y1 yREPAIR e The difference in the expected occupancy rates in the repair and nonrepair periods is y REPAIR 0 EMOTEL PCT COMPPCT y yREPAIR y y REPAIR 1 The null and alternative hypotheses are H0y0 H1y2lt0 The test statistic and its distribution assuming the null hypothesis is true are 172 t j 1 Se Y2 23 At a 5 signi cance level we reject H0 when tltt 2500 00123 Z The estimated regression model is MOTELPCT COMPPCT 168611141183REPAIR se 21092 39863 The calculated value of the Istatistic is t vi 2 141183 3542 sey 39863 Since 3542 lt 2500 we reject H 0 at a 5 signi cance level The regression estimates show that during the nonrepair period the motel enjoyed an occupancy rate 1686 higher than its competitors rate During the repair period this advantage fell by 1412 Our test shows that this decline is statistically signi cant at the 001 level of signi cance This test overcomes one of the potential problems of the test in part c namely if the repair period was a period in which demand was normally low then ignoring the competitor s occupancy rate could have led the low demand to be incorrectly attributable to the repairs Including the competitor s occupancy rate controls for normal uctuations in demand A 95 interval estimate for V2 is given by 2 i t09753sef2 141 18 i 20687 x 39863 2236 587 With 95 con dence we estimate that the effect of the repair period is to reduce the difference between the motel s occupancy rate and the competitors occupancy rate by a percentage between 587 and 2236 This interval is a relatively wide one we have not estimated the effect precisely but there does appear to have been a reduction in the motel s occupancy rate Chapter 3 Exercise Solutions Principles of Econometrics 4e 68 EXERCISE 37 8 We set up the hypotheses H0 2B 1 versus H1 3 751 The economic relevance of this test is to test Whether the return on the f1rm s stock is risky relative to the market portfolio Each beta measures the volatility of the stock relative to the market portfolio and volatility is often used to measure risk A beta value of one indicates that the stock s volatility is the same as that of the market portfolio The test statistic given H0 is true is rL1 z 130 The rejection region is tlt 1978 and tgt 1978 Where t0975 130 1978 The results for each company are given in the following table Stock Ivalue Decision rule Disney t 089794 10826 Since 1978lttlt1978 012363 fail to reject H0 089926 1 Since 1978lttlt1978 GE T 1020 0098782 fail to reject H0 GM 126141 11293 Since 1978lttlt1978 39 020222 fail to reject H IBM 118821 11489 Since 1978lttlt1978 0126433 fail to reject H0 M ft t131895 11984 Since tgt1978 O50 016079 39 reject H0 ExxonMobil t p 6532 Since tlt 1978 reject H 0 0089713 For Disney GE GM and IBM We fail to reject the null hypothesis indicating that the sample data are consistent with the conjecture that the Disney GE GM and IBM stocks have the same volatility as the market portfolio For Microsoft and ExxonMobil We reject the null hypothesis and conclude that these stocks do not have the same volatility as the market portfolio Chapter 3 Exercise Solutions Principles of Ec0n0meIrics 4e 69 Exercise 37 continued b C d We set up the hypotheses H0 13 21 versus H1213 lt1 Where j MobilExxon The relevant test statistic given H0 is true is Z91 Sebj 130 The rejection region is I lt 1658 Where IC I005D130 1657 The value of the test statistic is 2 041397 1 2 6532 0089713 Since I 6532 lt I6 1657 We reject H0 and conclude that MobilEXXon s beta is less than 1 A beta equal to 1 suggests a stock39s variation is the same as the market variation A beta less than 1 implies the stock is less volatile than the market it is a defensive stock We set up the hypotheses H0 13 S1 versus H1 13 gt1 where j Microsoft The relevant test statistic given H0 is true is 9 1 t 2 E N t130 The rejection region is I gt 16567 Where I095D130 16567 The value of the test statistic is t131895 1 016079 Since I 19836 gt I6 16567 We reject H0 and conclude that Microsoft s beta is greater than 1 A beta equal to 1 suggests a stock39s variation is the same as the market variation A beta greater than 1 implies the stock is more volatile than the market it is an aggressive stock 19836 A 95 interval estimator for Microsoft s beta is 19 iI0975D130 gtltseb Using our sample of data the corresponding interval estimate is 13190i 1978 x 016079 1001 1637 Thus We estimate with 95 con dence that Microsoft s beta falls in the interval 1001 to 1637 It is possible that Microsoft s beta falls outside this interval but We would be surprised if it did because the procedure We used to create the interval Works 95 of the time This result appears in line with our conclusion in both parts a and c Chapter 3 Exercise Solutions Principles of Econometrics 4e 70 Exercise 37 continued e The two hypotheses are H0 oz 0 Versus H1 oz 7 0 The test statistic given H0 is true is r if r N 130 SeaJ The rejection region is tlt 1978 and tgt 1978 Where t0975 130 1978 The results for each company are given in the following table Stock tValue Decision rule D t 000115 0193 Since l978lttltl978 lsney 0005956 fail to reject H0 GE 0001167 2 245 Since 1978lttlt1978 0004759 fail to reject H0 GM t0 011551185 Since 1978lttlt1978 0009743 fail to reject H0 IBM 0005851 0961 Since l978 lttlt1978 0006091 fail to reject H0 0006098 Microsoft t T 0787 Since l978 lt t lt 1978 0007747 fail to reject H0 000788 1 823 Since 1978 lttlt1978 t Mobll Exxon 0004322 fail to reject H0 We do not reject the null hypothesis for any of the stocks This result indicates that the sample data is consistent with the conjecture from economic theory that the intercept term equals 0 Chapter 3 Exercise Solutions Principles of Econometrics 4e 71 EXERCISE 38 8 b The estimated linear regression is z EE 28408 73772SQFT se 5728 2301 The hypotheses are H0 B2 0 versus H1 B gt 0 The test statistic given H0 is true is Z L N t580 seb2 With OL 001 the rejection region is I gt 2333 t The value of the test statistic is 099580 t 73772 3206 2301 Since I 3206 gt 2333 We reject the null hypothesis that 32 0 and accept the alternative that B gt 0 We conclude that the slope is not zero and that there is a statistically signi cant relationship between house size in square feet and house sale price For testing H0 E PRICE SQF T 2000 31 B2SQF T S 120000 against an alternative that the expected price is greater than 120000 We set up the hypotheses H0 13 20000 120000 H1 13 20000 gt 120000 The test statistic given H0 is true is 91 2000 120000 seb1 2000b2 N t580 To obtain the standard error seb1 2000192 We first calculate the estimated variance v 20002 xvIo2x2000xc5v za7 32811823 4000000 gtlt 5294462 4000 gtlt 1233534 464831 1 The corresponding standard error is seb1 200019 j x464831 2156 The rejection region ist gt 2333 t The value of the test statistic is 099580 t 2840756 20007377195 120000 2156 0401 Since 0400 lt 2333 We do not reject the null hypothesis There is not enough evidence to suggest that the expected price of a house of 2000 square feet is greater than 120000 Chapter 3 Exercise Solutions Principles of Econometrics 4e 72 Exercise 38b continued 0 d The pValue ofthe test is p PR gt 0401 0656 580 11 Figure xr38b pvalue A 95 interval estimate for the expected price of a house of 2000 square feet is 91 2000 4 t075580seb1 2000b2 2840756 2000 gtlt 7377195 i 1964 x 2156 1191363 i 42344 114902 123371 We estimate with 95 con dence that the expected house price of a 2000 square foot house lies between 114902 and 123371 The estimated quadratic regression is l l 68710 0012063SQFT2 se 2873 0000346 The marginal effect of an additional square foot of living area is 20 2SQFT dSQF T Its estimates for houses of 2000 and 4000 square feet are CZPRICE 2 x 0012063 x 2000 48253 dSQF T SQFT2000 CZPRICE 2 x 0012063 x 4000 96506 dSQF T SQFT4000 Chapter 3 Exercise Solutions Principles of Econometrics 4e 73 Exercise 38d continued 6 For the case of a 2000 square foot house We Wish to test the hypotheses H 0 24000012 75 against the alternative H1 I4OOOOL2 lt 75 The test statistic given H0 is true is 4000612 75 t T N t580 se4000a2 For OL 001 the rejection region is I lt 2333 t The value of the test statistic is 001580 48253 75 26747 19 31 4000gtlt 000034626 1385 Since t 1931 lt 2333 We reject the null hypothesis that 4000012 75 and accept the alternative that 4000012 lt 75 We conclude that the marginal effect of an additional square foot of living area in a home with 2000 square feet is less than 75 For the case of a 4000 square foot house We Wish to test the hypotheses H0 28000012 75 against the alternative H1 28000062 lt 75 The test statistic given H0 is true is 800002 75 T N t580 se8000a2 The rejection region is I lt 2333 t The value of the test statistic is 001580 t 96506 75 21506 7 76 8000gtlt 000034626 2770 Since t 776 gt 2333 we do not reject the null hypothesis 8000012 75 in favor of the alternative that 8000012 lt 75 There is no evidence to suggest that the marginal effect of an additional square foot of living area in a home with 4000 square feet is less than 75 The two different hypothesis test outcomes occur because the marginal effect of an additional square foot is increasing as the house size gets larger The estimated loglinear model is ln7 E 1079894 0000413235SQFT se 003467 0000013927 The marginal effect of an additional square foot of living area is dPRICE yPR1CE dSQFT The estimated value of PRICE when SQF T 2000 is PRICE eXpf1 9sQFT eXpl079894 0000413235gtlt 2000 1119055 Chapter 3 Exercise Solutions Principles of Econometrics 4e 74 Exercise 38e continued For a 2000 square foot house We Wish to test the hypotheses H0 1119055y2 75 against the alternative H1 1119055y2 lt 75 The test statistic given H0 is true is 1119055302 75 1 TA N t580 se1119055y2 With OL 001 the rejection region is I lt 2333 t The value of the test statistic is 001580 t1119055 gtlt 0000413235 75 28757 1119055gtlt0000413235 1559 1845 Since t 1845 lt 2333 We reject the null hypothesis that 111905572 75 and accept the alternative that 111905537 lt 75 We conclude that the marginal effect of an additional square foot of living area in a home with 2000 square feet is less than 75 For the case of a 4000 square foot house the estimated price is z exp9 9sQFT eXp1079894 0000413235gtlt 4000 255731 Thus We Wish to test the hypotheses H0 225573172 75 against the alternative H1 2255731372 lt 75 The test statistic given H0 is true is 25573137 75 1 TA N t580 se255731y2 For OL 001 the rejection region is I lt 2333 1 The value of the test statistic is 001580 t 255731gtlt 0000413235 75 30677 8613 255731gtlt0000413235 3562 Since t 8613 gt 2333 We do not reject H0 255731y2 75 in favor of the alternative that 25573172 lt 75 There is no evidence to suggest that the marginal effect of an additional square foot of living area in a home with 4000 square feet is less than 75 Like in part d the two different hypothesis test outcomes occur because the marginal effect of an additional square foot is increasing as the house size gets larger Note The above solution to part e assumes that the predicted values of price for SQF T 2000 and SQF T 4000 are known with certainty it assumes there is no sampling error associated with these predictions Because 1 UCE eXpj1 f2SQF T and 1 and 92 contain sampling error l laj will also be subject to sampling error To accommodate this sampling error in part e We need to test the hypothesis H0 y expy ysQFT 75 Techniques for testing nonlinear functions of parameters such as this one are considered in Chapter 563 Chapter 3 Exercise Solutions Principles of Ec0n0meIrics 4e 75 EXERCISE 39 8 b C We set up the hypotheses H0 B2 0 versus H1 B2 gt 0 The alternative B2 gt 0 is chosen because we assume that growth if it does in uence the vote will do so in a positive way The test statistic given H0 is true is A N t22 sew The rejection region is I gt 1717 I The estimated regression model is 09522 IEFE 508484 O8859GROWTH se 10125 01819 The value of the test statistic is t 08859 01819 4870 Since I 4870 gt 1717 we reject the null hypothesis that B2 0 and accept the alternative that B2 gt 0 We conclude that economic growth has a positive effect on the percentage vote earned by the incumbent party A 95 interval estimate for B2 from the regression in part a is 192 7 seb2 08859 2074 X 01819 0509 1263 097522 This interval estimate suggests that with 95 confidence the true value of B2 is between 0509 and 1263 Since B2 represents the change in percentage vote due to economic growth we expect that an increase in the growth rate of 1 will increase the percentage vote by an amount between 0509 and 1263 We set up the hypotheses H0 B2 0 versus H1 B2 lt 0 The alternative B2 lt 0 is chosen because we assume that in ation if it does in uence the vote will do so in a negative way The test statistic given H0 is true is 2 Z sew Selecting a 5 signi cance level the rejection region is Ilt 1717I The 00522 estimated regression model is I75FE 534077 O4443INFLA TION se 22500 05999 The value of the test statistic is t 04443 05999 0741 Since 0741 gt 1717 we do not reject the null hypothesis There is not enough evidence to suggest in ation has a negative effect on the vote Chapter 3 Exercise Solutions Principles of Econometrics 4e 76 Exercise 39 continued d 6 f A 95 interval estimate for B from the regression in part c is Z i t0975seb 04443 i 2074 x 05999 1688 0800 This interval estimate suggests that with 95 confidence the true value of B is between 1688 and 0800 It suggests that an increase in the in ation rate of 1 could increase or decrease or have no effect on the percentage vote earned by the incumbent party When INFLATION 0 the expected vote in favor of the incumbent party is EVOTE INFLATION 013 13 x 0 13 Thus We Wish to test H 0 2B 2 50 against the alternative H1 13 lt 50 The test statistic assuming H0 is true at the point B 50 is 2 E se19 lt2 The rejection region is tlt 1717 t005 The value of the test statistic is t 534077 50 22500 Since 1515 gt 1717 We do not reject the null hypothesis There is no evidence to suggest that the expected vote in favor of the incumbent party is less than 50 when there is no in ation 1515 A point estimate of the expected vote in favor of the incumbent party when INFLATION 2 is l 19 219 534077 2x 114443 1 525191 The standard error of this estimate is the square root of vet119219vE1922v 1922cW1919 50625 403599 410592 22653 The 95 interval estimate is therefore 19 219 1 9se19 219 525191 2074 525191i31216 4940 5564 We estimate with 95 confidence that the expected vote in favor of the incumbent party when in ation is at 2 is between 4940 and 5564 In repeated samples of elections with in ation at 2 We expect the mean vote to lie Within 95 of the interval estimates constructed from the repeated samples Chapter 3 Exercise Solutions Principles of Econometrics 4e 77 EXERCISE 310 3 b The estimated equation using a sample of small and regularsized classes Without aide is 918043 13899SMALL se 1667 2447 The null and alternative hypotheses are H0B2S0 H1B2gt0 The test statistic and its distribution when the null hypothesis is true are A 3741 sew We reject H 0 When tgt t095 374D 1645 The calculated value of the test statistic is t13899 5681 24466 Since 5681 gt 1645 We reject H0 The mean score of students in small classes is signi cantly greater than that of students in regularsized classes It suggests that governments should invest in more teachers and classrooms so that class sizes can be smaller The pvalue of the test is p Pt3741 gt5681721gtlt10 9 70 Figure xr310a Illustration of pvalue A 95 interval estimate for 32 is 92 i t0975p3741seb2 13899 i 19606 x 24466 9101870 With 95 con dence We estimate that the average score for students in small classes is between 910 and 1870 points higher than the average score for students in regularsized classes Chapter 3 Exercise Solutions Principles of Econometrics 4e 78 Exercise 310 continued 0 d For READSCORE the estimated equation is R 434733 5819SMILL se 0707 1 03 8 Using the same hypotheses test statistic and rejection region as in part a the value of the test statistic is t 9 25819125605 seb2 10382 Because 5605 gt 1645 We reject H0 B2 3 0 in favor of H1 2B2 gt 0 The mean reading score of students in small classes is significantly greater than that of students in regular sized classes The pvalue diagram is similar to that given in Figure Xr310a For MA T HSCORE the estimated equation is A 483310 8080SMILL se 1081 1586 Using the same hypotheses test statistic and rejection region as in part a the value of the test statistic is t 6 80799 s66 15865 39 Because 5093 gt 1645 We reject H0 2B2 S 0 in favor of H1 2B2 gt 0 The mean math score of students in small classes is signi cantly greater than that of students in regularsized classes The pvalue diagram is similar to that given in Figure Xr310a No differences are uncovered if scores in math and reading tests are considered separately Having a smaller class has a positive effect on the learning of both math and reading The estimated equation using regularsized classes with and Without a teacher aide is 918043 0314AIDE se 1613 2270 The null and alternative hypotheses are H0y2S0 H1y2gt0 The test statistic and its distribution when the null hypothesis is true are L S6062 3741 We reject H 0 When tgt t095374D 1645 The calculated value of the test statistic is 2 Chapter 3 Exercise Solutions Principles of Econometrics 4e 79 Exercise 310d continued 9 f t 03139 22704 0138 Since 0038 lt 1645 We do not reject H0 The mean score of students in classes with a teacher aide is not signi cantly greater than that of students in classes Without a teacher aide It suggests that governments should not invest in providing more teacher aides in classrooms A 95 interval estimate for y is 2 i t09753741sef2 03139 i 19606 x 22704 414 477 With 95 con dence We estimate that the difference in average scores for students from classes with and Without a teacher aide lies between 414 and 477 In other Words having an aide may improve scores it may lead to scores that are Worse or it may have no effect For READSCORE the estimated equation is R 434733 0705IIDE se 0697 0982 Using the same hypotheses test statistic and rejection region as in part d the value of the test statistic is t 172 07054 se9 09817 2 0719 Because 0719 lt 1645 We do not reject H0 17 0 in favor of H1 17 gt0 The mean reading score of students in classes with a teacher aide is not signi cantly greater than that of students in classes Without a teacher aide For MA T HSCORE the estimated equation is A 483310 0391IIDE se 1043 1469 Using the same hypotheses test statistic and rejection region as in part d the value of the test statistic is 7 V2 390393915 0267 sef2 14687 Because 0267 lt 1645 We do not reject H0 17 S 0 in favor of H1 17 gt 0 The mean math score of students in classes with a teacher aide is not signi cantly greater than that of students in classes Without a teacher aide No differences are uncovered Having a teacher aide improves neither the average reading score nor the average math score Chapter 3 Exercise Solutions Principles of Econometrics 4e 80 EXERCISE 311 a The estimated equation is ME 182577 0089OEXPER se O9273 00315 I 196885 28257 The estimated equation tells us that with every additional year of experience the associated increase in hourly Wage is 00890 Furthermore it tells us that the average Wage for those Without experience is 1826 The relatively large tvalues suggest that the least squares estimates are statistically signi cant at a 5 level of signi cance 80 70 60 50 40 NAGE 3oo 20 3 quot 1O O 0 EXPER Figure xr311a Fitted regression line and observations b We set up the following hypothesis test H0B20 H1B2gt0 The alternative hypothesis is set up as 32 gt0 because We expect experience to have a positive effect on Wages The test statistic given H0 is true is b T2 N t998 sec The rejection region is I gt 1646 t The value of the test statistic is 095998 39 t 00890 00315 Decision Reject H0 because 2826gtl646 2826 We conclude that the estimated slope of the relationship 92 is statistically signi cant There is a positive relationship between the hourly Wage and a Worker s experience Chapter 3 Exercise Solutions Principles of Econometrics 4e 81 Exercise 311 continued ci For females the estimated equation is ME 178413 00497EXPER se 12735 00427 1 140096 11650 With every extra year of experience the associated increase in average hourly Wage for females is 00497 This estimate is not signi cantly different from zero however The average Wage for females Without experience is 1784 80 70 69 50 0 9 0 40 0 O O 00 00 WAG E o 0 o o o 0 30 O o 3 0 00 8 E go8o o EXPER Figure xr311ci Fitted regression line and observations for females cii For males the estimated equation is WE 184511 O1407EXPER se 13349 00460 1 138222 30619 With every extra year of experience the associated increase in average hourly Wage for males is O1407 The average Wage for males Without experience is 1845 80 70 60 50 40 WAG E 30 20 10 0 EXPER Figure xr311cii Fitted regression line and observations for males Chapter 3 Exercise Solutions Principles of Econometrics 4e 82 Exercise 311c continued ciii For blacks the estimated equation is ME 157893 0073 SEXPER se 25319 00834 1 62362 O8858 With every extra year of experience the associated increase in average hourly Wage for blacks is 00738 This estimate is not signi cantly different from zero however The average Wage for blacks Without experience is 1579 80 70 60 WMGE A o I EXPER Figure xr311ciii Fitted regression line and observations for blacks civ For White males the estimated equation is V 186556 O1455EXPER se 14607 00499 1 12771529146 With every extra year of experience the associated increase in average hourly Wage for White males is O1455 The average Wage for White males Without experience is 1866 80 70 60 50 C 40 o O 0 0 O 8 O MMGE EXPER Figure xr311civ Fitted regression line and observations for white males Chapter 3 Exercise Solutions Principles of Econometrics 4e 83 Exercise 311c continued d Comparing the estimated wage equations for the four categories we nd that experience counts the most or leads to the largest increase in wages for white males The effect is only slightly less for males in general It is approximlately halved for blacks and is less still for females For those with no experience the wage ranking is white males males females and blacks The residual plots appear in the gures below The main observation that can be made from all the residual plots is that the pattern of positive residuals is quite different from the pattern of negative residuals There are very few negative residuals with an absolute magnitude larger than 20 whereas the magnitude of the positive residuals cover a much greater range These characteristics suggest a distribution of the errors that is not normally distributed but skewed to the right 60 50 oo 40 o o O O RESID oo ltBgto8 o o 9 0 ago co 0 03 o ego B3ltgt9g88 e 8 68 o 0 A 0 co 0 O EXPER Figure xr311d Plotted residuals for full sample regression 60 50 40 O 0 0 0 o 30 0 0 o 0 0 E 0 00 8 o 0 LG 20 0 0 08 O 39 o 3 M 0 0 00 0 0 0 0 0 o EXPER Figure xr311di Plotted residuals for female regression Chapter 3 Exercise Solutions Principles of Econometrics 46 Exercise 311d continued o 50 o 0o o 00 O 40 00 0 Do 0 30 0 o O 0 80 o 0 oo 0 0 D 20 O o 0 0 0 2 O D o o 08800 8 O o o 9 o 0o 0 LU 0 oo 000 o I 0 09 oo o88 0 6 oo8 9 8 0 80 Z goo 8 88o8 o 88 o 0 8 ogo ltgt8 39 a 8 e g e88 o 0888 g 000 3 080 9 o go 8 o8 0 8 960900 0 og g 0 9o8oo g 9 ooo 10 g 96 o 8 00 o oooo g80oo o o8o3 0 0 oo oo o o 00 680 O 9 08 o 00 O 20 O o O o 0 0 O 3930 I I I I I I EXPER Figure xr311dii Plotted residuals for male regression 60 O O 50 40 o o 30 9 0 O E 20 0 0 Ir 0 O 10 0 0 0 0 o 0 oo 0 0 0 0 8 0 coo o 8 o o 0 000 80 0 0 0 oo o 0 oo 0 00 0 o O 10 ago oooo O0 089 08 0000 O o 3920 I I I I I I EXPER Figure xr311diii Plotted residuals for black regression 60 50 o O o o 0 oo 40 o 00 O 0 00 3o 0 0 0 80 0 0 00 0 OO O 20 0 0 0 0 7 0 00 0980 O 0 Lu 0 o D 10 o o o o 0 6000 00 08 goo 0 0 80 o oo Ego gsooogogo o 0 oogooggoo 85900 o3Se 8o 0 088 908 90 9 O 80 0 3398 080 0 008 0 0 08 ooo 0800 6 O o g 88g 8 9 03 3910 W53 3 i 3 9 0 o s89 o o O0 o Oo8O O OO 0 0 20 O O 0 3930 I I I I I I EXPER Figure xr311div Plotted residuals for white male regression 84 Chapter 3 Exercise Solutions Principles of Econometrics 4e 85 EXERCISE 312 8 b The required scatter diagram is displayed in Figure Xr312a There are no distinct patterns evident The few observations with the largest experience have a low wage and those with the highest wages tend to be those where EXPER30 lies between 10 and 15 but it is hard to discern a strong relationship The distribution of wages is skewed to the right with the majority of people having a wage less than 30 and with a small number having wages more than double this amount 80 70 60 50 40 39 O I 0 0 g 5 39 f 3 ii 39 so 39 393939 3 quot 39 W1 39 0 0 0 20 1 9 tt 0 39 3 quot039 P 1 5 um B 1 0 u I I I I I 30 20 10 0 10 20 so 40 EXPER30 Figure xr312a Scatter diagram for WAGE and EXPER30 The estimated equation is M55 23067 0013828EXPER302 se 0527 0001956 t 4380 7068 The Ivalues for both coef cient estimates are greater than 2 indicating that they are signi cantly different from zero at a 5 signi cance level To test H 0 y 2 0 against the alternative H1 y lt 0 we use the test statistic Y2 t Se2 t998 The rejection region is I lt I 1646 The calculated value of the test statistic is 005998 t 9 0013828 se9 0001956 7068 Because 7068 lt 1646 we reject H0 2y 2 0 Accepting the alternative H1 zyz lt 0 implies a signi cant quadratic relationship in the shape of an inverted U Chapter 3 Exercise Solutions Principles of Econometrics 4e 86 Exercise 312 continued c Noting that EWA GE y y EXPER2 6OEXPER 900 the marginal effect of experience on wage is given by dEWAGE d EXPER y ZEXPER 60 Using 3 00138283 the estimated marginal effects for persons with 10 30 and 50 years experience are IWVTC mew dEXPER 0013828320 60 05531 EXPER 1 0 JWVZGE me30 2 dEXPER 0013828360 60 00 EXPER 3 0 me 0 00138283100 60 05531 5 dEXPER EXPER50 Their standard errors are semew semew 40 x sef2 40 x 00019564 007826 semew 0 The marginal effect at 30 years of experience is not significantly different from zero since it is zero for all possible values of y2 Both mew and mew are signi cantly different from zero at a 5 significance level because the values t i 0553 1 007 826 i7 068 do not lie between tw05998 1962 and tw975998 1962 d The 95 confidence intervals for the slopes are as follows mew i1962 gtlt semew 05531i1962 gtlt 007826 0400 0707 mew i 1962 x semew 00 i1962 x 00 00 00 mew 411962 x semew 05531i1962 gtlt 007826 o707 0400 The marginal effect for EXPER 30 is exact No estimation was necessary The marginal effects for EXPER 10 and 50 are relatively precise They suggest an extra year of experience will change the wage by an amount between 071 and 040 Chapter 3 Exercise Solutions Principles of Econometrics 4e 87 Exercise 312 continued e A plot of the actual and tted WAGE appears in Figure Xr312e The estimates in part c are consistent with the tted values The slope is positive when EXPER30 20 it is zero when EXPER30 0 and negative when EXPER30 20 80 CD 0 0 00 00CP 70 60 50 WAGE 40 0 Fitted WAGE EXPER30 Figure xr312e Plot of tted and actual values of WAGE i The two estimated regressions are M55 20926 0088953EXPER30 se 0419 0031480 WEE 18258 0088953EXPER se 0927 0031480 The two equations have the same slope coef cient but different intercepts To reconcile the two intercepts we note that the righthand side of the rst equation can be written as 209263 00889534EXPER 30 209263 00889534 x 30 00889534EXPER 18258 0088953EXPER which agrees with the second equation To derive the standard error of 6L1 from the covariance matrix of the estimates from the rst equation we note that 611 191 30192 and hence that se6tv za302v za 2x30xcov7a7 0l7567076 900 x 000099l00088 60 x 000346057507 0927 Chapter 3 Exercise Solutions Principles of Econometrics 4e 88 Exercise 312f continued g The estimated marginal effect of experience on Wage from the two regressions is 008895 The assumption of a constant slope does not appear to be a good one The results from parts b to d suggest the slope will decline with experience and eventually become negative The marginal effect of experience is greatest when a Worker has little or no experience Using the larger data set in cps4dat We obtain the following results The estimated equation is ME 22355 O012393EXPER302 se 0237 0000879 t 9448 14098 The tvalues for both coefficient estimates are very large indicating that they are significantly different from zero at a 5 significance level To test H 0 v2 2 0 against the alternative H1 zyz lt 0 We use the test statistic Y2 A sey2 39 t4836 The rejection region is I lt t0054836 1645 The calculated value of the test statistic is t 92 00l239307 A 14098 sey 000087909 Because 14098 lt 1645 We reject H0 23 2 0 Accepting the alternative H1 y2 lt 0 implies a signi cant quadratic relationship in the shape of an inverted U The marginal effect of experience on Wage is given by dEWAGE L y ZEXPER 60 dEXPER Using 372 00l239307 the estimated marginal effects for persons with 10 30 and 50 years experience are dlt7ltmeTgtgt mew 4 00123930720 6004957 dEXPER EXPERl 0 o711 lt7V76757 me30 dEXPER 00l23930760 60 00 EXPER30 o711 lt7V76797 me50 001239307100 60 04957 dEXPER EXPER5 0 Chapter 3 Exercise Solutions Principles of Econometrics 4e 89 Exercise 312g continued Their standard errors are seme10 seme50 40gtlt sef2 40gtlt 000087909 003516 seme30 0 The marginal effect at 30 years of experience is not signi cantly different from zero since it is zero for all possible values of y2 Both mew and me50 are signi cantly different from zero at a 5 signi cance level because the values t Jr04957003516 414098 do not lie between t00254836 1960 and t0975p4836 1960 The 95 con dence intervals for the slopes are as follows mew i1962 gtlt seme10 04957 i 196 X 003516 0427 0565 me30 i 1962 x seme30 00 i196 x 00 00 00 me50 i1962gtlt seme50 04957 i196gtlt 003516 O565 0427 The larger sample has increased the precision of estimation by reducing the Width of the con dence intervals by more than half from 0307 to 0138 Chapter 3 Exercise Solutions Principles of Econometrics 4e 91 Exercise 313 continued c Using the hint We have a WAGE T22 EXPER 30 WAGE d EXPER M The predicted Values for WA GE when EXPER 10 30 and 50 are WAGE10 exp2982638 0000708822 x 10 302 148665 WAGE30 exp2982638 0000708822gtlt 30 302197398 WAGE50 6xp2982638 0000708822gtlt 50 302 148665 Using these Values and 2 0000708822 We can compute the following estimates for the marginal effects M1 mew A 2 x 0000708822 x 10 30 X 148665 04215 dEXPER EXPER1 0 6W1 A 2x 0000708822 X 30 30gtlt 197398 00 dEXPER EXPER30 me30 2 W1 A 2 x 0000708822 X 50 30 x 148665 04215 dEXPER EXPER50 me50 2 d A plot of the actual and fitted WA GE appears in Figure Xr313d The estimates in part c are consistent with the tted Values The slope is positive when EXPER30 20 it is zero when EXPER30 O and negative when EXPER30 20 80 0 7O 0 ca lt10 lt1 cm 0 0 lt1 0 o 60 8 6 6 lt9 cm lt1 00 30 q 9 0 P 50 q Cquot q Q0 0 C 30 pwo 80 to CD Q quot9 Q9 lt3 WAGE 40 lt1 lt1 393 2 9 8 ofttdWAGE 3 quotquot 3 W I e 3 q Q qfvg qgcp CP gcpg P 8 P P 30q P0 38ltvq 3903 93 q 3 3q q g 3agt lt9 8 2 98 lt9 o g quoti E39 iilt 890 o 0 20 F D I 3 quot 1139 4 1 1 P p F P I o N2 via l0 E23 lg cpgac 06 rho CP Q V0 9 9 Qq O 9 lt9 q q lt9 6 I I I I I I 30 20 10 O 10 20 30 40 EXPE R30 Figure xr313d Plot of tted and actual values of WAGE Chapter 3 Exercise Solutions Principles of Econometrics 4e 92 EXERCISE 314 8 b C The relationship between sales SAL and the relative price variables is expected to be a negative one Since brands 2 and 3 are substitutes for brand 1 an increase in the price of brand 1 relative to the price of brand 2 or relative to the price of brand 3 will lead to a decline in the sales of brand 1 The estimated loglinear regression is lnSAL1 102758 18581RPRICE2 se 05185 05139 The typical interpretation of B2 in a loglinear model is that 1unit increase in x will lead to a 100 B2 increase in y In this particular case where RPRICE2 is a unitfree relative price variable it is not so meaningful to talk about a 1unit increase in RPRICE2 Instead we consider the elasticity dSAL1 dRPRICE2 RPRICE2 SAL I B2RPRICE2 We can interpret B2 as the percentage change in sales from a 1 increase in the relative price when the prices of the two brands are identical RPRICE2 1 In terms of our estimate and considering a price change of a realistic magnitude If the prices of brands 1 and 2 are the same and the relative price of brand 1 to brand 2 increases by 10 the sales of brand 1 will decline by 1858 Demand is elastic A 95 interval estimate for B2 from the regression is 92 i t097550seb2 1 85807 i 2009 x 05139 2890 0826 This interval estimate suggests that with 95 con dence when the two prices are the same a 10 increase in the relative price of brand 1 tuna to brand 2 tuna will decrease sales of brand 1 by between 826 and 2890 We set up the following hypothesis test H0B20 H1B2lt0 The test statistic given H0 is true is A N t50 seb2 The rejection region is I lt 2403 t The value of the test statistic is 00150 18581 2 3616 05139 Chapter 3 Exercise Solutions Principles of Econometrics 4e 93 Exercise 314c continued d Decision Reject H0 because 3616lt 2403 A sketch of the rejection region is displayed in Figure Xr314c We conclude that there is a statistically signi cant inverse relationship between the unit sales of brand 1 tuna and the relative price of brand 1 tuna to brand 2 tuna This result is consistent with economic theory as it is expected that demand for a good should be inversely related to the relative price of that good to a substitute good if Wquot Rejection region Figure xr314c Rejection region for hypothesis test The estimated loglinear regression is lnSAL1 114810 3 0543RPRICE3 se 05347 05291 The estimate of y can be interpreted as follows If the prices of brands 1 and 3 are the same and the relative price of brand 1 to brand 3 increases by 10 the sales of brand 1 will decline by 3054 Demand is elastic A 95 interval estimate for y from the regression is 372 i t097550sef2 30543 i 2009 x 05291 4117 1991 This interval estimate suggests that with 95 con dence when the two prices are the same a 10 increase in the relative price of brand 1 tuna to brand 3 tuna will decrease sales of brand 1 by between 1991 and 4117 Chapter 3 Exercise Solutions Principles of Econometrics 4e 94 Exercise 314 continued e We set up the following hypothesis test H0y2O H1y2ltO The test statistic given H0 is true is sewn The rejection region is I lt 2403 t001p50 The value of the test statistic is t 3O5425 5772 052913 Decision Reject H0 because 5772lt 2403 A sketch of the rejection region is displayed in Figure xr314e We conclude that there is a statistically signi cant inverse relationship between the unit sales of brand 1 tuna and the relative price of brand 1 tuna to brand 3 tuna This result is consistent with economic theory as it is expected that demand for a good should be inversely related to the relative price of that good to a substitute good W 391 2103 Rejection 1 region 9772 Figure xr314e Rejection region for hypothesis test Chapter 3 Exercise Solutions Principles of Econometrics 4e 95 EXERCISE 315 3 b C The estimated loglinear regression using data from 1987 is LCRMRTE 29854 18844PRBARR se 01218 03744 If the probability of arrest increases by 10 or 01 the crime rate will decrease by 01gtlt1884gtlt1001884 A 95 interval estimate for B from the regression is 92 it seb2 18844 i 19873 x 03744 2628 1140 097588 Thus a 95 interval estimate for the percentage change in the crime rate after an increase in the probability of arrest of 01 is 2628 1140 We set up the following hypothesis test H0B20 H1B2lt0 The test statistic given H0 is true is L quot sew 81 The rejection region is t lt 2369 t00188 The value of the test statistic is 1 t L844 5033 03744 Decision Reject H0 because 5033 lt2374 We conclude that there is a statistically signi cant relationship between the crime rate and the probability of arrest and that this relationship is an inverse relationship The estimated loglinear regression using data from 1987 is LCRMRTE 31604 06922PRBCONV se 00966 01478 If the probability of conviction increases by 10 or 01 the crime rate will decrease by 01gtlt 0692 x 100 692 A 95 interval estimate for B2 from the regression is 2 it seb2 069224 i19873 gtlt 014775 09859 03986 097588 Thus a 95 interval estimate for the percentage change in the crime rate after an increase in the probability of conviction of 01 is 986 399 Chapter 3 Exercise Solutions Principles of Econometrics 4e 96 Exercise 315c continued To test the relationship between crime rate and the probability of conviction at the 1 signi cance level we set up the following hypothesis test H0B2O H1B2ltO The test statistic given H0 is true is A N t88 seb2 The rejection region is I lt 2369 IWOL88 The value of the test statistic is t O69224 014775 Decision Reject H0 because 4685 lt 2374 4685 We conclude that there is a statistically signi cant relationship between the crime rate and the probability of conviction and that this relationship is an inverse relationship CHAPTER 4 Exercise Solutions 97 Chapter 4 Exercise Solutions Principles of Econometrics 4e EXERCISE 41 A2 21 R2 1 i1 1823985 o71051 Zyl yf 63163 b To calculate R2 We need yl 72 Zy y2 Zyf N172 593094 2Ogtlt16O352 7885155 Therefore 2 SSR 66672 SST 7885155 08455 C From R2 1 P 1 0 SST SST We have 6 SST1 R2 55236gtlt 1 07911 N K 202 64104 98 Chapter 4 Exercise Solutions Principles of Econometrics 4e 99 EXERCISE 42 a y583l738x39 Where x 2210 123 234 01166 001738 A b X Where y 1 00246 000234 50 ltgt 02915 0869 A y X Where y and x 0 00615 0117 20 20 The Values of R2 remain the same in all cases Chapter 4 Exercise Solutions Principles of Econometrics 46 100 EXERCISE 43 a j20b1b2x05 13gtlt4 02 2 2 15 Varf e21 j 1433331j 2293333 sef 229333 15144 c Using se f from part b and re t 31824 09753 yo ire sef O2i31824gtlt15144 50194619 d Using 2 x0 2 the prediction is 320 5 13 x 2 24 and A2 1 x0 T2 12 22 Varf G 1Nxif2 14333315j1O 172 sefml3115 yo itcsef 24i31824gtlt13115 17746574 Width in part c 4619 5019 9638 Width in part d 6574 1774 8348 The Width in part d is smaller than the Width in part c as expected Predictions are more precise when made for x Values close to the mean Chapter 4 Exercise Solutions Principles of Econometrics 46 101 EXERCISE 44 3 b C d Graphs for each of the models are given below Figure Xr44a 35 I Ii lt0 32 I RATING 3 I RATING 28 3 I 25 I 26 I I I I 10 20 30 40 1390 2390 3390 4390 EXPER EXPER Model 1 the quadratic model Model 2 the linearlog model The predicted ratings for a Worker with 10 years of experience are Model 1 174717 34464 OOO14591O 352 25345 Model 2 l71N7 14276 O5343ln10 26579 Estimates of the marginal effects at EXPER 10 are Model 1 dRLVG 00014592 gtlt EXPER 70 dEXPER 00014592 gtlt10 70 007295 Model 2 MBALJW 05343 x 1 05343 x L 005343 dEXPER EXPER 10 The 95 interval estimates for the marginal effect from each model are Model 1 me t0975p48senfe 007295 20106 x 00000786 X 50 0065000809 Model 2 00433 lTe it097547sen1e 005343 201 17 X 00447 00621 Chapter 4 Exercise Solutions Principles of Econometrics 46 102 EXERCISE 45 8 b If we multiply the x Values in the simple linear regression model y 51 Bzx e by 20 the new model becomes 3 yB12 2 xgtlt20e B1Bxe where BB220 and xxgtlt2O The estimated equation becomes 9 y 19 2 jxx2O Thus 31 and b1 do not change and 32 and 9 become 20 times smaller than their original Values Since e does not change the Variance of the error term Vare 62 is unaffected Multiplying all the 2 Values by 50 in the simple linear regression model y 51 Bzx e gives the new model ygtlt5OB1gtlt5OB2 gtlt50xegtlt50 or y Bi BZXe where 3 13 X50 3 13 x50 e egtlt5O The estimated equation becomes 9 3 gtlt5Ob1gtlt5Ob2gtlt5Ox yygtlt50 Thus both 51 and 32 are affected They are 50 times larger than their original Values Similarly b1 and 92 are 50 times larger than their original Values The Variance of the new error term is Vare Vare x 50 2500 gtlt Vare 250062 Thus the Variance of the error term is 2500 times larger than its original Value Chapter 4 Exercise Solutions Principles of Econometrics 46 103 EXERCISE 46 8 b The least squares estimator for Bl is bl 37 b2 Thus 7 bl bl and hence 9 lies on the tted line Consider the fitted line 3 bl xlbz Averaging over N We obtain 7 1 1 y amp Zblxb blNbZxbl19C blbx N N From part a We also have 7 bl bl Thus 7 y Chapter 4 Exercise Solutions Principles of Econometrics 46 104 EXERCISE 47 a The least squares predictor in this model is 330 bzxo b Using the solution from Exercise 24 part f SSE Z f 206592 213192 119782 O73632 O67O32 O6O442 116044 Z12 4262727292112 352 R21116O44 u 0967 352 c The squared correlation between the predicted and observed Values for y is fpZ V2 425492 20943 626 Zyl2Z l2 65461gtlt29333 The two alternative goodness of t measures R3 and ry are not equal d Calculations reveal SST Z0Z if 29333 and SSR Z0 if 67370 Thus SSR SSE 67370 116044 78974 7 SST 29333 The decomposition does not hold Chapter 4 Exercise Solutions Principles of Econometrics 46 105 EXERCISE 48 a Linear regression results 9 06954 00150 R2 04245 se 0071900025 Linearlog regression results jt 05623 01696 lnt R2 02254 gtlltgtlltgtllt se 01425 00469 Quadratic regression results 9 07994 000033812 R2 05252 se 004850000048 b i ii 24 20 16 12 ljiwn A Za Actual Fitted Residual Figure xr48b Fitted line and residuals for the simple linear regression 24 20 08 O8 l 1 00 KV l A V M V V J V 04 V U 3908 393939393939393939393939393939393939393939393939393939393939393939393939 5 10 15 20 25 so 35 40 45 Residual Actual Fitted Figure xr48b Fitted line and residuals for the linearlog regression Chapter 4 Exercise Solutions Principles of Econometrics 4e 106 Exercise 48b continued b i ii 24 2o 1e 8 12 6 08 4 2 AK A i 04 39 W LI vV m A V 2 F MW 4 V U V quot396 quotlquotquotlquotquotl39quot39lquotquotlquotquotlquotquotlquotquotlquotquotl39 5 10 15 20 25 so 35 40 45 Residual Actual Fitted Figure xr48b Fitted line and residuals for the quadratic regression iii Residual histograms and J arqueBera error normality tests Figure xr48 Residual histogram linear relation Figure xr48 Residual histogramzlinearlog relation 2 gr 8 39 0 O o 1 Residuals Residuals Figure xr48 Residual histogram quadratic relation Linear JB 0878 pValue 0645 3 Linear log JB 2778 pValue 0249 8 Quadratic JB 0416 pValue 0812 0 Residuals iv Values of R2 are given in part a Chapter 4 Exercise Solutions Principles of Econometrics 46 107 Exercise 48b continued To choose the preferred equation we consider the following 1 The signs and signi cance of the estimates of the response parameters B2oc2 and y We expect them to be positive because we expect yield to increase over time as technology improves All estimates have the expected signs and are signi cantly different from zero at a 1 signi cance level R2 The value of R2 for the third equation is the highest namely 05685 The plots of the tted equations and their residuals The upper parts of the gures display the tted equation while the lower parts display the residuals Considering the plots for the tted equations the one obtained from the third equation seems to t the observations best In terms of the residuals the rst two equations have concentrations of positive residuals at each end of the sample The third equation provides a more balanced distribution of positive and negative residuals throughout the sample The residual histograms and JarqueBera tests Normality of the residuals is not rejected in any of the cases However visual inspection of the histograms suggests those from the linear and quadratic equations more closely resemble a normal distribution Considering all these factors the third equation is preferable EXERCISE 49 a Equation 1 Equation 2 Equation 3 b C d Chapter 4 Exercise Solutions Principles of Econometrics 46 108 90 069538 0015025gtlt 48 1417 Using computer software We nd the standard error of the forecast error is se f 025293 Then the 95 prediction interval is given by y t097545se f 14166 20141 x 025293 O9071926 9 056231 016961gtlt ln48 1219 The standard error of the forecast error is se f 028787 The 95 prediction interval is given by 120 i t097545sef 12189 i 20141 x 028787 06391799 9 079945 0000337543 x 482 1577 The standard error of the forecast error is se f 023454 The 95 prediction interval is given by 120 i t097545sef 1577145 i 20141gtlt 0234544 1105 2050 The actual yield in Chapman was 1844 which lies Within the interval estimates from the linear and quadratic models but outside the interval estimate from the linearlog model Equation 1 Equation 2 Equation 3 ZS dy A 00150 dz B2 431 01696 dt Z 48 00035 291 2 X 00003375 x 48 00324 Evaluating the elasticities at t 48 and the relevant value for 330 We have Equation 1 Equation 2 Equation 3 dy if3 001502x 48 0509 dry y0 14166 O1696O139 dt y y 1219 Q 2912 2gtlt00003375gtlt482 dt y 90 1577 0986 The slopes dydt and the elasticities dydtgtltty give the marginal change in yield and the percentage change in yield respectively that can be expected from technological change in the next year The results show that the predicted effect of technological change is very sensitive to the choice of functional form Chapter 4 Exercise Solutions Principles of Econometrics 46 109 EXERCISE 410 a For households with 1 child WEE 10099 01495lnTOTEXP se 00401 00090 R2 03203 1 2519 1670 For households with 2 children 659 09535 01294lnTOTEXP se 00365 00080 R2 02206 I 2610 1616 For 52 We would expect a negative value because as the total expenditure increases the food share should decrease with higher proportions of expenditure devoted to less essential items Both estimations give the expected sign The standard errors for b1 and 92 from both estimations are relatively small resulting in high values of 1 ratios and signi cant estimates b For households with 1 child the average total expenditure is 94848 and b1b21nT0TEXP1 10099 01495gtltln94848 1 j 05461 1 21T0TEXp 10099 01495gtlt ln94848 For households with 2 children the average total expenditure is 101168 and b1b21nT0TEXP1 09535 012944gtltln1011681 T 06363 121mTEXp 09535 012944gtlt ln101168 Both of the elasticities are less than one therefore food is a necessity C Figure xr410c Plots for 1child households 08 0 04 06 02 0030 otmooo 0 5 35 g 88 i s 8 0 a ee9s 00 e 2818 cg gg 3 0 5 o e Egg E33 8 8 05 B8 o e O 04 2 3 LI 0 D E 8 02 02 00 04 3 4 5 6 3 4 5 6 X1 X1 Fitted equation Residual plot Chapter 4 Exercise Solutions Principles of Econometrics 46 110 Exercise 410c continued 0 d The tted curve and the residual plot for households with 1 child suggest that the function linear in WFOOD and lnT OT EXP seems to be an appropriate one However the observations vary considerably around the tted line consistent with the low R2 value Also the absolute magnitude of the residuals appears to decline as lnT OT EXP increases In Chapter 8 We discover that such behavior suggests the existence of heteroskedasticity The plots of the tted equation and the residuals for households with 2 children lead to similar conclusions The values of JB for testing H0 the errors are normally distributed are 107941 and 63794 for households with 1 child and 2 children respectively Since both values are greater than the critical value X89592 5991 We reject H0 The pvalues obtained are 00045 and 00412 respectively con rming that H0 is rejected We conclude that for both cases the errors are not normally distributed Figure xr410c Plots for 2child households 08 04 02 O E 3 00 O 2 I 8 02 0 0 0 4 35 40 45 50 55 50 35 40 45 50 55 50 x2 x2 Fitted equation Residual plot The estimated equation for the fuel budget share is WFIEL 03009 004641nTOTEXP se 00198 00043 R2 01105 1 1522 1071 The estimated slope coef cient is negative and statistically signi cant at the 5 level The negative sign suggests that as total expenditure increases the share devoted to fuel will decrease Chapter 4 Exercise Solutions Principles of Econometrics 46 111 Exercise 410d continued 6 The estimated equation for the transportation budget share is 9 00576 004101nTOTEXP se 00414 00091 1 139 451 R2 00216 The estimated slope coefficient is positive and statistically signi cant at the 5 level The positive sign suggests that as total expenditure increases the share devoted to transportation will increase The elasticity for quantity of fuel with respect to total expenditure evaluated at median total expenditure is 0300873 0046409gtlt ln90l 04958 0300873 0046409 gtlt ln90 and at the 95th percentile of total expenditure it is A 0300873 0046409gtltln1 801 8 02249 0300873 0046409gtlt ln180 These elasticities are less than one indicating that fuel is a necessity The share devoted to fuel declines as total expenditure increases At the higher expenditure level the elasticity is smaller indicating that for these households additional percentage increases in total expenditure lead to smaller percentage increases in the quantity of fuel used Using similar calculations We nd that the elasticity for transportation at median total expenditure is 13232 and at the 95 percentile of total expenditure it is 12640 These elasticities are greater than one indicating that transportation is a luxury The share devoted to transportation increases as total expenditure increases At the higher expenditure level the elasticity is slightly smaller indicating that for these households additional percentage increases in total expenditure lead to smaller percentage increases in the quantity of transportation used These results for fuel are consistent with economic reasoning Fuel need to heat houses would be considered essential and those households with higher incomes higher total expenditures are likely to make a smaller adjustment because they would be using an amount closer to What they consider necessary Classifying transportation as a luxury is consistent with households moving to more expensive and quicker modes of transportation as their incomes increase One might expect the elasticity to be higher for the higher level of total expenditure but there is not a big difference in their magnitudes at 90 and 180 pounds Chapter 4 Exercise Solutions Principles of Econometrics 46 112 EXERCISE 411 3 b C d The estimated regression model for the years 1916 to 2008 is T50848408859GROWTH R2 05189 se 10125 01819 The predicted value of VOTE in 2008 is IEEZOO8 508484 08859 x 0220 51043 The least squares residual is VOTE2008 TE2008 46600 51043 4443 The estimated regression model for the years 1916 to 2004 is I7039FE 510533 08780GROWTH R2 05243 se 10379 01825 The predicted value of VOTE in 2008 is IEEZOO8 5105325 087798 X 022 51246 The prediction error is f VOTE2008 2008 46600 51246 4646 This prediction error is larger in magnitude than the least squares residual This result is expected because the estimated regression in part b does not contain information about VOTE in the year 2008 The 95 prediction interval is VOTE2008 Jrt0975721 gtlt se f 512464 i 20796 x 49185 4101861475 The actual 2008 outcome VOTE2008 466 falls Within this prediction interval The estimated value of GROWTH that would have given the incumbent party 501 of the vote is that value of GROWTH for which 501 5105325 0877982 gtlt GROWTH Solving for GROWTH yields WLWW 1686 0877982 We estimate that real per capita GDP would have had to decrease by 1086 in the rst three quarters of the election year for the incumbent party to Win 501 of the vote GROWTH Chapter 4 Exercise Solutions Principles of Econometrics 46 113 EXERCISE 412 a The estimated reciprocal model is Q 60244 4836501P R2 08770 se 20592 25612 A plot of this equation appears below The reciprocal model ts the data relatively Well There is some tendency to underestimate quantity in the middle range of prices and overestimate quantity at the low and high extreme prices 50 I 40 I 30 I 20 I 10 I I I 2 25 3 Price of Chicken Figure xr412a Scatter of data points and tted reciprocal model Fitted values 0 Quantity of Chicken l b The derivative of the reciprocal model is Q 48365i dP P Thus the elasticity is given by 8dQ 48365 dP Q PQ When P131 Q 60244 48365 x T131 30895 and 8 48365 24195 131gtlt 30895 The elasticity found using the loglog model was 8 ll2l a similar but slightly smaller absolute value than that for the reciprocal model Chapter 4 Exercise Solutions Principles of Econometrics 46 114 Exercise 412 continued c The estimated linearlog model is Q412111 319078lnP R2 08138 se 09898 21584 A plot of this equation appears below Like the reciprocal model this loglinear model tends to over predict for low and high prices and under predict for midrange prices Also its t appears slightly worse than that of the reciprocal model 50 I 40 I 30 I 20 I 10 I I I I I 1 15 25 3 Figure xr412c Scatter of data points and tted reciprocal model 2 Price of Chicken Fitted values 0 Quantity of Chicken 1 d The derivative of the linearlog model is Q 4190781 dP P Thus elasticity when P 131 is given by 8 dQ 5 319078 319078 0979 dP Q Q 412111 319078ln131 The elasticities for the loglog and reciprocal models were 1121 and 1195 respectively Thus the linearlog model yields a lower elasticity in absolute value than the other models e After considering the data plots in parts a and c and Figure 416 in the text we can conclude that the loglog model ts the data best As shown in the plots it exhibits the least variation between the actual data and its tted values This is con rmed by comparing the R2 values for each model R 1Og1Og 08817 I feciprocal 08770 R2 linearlog 08138 Chapter 4 Exercise Solutions Principles of Econometrics 46 115 EXERCISE 413 a The regression results are lnPRICE 105938 0000596 SQFT se 002190000013 1 48484 4630 The intercept 105938 is the Value of lnPRICE when the area of the house is zero This is an unrealistic and unreliable Value since there are no prices for houses of zero area The coef cient 0000596 suggests an increase of one square foot is associated with a 006 increase in the price of the house To nd the slope 01 PRICE d SQF T We note that d1nPRICE d1nPRICEX dPRICE 1 dPRICE dSQFT dPRICE dSQFT PRICE X dSQFT B 2 Therefore 32 X PRICE dSQFT At the mean dPRICE 8 x PRICE 000059596 gtlt1128108l 6723 dSQFT The Value 6723 is interpreted as the increase in price associated with a 1 square foot increase in living area at the mean The elasticity is calculated as 1 dPRICE dPRICE PRICE APRICE 5 FT 5 FT BZX Q PRICEX dSQFTX Q dSQFTSQFT ASQFT At the mean elasticity 8 X SQFT 000059596 x 161 19682 09607 This result tells us that at the mean a 1 increase in area is associated with an approximate 1 increase in the price of the house Chapter 4 Exercise Solutions Principles of Econometrics 46 116 Exercise 413 continued b C The regression results are 1nPRICE 41707 100661nSQFT se 01655 00225 7 2520 4465 The intercept 41707 is the Value of lnPRICE when the area of the house is 1 square foot This is an unrealistic and unreliable Value since there are no prices for houses of 1 square foot in area The coefficient 10066 says that an increase in living area of 1 is associated with a 1 increase in house price The coefficient 10066 is the elasticity since it is a constant elasticity functional form To nd the slope 01 PRICE d SQF T note that d 1nPRICE SQFT dPRICE d1nSQFT PRICE dSQFT B2 Therefore dPRICE B X PRICE dSQFT 2 SQFT At the means R432 XE X7o444 dSQFT SQFT 161 19682 The Value 70444 is interpreted as the increase in price associated with a 1 square foot increase in living area at the mean From the linear function R2 0672 From the loglinear function in part a c0Vyy2 199573x1092 R2 2 0715 g comy y 1 VaryVarf 278614x109 gtlt199996gtlt109 From the loglog function in part b 2 9 2 157631gtlt10 R corryf2 COVO y L J 0673 VaryVarf 278614gtlt109gtlt132604gtlt109 The highest R2 Value is that of the loglinear functional form In other Words the linear association between the data and the fitted line is highest for the loglinear functional form In this sense the loglinear model ts the data best Chapter 4 Exercise Solutions Principles of Econometrics 46 117 Exercise 413 continued d 120 100 80 J arqueBera 7885 p Value 00000 60 2 WWW HHHHH00 075 050 025 000 025 050 075 Figure xr413d Histogram of residuals for loglinear model 120 100 80 J arqueBera 5274 p Value 00000 6 40 20 lg ml i i HHHHHHH 5 lil II I I 075 050 025 000 025 050 075 C Figure xr413d Histogram of residuals for loglog model 200 160 120 J arqueBera 2456 p Value 00000 8 40 0 I I i iiilllIlI 100000 0 I I I I1I0o39oIo0I I I I I 0o39oIoo Figure xr413d Histogram of residuals for simple linear model All JarqueBera Values are signi cantly different from 0 at the 1 level of signi cance We can conclude that the residuals are not compatible with an assumption of normality particularly in the simple linear model Chapter 4 Exercise Solutions Principles of Econometrics 46 118 Exercise 413 continued 6 08 residual o 39r 04 39 39 39 39 0 1000 2000 3000 4000 5000 SQFT Figure xr413e Residuals of loglinear model 12 08 residual 2 2 04 o39 0 0 1000 2000 3000 4000 5000 SQFT Figure xr413e Residuals of loglog model 250000 200000 150000 100000 50000 residaul 0 50000 100000 150000 0 1000 2000 3000 4000 5000 SQFT Figure xr413e Residuals of simple linear model The residuals appear to increase in magnitude as SQF T increases This is most evident in the residuals of the simple linear functional form Furthermore the residuals for the simple linear model in the area less than 1000 square feet are all positive indicating that perhaps the functional form does not fit Well in this region Chapter 4 Exercise Solutions Principles of Econometrics 46 119 Exercise 413 continued 0 8 Prediction for loglinear model WET eXpb1 ZDZSQFT 522 exp10593790000595963 gtlt 2700 0203032 2 203516 Prediction for loglog model z exp4170677 1006582 gtlt log2700020825122 188221 Prediction for simple linear model z 1838565 813890gtlt 2700 201365 The standard error of forecast for the loglinear model is 2 sefc s21 x xg2 Zx139x 2700 16119682 880 2487689331 020363 02030341 The 95 con dence interval for the prediction from the loglinear model is eXP17 i t0975878Sef eXp10593790000595963 X 2700 i196267 gtlt 020363 133683 297316 The standard error of forecast for the loglog model is 790101 73355 2 se f 02082511L 880 8534453 020876 The 95 con dence interval for the prediction from the loglog model is eXP17 i t0975878Sef eXp4170677 1006582 gtlt log2700 i 196267 gtlt 020876 122267 277454 Chapter 4 Exercise Solutions Principles of Econometrics 46 120 Exercise 413g continued h The standard error of forecast for the simple linear model is 1 27OO 16119682 Z 3034826 880 248768933l sef 3025921 The 95 con dence interval for the prediction from the simple linear model is 390 i t0975878sef 20136462 i 196267 gtlt 3034826 141801 260928 The simple linear model is not a good choice because the residuals are heavily skewed to the right and hence far from being normally distributed It is difficult to choose between the other two models the loglinear and loglog models Their residuals have similar pattems and they both lead to a plausible elasticity of price with respect to changes in square feet namely a 1 change in square feet leads to a 1 change in price The log linear model is favored on the basis of its higher R value and its smaller standard deviation of the error characteristics that suggest it is the model that best fits the data Chapter 4 Exercise Solutions Principles of Econometrics 46 121 EXERCISE 414 3 b C Figure xr414a 80 140 25 30 35 Histogram of lnWAGE Histogram of WAGE Neither WAGE nor lnWAGE appear normally distributed However lnWAGE more closely resembles a normal distribution While the distribution for WAGE is positively skewed that for lnWAGE exhibits a more symmetric normal shape This conclusion is con rmed by the JarqueBera test results which are JB 77373 pvalue 00000 for WAGE and JB 06349 pvalue 07280 for lnWAGE The regression results for the linear model are WAGE 6710311980EDUC R2 01750 se 19142 01361 6 1OO 19803 The estimated return to education at mean wage x gtlt100 961 WA GE 206157 The results for the loglinear model are 1nV lt7E 16094 00904EDUC R2 01782 se 00864 00061 The estimated return to education 9 x 100 904 The histograms of residuals are displayed in Figure Xr414c The J arqueBera test results are JB83982 pvalue 00000 for the residuals from the linear model and JB2753 pvalue 00000 for the residuals from the loglinear model Both the histograms and the J arqueBera test results suggest the residuals from the loglinear model are more compatible with normality However in both cases a null hypothesis of normality is rejected at a 1 level of signi cance Chapter 4 Exercise Solutions Principles of Econometrics 46 122 Exercise 414c continued Figure xr414c Histograms of residuals 200 140 120 K 160 100 8O 120 so 80 40 J i 400 Oquot39lquot quotquot39quot39quot39quot39quoti O393939 393939 30 20 10 O 10 20 30 40 50 25 20 15 10 05 00 05 10 15 Simple linear regression loglinear regression d Linear model R2 01750 R 2 Loglinear model R corrWAGEWAGE 01859 Where ME eXpb1 bED UC Since R gt R2 we conclude that the loglinear model ts the data better 6 Figure xr414e Residuals plotted against EDUC so 2 50 o 3 0 40 O o O O 1 3 o O 8 8 2 so 9 9 9 o g E 20 O 0 g g To 0 e 0 g 2 5 0 a E 5 5 3 Q 10 8 8 0 E g e E 0 9 0 g 0 1 g 0 3 3 E E S 8 8 8 g 0 9 10 O 8 E 8 2 20 8 3 3930 I I I I 0 I O 393 I I I I I O 4 8 12 16 20 24 O 4 8 12 16 20 24 educ educ Simple linear model Loglinear model The absolute value of the residuals increases in magnitude as EDUC increases suggesting heteroskedasticity which is covered in Chapter 8 It is also apparent for both models that there are only positive residuals in the early range of EDUC This suggests that there might be a threshold effect education has an impact only after a minimum number of years of education We also observe the nonnormality of the residuals in the linear model the positive residuals tend to be greater in absolute magnitude than the negative residuals Chapter 4 Exercise Solutions Principles of Econometrics 46 123 Exercise 414 continued f Prediction for the simple linear model VWFEO 671028 198029gtlt 16 24974 Prediction for loglinear model VWFEC exp160944 0090408gtlt 16 052661 12 2 24401 Actual average Wage of all Workers with 16 years of education 25501 g The loglinear function is preferred because it has a higher goodnessof t value and its residuals are more consistent with normality However when predicting the average age of Workers with 16 years of education the linear model had a smaller prediction error Chapter 4 Exercise Solutions Principles of Econometrics 46 124 EXERCISE 415 Results using cps4smalldat 61 b Summary statistics for WAGE C Subsample Mean Std Dev Min Max CV i all males 22142 12744 230 7213 576 ii all females 19172 12765 197 7639 666 iii allwhites 20839 12851 197 7639 617 iv all blacks 17780 12339 650 7213 694 v whitemales 22500 12965 230 7213 576 vi white females 19206 12539 197 7639 653 vii blackmales 17150 10368 745 5250 605 viii black females 18218 13606 650 7213 747 These results show that on average white males have the highest wages and black males the lowest The wage of white females is approximately the same as that of all females Black females have the highest coefficient of variation and all males and white males have the lowest Regression results Subsample Constant ED UC return R2 i all males 18778 00796 796 01716 se 01092 0007 9 ii all females 11095 01175 1175 02437 se 01314 00092 iii all whites 16250 00904 904 01770 se 00941 00067 iv all blacks 11693 01147 1147 02310 se 02716 00200 v white males 19345 00770 770 01612 se 01176 00086 vi white females 10197 01243 1243 02656 se 0 1439 00100 vii black males 18068 00692 692 00933 se 04244 00325 viii black females 05610 01560 1560 03712 se 03552 00254 The return to education is highest for black females 1560 and lowest for black males 692 It varies approximately from 8 to 125 for all other subsamples Chapter 4 Exercise Solutions Principles of Econometrics 46 125 Exercise 415 continued Results using cps4smalldat d e The model does not t the data equally Well for each subsample The best ts are for black females and White females Those for White males and black males are particularly poon The Ivalue for testing H0 8 010 against H1 3 at 010 is given by t 192 01 seb2 We reject H0 if t gt tc or tlt tc Where I6 t0975 dfo The results are given in the following table Test results for H0 5 010 versus H15 at 010 Subsample tvalue df 1 pvalue Decision i all males 2569 484 1965 0011 Reject H 0 ii all females 1917 512 1965 0056 Fail to reject H0 iii all whites 1425 843 1963 0155 Fail to reject H0 iv all blacks 0736 110 1982 0463 Fail to reject H 0 v white males 2679 417 1966 0008 Reject H0 vi White females 2420 424 1966 0016 Reject H 0 vii black males 0947 44 2015 0349 Fail 10 reject H 0 viii black females 2207 64 1998 0031 Reject H0 The null hypothesis is rejected for males White males white females and black females suggesting that there is statistical evidence that the rate of retum is different to 10 For males and white males the Wage retum to an extra year of education is estimated as less than 10 while it is greater than 10 for the other two subsamples Where H0 was rejected In all other subsamples the data do not contradict the assertion that the Wage retum is 10 Chapter 4 Exercise Solutions Principles of Econometrics 4e EXERCISE 415 Results using cps4dat 3 b 0 Summary statistics for WAGE Subsample Mean Std Dev Min Max CV i all males 22258 13473 100 17300 605 ii all females 18054 11157 114 9617 618 iii all whites 20485 12638 114 17300 617 iv all blacks 16444 10136 100 7213 616 v white males 22834 13671 150 17300 599 vi white females 18119 11013 114 9617 608 vii black males 16213 9493 100 7213 586 viii black females 16621 10616 375 7213 639 126 These results show that on average white males have the highest wages and black males the lowest Overall males have higher average wages than females and whites have higher average wages than blacks The highest wage earner is a white male Black females have the highest coef cient of variation and black males have the lowest Regression results Subsample Constant ED UC return R2 i all males 17326 00884 884 02043 se 00499 00036 ii all females 12427 01064 1064 02312 se 00559 00039 iii all whites 15924 00911 911 01923 se 00411 00029 iv all blacks 12456 01052 1052 02033 se 0 127 8 00094 v white males 17909 00861 861 02059 se 00522 00037 vi white females 12541 01057 1057 02264 se 00617 00043 vii black males 16521 00762 762 00983 se 02105 00158 viii black females 09395 01262 1262 03024 se 01592 00115 The return to education is highest for black females 1262 and lowest for black males 762 For all other subsamples it varies from approximately 85 to 105 Chapter 4 Exercise Solutions Principles of Econometrics 46 127 Exercise 415 continued Results using cps4dat d e The model does not t the data equally Well for each subsample The best ts are for all females and black females That for black males is particularly poor The Ivalue for testing H0 8 010 against H1 3 at 010 is given by t 192 01 seb2 We reject H0 if t gt tc or tlt tc Where I6 t09757dfo The results are given in the following table Test results for H0 5 010 versus H15 at 010 Subsample tvalue df to pvalue Decision i all males 3263 2393 1961 00011 Rejeet H 0 ii all females 1629 2441 1961 01034 Fail to reject H0 iii all whites 3075 4114 1961 00021 Reject H0 iv all blacks 0551 491 1965 05816 Fail to reject H 0 v white males 3720 2063 1961 00002 Reject H0 vi white females 1326 2049 1961 01851 Fail to reject H 0 vii black males 1504 212 1971 01341 Fail 10 reject H 0 viii black females 2273 277 1969 00238 Rejeet H0 The null hypothesis is rejected for males all Whites White males and black females suggesting that there is statistical evidence that the rate of retum is different to 10 For males and all Whites the Wage retum to an extra year of education is estimated as less than 10 while it is greater than 10 for the other two subsamples Where H0 was rejected In all other subsamples the data do not contradict the assertion that the Wage retum is 10 Chapter 4 Exercise Solutions Principles of Econometrics 4e EXERCISE 416 81 128 By de nition yield is given as So the inverse of yield is YIELD PRODUCTION tonnes hectare AREA RYIELD 1 AREA hectares tonne YIELD PRODUCTION Thus RYIELD can be interpreted as the number of hectares needed to produce one tonne of Wheat b Figure xr416b Plots of the reciprocal of yield against time 28 36 32 24 0 28 2 Z Z 0 2 24 lt11 0 039 0 E g 16 E 20 I I 0 0 M II no O 16 O 12 E 0 0 0 39 o o 12 3900 0 O 0 0 0 39 39 0 0398 39 3939o 08 39 H 39 39 3939 o 0 o 0 04 I I I I I 04 I I I I I 1940 1950 1960 1970 1980 1990 2000 1940 1950 1960 1970 1980 1990 2000 YEAR YEAR Chapman Northampton 28 24 O O 24 20 O 20 I lt 39 lt9 E D 16 it 0 16 E 3 0 Lu 0 E 0 DC o m D 12 1 2 0 o 0 39 0 DC 0 0 0 O 0 O O Q Q C Q C 0 8 O 0 08 o 39 o o 39 39 0 quot0 0 0 0 0 04 I I I I I 04 I I I I I 1940 1950 1960 1970 1980 1990 2000 1940 1950 1960 1970 1980 1990 2000 YEAR YEAR Mullewa Greenough Chapter 4 Exercise Solutions Principles of Econometrics 46 129 Exercise 416b continued 0 There is an outlier in 1963 across all four shires implying that a greater number of hectares was needed to produce one tonne of wheat than in any other year There were similar but less extreme outliers in Mullewa in 1976 1977 and 1979 and in Chapman in 1976 and 1977 Wheat production in Westem Australia is highly dependent on rainfall and so one would suspect that rainfall was low in the above years A check of rainfall data at httpwwwbomgovauclimatedata reveals that rainfall was lower than usual in 1976 and 1977 but higher than normal in 1963 Thus it is dif cult to assess why 1963 was a bad year excess rainfall may have caused rust or other disease problems during the growing season or rain at harvest time may have led to a deterioration in wheat quality The estimated equations are Northampton RYE 13934 00169TIME R2 02950 se 0 1087 00039 Chapman I m 13485 00132TIME R2 02869 se 00862 00031 Mullewa 14552 00121TIME R2 01306 se 01300 00046 Greenough I m 13594 00164TIME R2 04954 se 00686 00024 In each case the estimate of 0c is an estimate of the average annual change in the number of hectares needed to produce one tonne of wheat For example for Greenough we estimate that the number of hectares needed declines by 00164 per year The test results for testing H 0 oc 0 against the alternative H1 zocz lt O are given in the table below A onetail test is used because if on is not zero we expect it to be negative since technological change will lead to a reduction in the number of hectares needed to produce one tonne of wheat The test statistic assuming the null hypothesis is true is St l 2 4 1 s99c2 46 We reject H0 if tltt 1678 or pvaluelt005 In all four cases the null 00546 hypothesis is rejected indicating that the required number of hectares is decreasing over time Chapter 4 Exercise Solutions Principles of Econometrics 46 Exercise 416c continued C1 Test results for H0 zocz 0 versus H zocz lt 0 Shire tvalue tc pvalue Decision i Northampton 4387 1679 00000 Reject H0 ii Chapman 4302 1679 00000 R j 650i H 0 iii Mullewa 2629 1679 00058 Reject H0 iv Greenough 6721 1679 00000 Reject H0 Figure 416c Residual plots from estimated equations 125 100 075 Z lt E lt 050 I E2 Q UJ 0 m 0 0 J5 0 O Q 000 0 e 0 5 025 0 0 0 0 quot050 7 I I I 1940 1950 1960 1970 1980 1990 2000 YEAR Chapman 16 12 39 0 lt 08 E j 39 I E 04 ml 0 d m 0 5 00 39 0 0 0 0 0 0 h 04 Q 0 O 0 3908 I I I I I 1940 1950 1960 1970 1980 1990 2000 YEAR Mullewa RESRNORTHAM PTON RESRGREENOUGH 130 25 0 20 15 10 0 05 0 0 00 e 39 5 a a 0000 0 JV 3 0 3905 I I I I 1940 1950 1960 1970 1980 1990 2000 YEAR Northampton 12 10 08 05 04 0 02 f W 0 u 00 g 39 0 O 0 02 0 0 3904 I I I I I 1940 1950 1950 1970 1980 1990 2000 YEAR Greenough The residual for 1963 is clearly much larger than all others for all shires except Mullewa con rming that this observation is an outlier In Mullewa this observation is also an outlier but in addition the residuals for 1976 197 8 and 1980 are relatively large Chapter 4 Exercise Solutions Principles of Econometrics 46 131 Exercise 416 continued 6 The estimated equations with the observation for 1963 omitted are Northampton ffrfi 12850 00144TIME se 00549 00019 R2 05515 Chapman FY75 12862 00117TIME R2 03429 se 00686 00024 Mullewa 13929 00107TIME R2 01222 se 01211 00043 Greenough 13010 00150TIME R2 06448 se 00472 00017 When We reestimate the reciprocal model Without data for the year 1963 in all cases the coefficient of time declines slightly in absolute Value suggesting that the earlier estimates may have exaggerated the effect of technological change Also the Value of R2 increases considerably for Northampton Chapman and Greenough but that for Mullewa shire decreases slightly The standard errors for the coefficient of interest it decrease for all four shires CHAPTER 5 Exercise Solutions 132 Chapter 5 Exercise Solutions Principles of Econometrics 46 133 EXERCISE 51 8 b C d 6 f g h 1 gt A I I gt gt 1gt xgt A gt tgt A I gt Zyfxfz 13 X96722 16 Xyfxfg 4 Xxff 10 1x3ltZx2x313gtlt10 4gtltO Zx Zxi Zxx22 W1002 yx2Zx 2 Zx Zx Zx2x2 qy 1 08125 2 ZyZ x 3Zx3x34x16 13x0O4 39 16x10 02 7 3 04 09875 0025 0375 14125 0025 06 04125 01875 62 2672 339837506396 N K 93 V Zx3923 C2xz3933 C3 Zx2x3 23 O Z022 6gt2Zltx 2712 Zx22Zx32 s66 vE6 Z 62 0613696 01999 X2 72gt2lt1 6igt SSEz f 38375 SSTZy y2 16 SSR 121625 SST 16 SSR SST SSE 121625 R2 07602 Chapter 5 Exercise Solutions Principles of Econometrics 46 134 EXERCISE 52 a A 95 con dence interval for B is 19 it seZ92 08125 2447gtlt 01999 0323313017 09756 b The null and alternative hypotheses are H0 2B 1 H1B2 7amp1 The calculated Ivalue is t 92 1 O8125 1 seb2 01999 At a 5 signi cance level We reject H0 if M gt t097576 2447 Since l O9377 lt 2447 O9377 We do not reject H 0 Chapter 5 Exercise Solutions Principles of Econometrics 4e 135 EXERCISE 53 8 b C 91 2 00091 2 0476 seb1 00191 i The Istatistic for 91 is 00276 ii The standard error for 192 is seb2 j 000418 66086 iii The estimate for B3 is 193 00002gtlt 69624 00014 iv To compute R2 We need SSE and SST From the output SSE 5752896 To find SST We use the result y ST1 00633 which gives ssr 1518 gtlt OO6332 608246 Thus R SSE 1 575290 200054 ssr 608246 A SSE 57 52896 v The estimated error standard deviation is G T 0061622 N K 1519 4 The value 192 00276 implies that if lnT OT EXP increases by 1 unit the alcohol share will increase by 00276 The change in the alcohol share from a 1unit change in total expenditure depends on the level of total expenditure Speci cally d WALC d T OT EXP 00276 T OT EXP A 1 increase in total expenditure leads to a 0000276 increase in the alcohol share of expenditure The value 193 00014 suggests that if the age of the household head increases by 1 year the share of alcohol expenditure of that household decreases by 00014 The value 194 00133 suggests that if the household has one more child the share of the alcohol expenditure decreases by 00133 A 95 con dence interval for B3 is 93 i t0975D1515 seI93 00014 i 196 X 00002 00018 00010 This interval tells us that if the age of the household head increases by 1 year the share of the alcohol expenditure is estimated to decrease by an amount between 00018 and 0001 Chapter 5 Exercise Solutions Principles of Econometrics 4e 136 Exercise 53 Continued d The null and alternative hypotheses are H 0 B4 0 H1 B4 7 O The calculated Ivalue is I b4 4075 S6b4 At a 5 signi cance level We reject H0 if it gtt09751515 196 Since 4075 gtl96 We reject H0 and conclude that the number of children in the household in uences the budget proportion on alcohol Having an additional child is likely to lead to a smaller budget share for alcohol because of the nonalcohol expenditure demands of that child Also perhaps households with more children prefer to drink less believing that drinking may be a bad example for their children Chapter 5 Exercise Solutions Principles of Econometrics 46 137 EXERCISE 54 8 b C d 6 The regression results are V 003 15 00414lnTOTEXP 00001AGE 00130 NK R2 00247 se 00322 00071 00004 00055 The value 122 00414 suggests that as lnTOTEXP increases by 1 unit the budget proportion for transport increases by 00414 Alternatively one can say that a 10 increase in total expenditure will increase the budget proportion for transportation by 0004 See Chapter 433 The positive sign of 122 is according to our expectation because as households become richer they tend to use more luxurious forms of transport and the proportion of the budget for transport increases The value 123 00001 implies that as the age of the head of the household increases by 1 year the budget share for transport decreases by 00001 The expected sign for 23 is not clear For a given level of total expenditure and a given number of children it is dif cult to predict the effect of age on transport share The value 24 00130 implies that an additional child decreases the budget share for transport by 0013 The negative sign means that adding children to a household increases expenditure on other items such as food and clothing more than it does on transportation Alternatively having more children may lead a household to turn to cheaper forms of transport The pvalue for testing H0 33 0 against the alternative H1B3 0 where 33 is the coef cient of AGE is 0869 suggesting that AGE could be excluded from the equation Similar tests for the coefficients of the other two variables yield pvalues less than 005 The proportion of variation in the budget proportion allocated to transport explained by this equation is 00247 For a onechild household V 1EZzW0 00315 00414lnT0TEXP0 00001IGEO 0013NK0 00315 00414gtltln987 00001gtlt36 0013gtlt1 01420 For a twochild household o 00315 00414lnT0TEXP0 00001AGE 0013NK0 00315 00414 x ln987 00001 x 36 0013 x 2 01290 Chapter 5 Exercise Solutions Principles of Ec0n0meIrics 4e 138 EXERCISE 55 3 b C d The estimated equation is TEE 284067 01834CRIME 228109NITOX 63715ROOMS 00478AGE se 53659 00365 41607 03924 00141 13353DIST 02723lCCESS 00126TAX 11768PTRATIO 02001 00723 0003 8 01394 The estimated equation suggests that as the per capita crime rate increases by 1 unit the home value decreases by 1834 The higher the level of air pollution the lower the value of the home a one unit increase in the nitric oxide concentration leads to a decline in value of 22811 Increasing the average number of rooms leads to an increase in the home value an increase in one room leads to an increase of 6372 An increase in the proportion of owneroccupied units built prior to 1940 leads to a decline in the home value The further the weighted distances to the ve Boston employment centers the lower the home value by 1335 for every unit of weighted distance The higher the tax rate per 10000 the lower the home value Finally the higher the pupilteacher ratio the lower the home value A 95 con dence interval for the coefficient of CRIME is 92 i I0975 497seb2 01834 i 1965 X 00365 0255 0112 A 95 confidence interval for the coefficient of ACCESS is 97 i I seI97 02723 i1965 x 00723 0130 0414 o975497 7 The value of the I statistic is I 00171 We want to test H0 B 7 against H1 B I 00171 b 7 63715 7 03924 16017 seb rooms At OL005 we reject H0 if the absolute calculated I is greater than 1965 Since j16017 lt1965 we do not reject H0 The data is consistent with the hypothesis that increasing the number of rooms by one increases the value of a house by 7000 We want to test H0 3 2 1 against H1 3 lt 1 The value of the I statistic is ptratio ptratio t 11768112683 01394 At a significance level of OL 005 we reject H 0 if the calculated I is less than the critical value IMSD497 1648 Since 12683 gt 1648 we do not reject H0 We cannot conclude that reducing the pupilteacher ratio by 10 will increase the value of a house by more than 10000 Chapter 5 Exercise Solutions Principles of Econometrics 4e 139 EXERCISE 56 In each case We use a twotail test with a 5 significance level The critical values are given by t002560 2000 and t097560 2000 The rejection region is t lt 2 or tgt 2 8 b C The value of the I statistic for testing the null hypothesis H 0 B2 0 against the alternative H1 2B 7 0 is 1 L i Seb2 Z Since 2 lt 15 lt 2 We fail to reject H0 and conclude that there is no sample evidence to suggest that 32 7 0 For testing H0 31 2B2 5 against the alternative H1 31 2B2 7 5 We use the statistic 19 219 5 seb1 2192 For the numerator of the tvalue We have 191 2192 5 2 2 x 3 5 3 The denominator is given by sebbv39a1T72bvEaa4xvE1a4xcW1aT 34gtlt44gtlt2 33166 09045 Therefore t 33166 Since 2 lt 09045 lt 2 We fail to reject H0 There is no sample evidence to suggest that B 2B 7 5 For testing H0 B1 32 33 4 against the alternative H1 B1 B 33 7 4 We use tb1Z92 b34 seb1 Z92 b3 Now blb2b3423146 and Seb1b2 473 3432gtlt22gtltl04 Thus t 615 4 Since 2 lt 15 lt 2 We fail to reject H0 and conclude that there is insuf cient sample evidence to suggest that 31 32 33 4 is incorrect Chapter 5 Exercise Solutions Principles of Econometrics 46 140 EXERCISE 57 The Variance of the error term is given by 6 SSE 1112389 005590 N K 2023 Thus the standard errors of the least square estimates 9 and 93 are 2 005590 1 1 G 000684 362 Var 1 r2Zx 72 1 01142552gtlt1210178 A2 1 b G 005590 000137 563 W 1 r2Zx 72 1 01142552gtlt3030757 Chapter 5 Exercise Solutions Principles of Econometrics 4e 141 EXERCISE 58 a Equations describing the marginal effects of nitrogen and phosphorus on yield are 8E YIELD j 8011 2gtlt l944gtlt NITRO 0567 x PHOS aN1TRO 801 1 3888NITRO 0567PHOS aE YIELD L 4800 2gtlt 0778gtlt PHOS 0567 X NITRO aPHOS 4800 l556PHOS 0567NITRO These equations indicate that the marginal effect of both fertilizers declines We have diminishing marginal products and these marginal effects eventually become negative Also the marginal effect of one fertilizer is smaller the larger is the amount of the other fertilizer that is applied b i The marginal effects When NIT R0 1 and PHOS 1 are aEY1ELD 8011 3888 O567 3556 aN1TRO aEY1ELD 4800 1556 O567 2677 aPHOs ii The marginal effects When NIT R0 2 and PHOS 2 are aEY1ELD 8011 3888gtlt2 O567gtlt2 0899 aN1TRO aEY1ELD g 4800 l556gtlt2 O567gtlt2 0554 aPHOs When NIT R021 and PHOS 1 the marginal products of both fertilizers are positive Increasing the fertilizer applications to NIT R0 2 and PHOS 2 reduces the marginal effects of both fertilizers with that for nitrogen becoming negative c To test these hypotheses the coef cients are de ned according to the following equation YIELD 31B2NITRO 33PH0S 34NITRO2 BSPHOSZ B6NITROgtlt PHOS e i The settings NIT R0 1 and PHOS 1 Will yield a zero marginal effect for nitrogen if B 2B4 B6 0 Thus We test H0 2B2 2B4 36 0 against the alternative H 2B 2B4 36 7 0 The value of the test statistic is b2b4b6 801l 2gtlt1944 O567 t seb 2194 196 40233 7367 Chapter 5 Exercise Solutions Principles of Econometrics 4e 142 Exercise 58ci Continued Since I gt 16 t0975D21 2080 We reject the null hypothesis and conclude that the marginal effect of nitrogen on yield is not zero when NIT R0 1 and PHOS 1 ii To test Whether the marginal effect of nitrogen is zero when NIT R0 2 and PHOS 1 We test H0 2B2 4B4 B6 0 against H1B2 4B4 B6 7 0 The value of the test statistic is 196 419 196 801l 4gtlt1944 0567 t seI92 4196 196 0040 Since t lt 2080 1606975321 We do not reject the null hypothesis A zero marginal yield with respect to nitrogen cannot be rejected when NIT R0 1 and PHOS 2 1660 iii To test Whether the marginal effect of nitrogen is zero when NIT R0 3 and PHOS 1 We test H0 2B2 6B4 36 0 against the alternative H1 2B2 6B4 B6 7 0 The value of the test statistic is b26b4b6 8011 6gtlt1944 O567 t seI92 6196 196 O233 Since t gt 2080 t097521 We reject the null hypothesis and conclude that the 8742 marginal product of yield to nitrogen is not zero when NIT R0 3 and PHOS 1 d The maximizing levels NIT RO and PHOS are those values for NIT R0 and PHOS such that the firstorder partial derivatives are equal to zero aEY1ELD 2 PHOS NITRO0 B3 B5 B6 aEY1ELD j 2 1v1TR0 PHOs0 a1v1TRO B2 B4 B6 The solutions and their estimates are NITR0 2B2B5 B306 2 x 801 1 x 0778 4800 x 0567 2 1701 B 4B4B5 05672 4 x l9440778 PHOS 21326136 3256 2gtlt 4800gtlt2 19448011gtlt0567 2465 136 4136136 0567 4 x 1 944 O778 The yield maximizing levels of fertilizer are not necessarily the optimal levels The optimal levels are those Where the marginal cost of the inputs is equal to the marginal value product of those inputs Thus the optimal levels are those for which aEY1ELD PRICEPHOS aEY1ELD PRICENITRO aPH0s PRICEPEANUTS a1v1TR0 PRICEPEANUTS Chapter 5 Exercise Solutions Principles of Econometrics 46 143 EXERCISE 59 8 b C d The marginal effect of experience on Wages is 6Wl GE 4 2 EXPER 6EXPER B 3 B4 We expect B2 to be positive as Workers with a higher level of education should receive higher Wages Also We expect B3 and B4 to be positive and negative respectively When Workers are relatively inexperienced additional experience leads to a larger increase in their Wages than it does after they become relatively experienced Also eventually We expect Wages to decline with experience as a Worker gets older and their productivity declines A negative B3 and a positive B4 gives a quadratic function with these properties Wages start to decline at the point Where the quadratic curve reaches a maximum The maximum is reached when the rst derivative is zero Thus the number of years of experience at which Wages start to decline EXPER is such that 13 2B4EXPER 0 EXPER 2 4 i A point estimate of the marginal effect of education on Wages is 2 22774 8EDUC A 95 interval estimate is given by 19 z8seb 22774 1962 x 01394 20039 25509 ii A point estimate of the marginal effect of experience on Wages when EXPER 4 is WEE Z93 2294 gtlt4 0682l 8gtlt00l0l 06013 To compute an interval estimate We need the standard error of this quantity which is given by seb3 8b4JT82 E2x8x x0010987185 64 X 0000003476 16 x 0000l89259 009045 A 95 interval estimate is given by 19 8194 t0975998seb3 8194 06013 1962 x 009045 O423807788 Chapter 5 Exercise Solutions Principles of Econometrics 4e 144 Exercise 59d continued iii A point estimate of the marginal effect of experience on Wages when EXPER 25 UV 1S i if alum 93 2b4 gtlt25 O682l 50gtlt00lOl 01771 To compute an interval estimate We need the standard error of this quantity which is given by sea5oagajQEEmY 5YEj2x5oxamp RZEj 0010987185 2500 gtlt 0000003476 100 x 0000189259 002741 A 95 interval estimate is given by 93 50194 i t0975998seb3 50194 0177141 1962 x 002741 o12334123o9 Using the equation derived in part c We nd 2a 2x00NH We estimate that Wages will decline after approximately 34 years of experience 3377 To obtain an interval estimate for EXPER We require se b3 2194 which in turn requires the derivatives 8EXPER i 8EXPER i 8B3 2B4 5B4 2B3 Then 5 2 5 2 VEXpER9w VaE V 8 3 80 aEXPER aEXPER 2 covb3b4 883 884 and mm 2 2 s l 9 l 9 varEXPER 2 0m varb3 varb4 2 T4j S covb3b4 Substituting into this expression yields Chapter 5 Exercise Solutions Principles of Econometrics 46 145 Exercise 59div continued T 1 2 06821 2 var EXPER 4 gtlt0010987185 42 gtlt0000003476 2gtlt00lOl 2gtlt00lOl 2gtlt 4 x is gtlt0000189259 2gtlt00lOl 2gtltOOlOl2 3l3l785 se x3131785 1770 A 95 interval estimate for EXPER is 131 z4444444444se1 x1 2 3377 1962gtlt 177 303 372 ii The above answers to part d are based on hand calculations using the estimates and covariance matrix values reported in Table 59 of the text If the computations are made using software and the le cps4csmaZZdat slightly different results are obtained These results do not suffer from the rounding error caused by truncating the number of digits reported in Table 59 The answers obtained using software for parts dii iii and iv are d 11 194 8194 z4444444444se194 8194 060137 i1962gtlt 0090418 0423 907789 iii 193 50194 z4444444444se194 50194 017756 1962 x 0027425 0123702314 iv EXPER t0975998se 33798 1962 x 17762 303 373 Chapter 5 Exercise Solutions Principles of Econometrics 4e EXERCISE 510 The EViews output for Verifying the answers to Exercise 51 is given in the following table 0 d 6 f g h Method Least Squares Dependent Variable Y Method Least Squares Included observations 9 146 Coefficient Std Error tStatistic Prob X1 1000000 0266580 3751221 00095 X2 0812500 0199935 4063823 00066 X3 0400000 0252900 1581654 01648 Rsquared 0760156 Mean dependent var 1000000 Adjusted Rsquared 0680208 SD dependent var 1414214 SE of regression 0799740 Akaike info criterion 2652140 Sum squared resid 3837500 Schwarz criterion 2717882 Log likelihood 8934631 HannanQuinn criter 1728217 The least squares estimates can be read directly from the table The residuals from the estimated equation are 04000 09875 00250 03750 14125 00250 06000 04125 01875 2 The estimate 6 is given by the square of SE of regression That is 82 0799742 0639584 The correlation matrix for the three Variables is X2 X3 Y X2 1000000 0000000 0812500 X3 0000000 1000000 0316228 Y 0812500 0316228 1000000 The correlation between x2 and x3 is zero The standard error for 192 can be read directly from the EViews output From the EViews output SSE Sum squared resid 38375 and R2 0760156 To obtain SST note that sj 14142142 2 Then SSTZyl 2 n 1s 8gtlt216 SSR SST SSE 16 38375 121625 Chapter 5 Exercise Solutions Principles of Econometrics 4e 147 EXERCISE 511 a Estimates standard errors and pvalues for each of the coef cients in each of the estimated share equations are given in the following table Explanatory Dependent Variable Variables Food Fuel Clothing Alcohol Transport Other Constant Estimate 08798 03179 02816 00149 00191 00881 Std Error 00512 00265 00510 00370 00572 00536 pValue 00000 00000 00000 06878 07382 01006 lnTOTEXP Estimate 0 1477 00560 00929 00327 00321 00459 Std Error 00113 00058 00112 00082 00126 00118 pValue 00000 00000 00000 00001 00111 00001 AGE Estimate 000227 000044 000056 000220 000077 000071 Std Error 000055 000029 000055 000040 000062 000058 pValue 00000 01245 03062 00000 02167 02242 NK Estimate 00397 00062 00048 00148 00123 00139 Std Error 00084 00044 00084 00061 00094 00088 pValue 00000 01587 05658 00152 01921 01157 An increase in total expenditure leads to decreases in the budget shares allocated to food and fuel and increases in the budget shares of the commodity groups clothing alcohol transport and other Households with an older household head devote a higher proportion of their budget to food fuel and transport and a lower proportion to clothing alcohol and other Having more children means a higher proportion spent on food and fuel and lower proportions spent on the other commodities The coef cients of lnT OT EXP are signi cantly different from zero for all commodity groups At a 5 signi cance level age has a signi cant effect on the shares of food and alcohol but its impact on the other budget shares is measured less precisely Signi cance tests for the coef cients of the number of children yield a similar result NK has an impact on the food and alcohol shares but we can be less certain about the effect on the other groups To summarize lnTOTEXP has a clear impact in all equations but the effect of AGE and NK is only signi cant in the food and alcohol equations Chapter 5 Exercise Solutions Principles of Econometrics 46 148 Exercise 511 continued b The tValues and pValues for testing H0 2B 3 0 against H1 2B2 gt 0 are reported in the table below Using a 5 level of signi cance the critical Value for each test is t095496 21648 tValue pValue decision WFOOD 13083 10000 Do not reject H0 WFUEL 9569 10000 Do not reject H0 WCLOTII 8266 00000 Reject H0 WALC 4012 00000 Reject H0 WTRANS 2548 00056 Reject H0 WOTER 3884 00001 Reject H0 Those commodities which are regarded as necessities 192 lt 0 are food and fuel The tests suggest the rest are luxuries While alcohol transportation and other might be luxuries it is dif cult to see clothing categorized as a luxury Perhaps a ner classi cation is necessary to distinguish between basic and luxury clothing Chapter 5 Exercise Solutions Principles of Ec0n0meIrz cs 46 149 EXERCISE 512 8 b C d 6 f The expected sign for 32 is negative because as the number of grams in a given sale increases the price per gram should decrease implying a discount for larger sales We expect B3 to be positive the purer the cocaine the higher the price The sign for B4 will depend on how demand and supply are changing over time For example a xed demand and an increasing supply will lead to a fall in price A xed supply and increased demand would lead to a rise in price The estimated equation is IWFE 908467 00600 QUANT O1162QUAL 23546 TREND R2 05097 se 85803 00102 02033 13861 1 10588 5892 05717 16987 The estimated values for 2133 and B4 are 00600 01162 and 23546 respectively They imply that as quantity number of grams in one sale increases by 1 unit the price will go down by 00600 Also as the quality increases by 1 unit the price goes up by 01162 As time increases by 1 year the price decreases by 23546 All the signs turn out according to our expectations with 34 implying supply has been increasing faster than demand The proportion of variation in cocaine price explained by the variation in quantity quality and time is 05097 For this hypothesis We test H0 13 20 against H1 13 lt0 The calculated Ivalue is 5892 We reject H0 if the calculated I is less than the critical I095752 1675 Since the calculated I is less than the critical I value We reject H 0 and conclude that sellers are Willing to accept a lower price if they can make sales in larger quantities We want to test H0 B3 3 0 against H1 2B3 gt 0 The calculated Ivalue is 05717 At OL 005 We reject H0 if the calculated I is greater than 1675 Since for this case the calculated I is not greater than the critical I We do not reject H 0 We cannot conclude that a premium is paid for better quality cocaine The average annual change in the cocaine price is given by the value of 94 23546 It has a negative sign suggesting that the price decreases over time A possible reason for a decreasing price is the development of improved technology for producing cocaine such that suppliers can produce more at the same cost Chapter 5 Exercise Solutions Principles of Econometrics 46 150 EXERCISE 513 8 b The estimated regression is z 41948 90970SQFT 75504IGE se 6990 2403 14089 i The estimate 92 9097 implies that holding age constant on average a one square foot increase in the size of the house increases the selling price by 9097 dollars The estimate 93 75504 implies that holding SQFT constant on average an increase in the age of the house by one year decreases the selling price by 75504 dollars The estimate 91 could be interpreted as the average price of land if its value was meaningful Since a negative price is unrealistic We view the equation as a poor model for data values in the vicinity of SQF T 0 and AGE 0 EYJEIFE ii A point estimate for the price increase is SQF T 6 909698 A 95 interval estimate for B2 given that tc t0975p1077 1962 is 92 i re seb2 909698 i1962 x 24031 86259569 iii The Ivalue for testing H0 33 2 1000 against H1 B3 lt 1000 is l7386 b 1000 7550414 1000 2 3 seb3 1408936 The corresponding pvalue is Pt1077 lt 173 86 0959 The critical value for a 5 signi cance level is t005 1077 1646 The rejection region is is 1646 Since the Ivalue is greater than the critical value and the pvalue is greater than 005 We fail to reject the null hypothesis We conclude that the estimated equation is compatible with the hypothesis that an extra year of age decreases the price by 1000 or less The estimated regression is 11FE 170150 55784SQFT 0023153SQFT2 27978IGE 30160AGE2 se 10432 6389 0000964 3051 5071 For the remainder of part b We refer to these estimates as b1b2b3b4b5 in the same order as they appear in the equation with corresponding parameters B1B2B3B 4135 i The marginal effect of SQF T on PRICE is given by 8PRICE aSQFT Z132 2B3SQFT Chapter 5 Exercise Solutions Principles of Econometrics 46 151 Exercise 513bi continued ii The estimated marginal effect of SQF T on PRICE for the smallest house where SQFT 662 is 8PRICE asQFT 557842 2gtlt 0023153gtlt 662 2513 The estimated marginal effect of SQF T on PRICE for a house with SQF T 2300 is 8PRICE asQFT 557842 2x 0023153 x 2300 5072 The estimated marginal effect of SQF T on PRICE for the largest house where SQF T 7897 is 8PRICE asQFT 557842 2x 0023153 x 7897 30989 These values suggest that as the size of the house gets larger the price or cost for extra square feet gets larger and that for small houses extra space leads to a decline in price The result for small houses is unrealistic However it is possible that additional square feet leads to a higher price increase in larger houses than it does in smaller houses The marginal effect of AGE on PRICE is given by 8PRICE j 2 AGE 8lGE B B5 The estimated marginal effect of AGE on PRICE for the oldest house AGE 80 is 8PRI CE 8l GE The estimated marginal effect of AGE on PRICE for a house when AGE 20 is 2797788 2x 3016033 X 80 202786 8PRI C E 8A GE The estimated marginal effect of AGE on PRICE for the newest house A GE 1 is 2797788 2x 3016033 X 20 159138 8PRI CE 8l GE 2797788 2x 3016033 gtlt1 273747 When a house is new extra years of age have the greatest negative effect on price Aging has a smaller and smaller negative effect as the house gets older This result is as expected However unless a house has some kind of heritage value it is unrealistic for the oldest houses to increase in price as they continue to age as is suggested by the marginal effect for AGE80 The quadratic function has a minimum at an earlier age than is desirable Chapter 5 Exercise Solutions Principles of Econometrics 46 152 Exercise 513b continued iii A 95 interval for the marginal effect of SQF T on PRICE When SQF T 2300 and using tc t0975 1075 1962 is S me 7 seltn1e 50719 451962 x 25472 45725572 The standard error for n1e can be found using software or from sen1e varb2 46002 varb3 2x 4600covb2b3 4082499 46002 gtlt 9296015 gtlt10 7 9200 x 0005 870334 25472 iv The null and alternative hypotheses are H0B440B5 21000 H1B440B5 lt1000 The Ivalue for the test is t 19 4019 1000 591375 seb4 4005 139554 4238 The corresponding pvalue is Pt lt423800000 The critical value for a 1075 5 significance level is t0051075 1646 The rejection region is 1 1646 Since the Ivalue is less than the critical value and the pvalue is less than 005 We reject the null hypothesis We conclude that for a 20year old house an extra year of age decreases the price by more than 1000 The standard error seb4 40 Z25 can be found using software or from seb4 40195 varb4 402 varb5 2gtlt 40covb4b5 93095481600gtlt 2571554 80gtlt 1434561 13955 c The estimated regression is WE 114597 30729SQFT 0022185sQFT2 se 12143 6898 0000943 44203AGE 26519AGE2 093062SQFTgtlt AGE 41061 4939 011244 For the remainder of part c We refer to these estimates as b1b2b3b4b5b6 in the same order as they appear in the equation with corresponding parameters B1B2B3B4B5B6 Chapter 5 Exercise Solutions Principles of Econometrics 4e 153 Exercise 513c continued i ii The marginal effect of SQF T on PRICE is given by 8PRICE aSQFT 32 2B3SQFT B6AGE When AGE 20 the estimated marginal effect of SQF T on PRICE for the smallest house Where SQF T 662 is 8PRICE asQFT 307289 2x 0022185 x 662 093062gtlt 20 1997 When AGE 20 the estimated marginal effect of SQF T on PRICE for a house with SQF T 2300 is 8PRICE asQFT 307289 2 x 0022185 x 2300 093062 x 20 5271 When AGE 20 the estimated marginal effect of SQF T on PRICE for the largest house Where SQFT 7897 is 8PRICE asQFT 307289 2x 00221846 x 7897 0930621gtlt 20 30104 These Values lead to similar conclusions to those obtained in part b As the size of the house gets larger the price or cost for extra square feet gets larger For small houses extra space appears to lead to a decline in price This result for small houses is unrealistic It would be more realistic if the quadratic reached a minimum before the smallest house in the sample The marginal effect of AGE on PRICE is given by 8PRI C E 8A GE When SQF T 2300 the estimated marginal effect of AGE on PRICE for the oldest house AGE 80 is 34 ZBSAGE B6SQFT 8PRI CE 8A GE When SQF T 23 00 the estimated marginal effect of AGE on PRICE for a house of AGE 20 is 4420336 2x 26519gtlt 80 093062 X 2300 16606 8PRI CE 8A GE 4420336 2x 26519gtlt 20 093062gtlt 2300 15217 Chapter 5 Exercise Solutions Principles of Econometrics 46 154 Exercise 513cii continued d When SQFT 2300 the estimated marginal effect of AGE on PRICE for the newest house AGE 1 is 8PRI CE 8A GE 4420336 2 X 26519gtlt 1 093062gtlt 2300 25294 These results lead to similar conclusions to those reached in part bWhen a house is new extra years of age have the greatest negative effect on price Aging has a smaller and smaller negative effect as the house gets older This result is as expected However unless a house has some kind of heritage value the positive marginal effect for AGE 80 is unrealistic We do not expect the oldest houses to increase in price as they continue to age iii A 95 interval for the marginal effect of SQF T on PRICE when SQF T 2300 and AGE 20 and using tc t0975 1074 1962 is S me 1 seltn1e 52708 i1962gtlt 24825 47845758 The standard error for me was found using software iv The null and alternative hypotheses are H0 13 4013 2300136 2 1000 H1B4 400 230006 lt 1000 The Ivalue for the test is t 19 4019 2300196 1000 521701 3847 seb4 4019 23OOb6 135630 1074 5 significance level is t00571074 1646 The rejection region is 1 1646 Since The corresponding pvalue is Pt lt3847 00001 The critical value for a the Ivalue is less than the critical value and the pvalue is less than 005 we reject the null hypothesis We conclude that for a 20year old house with SQF T 2300 an extra year of age decreases the price by more than 1000 The results from the two quadratic speci cations in parts c and d are similar but they are vastly different from those from the linear model in part a In part a the marginal effect of SQF T is constant at 91 whereas in parts b and c it varies from approximately 20 to 300 The marginal effect of AGE is constant at 755 in part a but varies from approximately 2600 to 1800 in parts b and c with a similar pattern in b and c but some noticeable differences in magnitudes These differences carry over to the interval estimates for the marginal effect of SQF T and to the hypothesis tests on the marginal effect of AGE The marginal effects are clearly not constant and so the linear function is inadequate Both quadratic functions are an improvement but they do give some counterintuitive results for old houses and small houses It is interesting that the intercept is positive in the quadratic equations and hence has the potential to be interpreted as the average price of the land Both estimates seem large however relative to house prices Chapter 5 Exercise Solutions Principles of Econometrics 4e 155 EXERCISE 514 a The estimated regression is ln 111196 0038762SQFT100 0017555IGE 0000l7336IGEZ se 00274 0000869 0001356 000002266 b The estimate 612 003876 suggests that holding age constant an increase in the size of the house by one hundred square feet increases the price by 388 on average c The required derivative is given by a3 L 8lGE When AGE 5 gIlfE 0017555 2x 000017336gtlt5 001582 This estimate implies that holding SQF T constant the price of a 5year old house will decrease at a rate of 158 per year When AGE 20 00l7555 2x 0000l7336 x 20 00l062 This estimate implies that holding SQF T constant the price of a 20year old house will decrease at a rate of 106 per year d The required derivatives are given by 8PRI CE 8l GE 01 2014lGEgtlt PRICE 013 2014lGEgtlt eXpoL1 asQFT100 013lGEoc4IGE2 apli 0 2pR1CE asQFT100 012 X eXpoL1 asQFT100 013lGE 014IGE2 Where eXpx is notation for the exponential function e e To estimate these marginal effects We rst nd z expamp ampsQFT100 613IGE 614IGEZ eXp1111959 00387624gtlt 23 0017555 x 20 000017336gtlt 202 124165 Then Chapter 5 Exercise Solutions Principles of Econometrics 46 156 Exercise 514e continued a 1 0017555 2x 000017336gtlt 20gtlt 124165 13187 00387624gtlt124165 4813 f We require the standard errors of 53 61 40014gtltexp01123012 2061 400614 61 x exp011 2361 2061 40061 These expressions are nonlinear functions of the least squares estimators for the 01 s To compute their standard errors we need the delta method introduced on pages 1934 of the text Using computer software we nd the standard errors are se apjj 72671 se Wi 121637 8lGE 8SQFT100 g A 95 interval estimate for the marginal effect of SQF T I 00 is S me z6 sen1e 48129 i1962 gtlt 121637 45745o52 h The null and alternative hypotheses are H0 013 40014gtlt exp011 23012 20013 400014 2 1000 H1013 40014gtlt exp011 23012 20013 400014 lt 1000 The calculated value of the Istatistic is t 13187 1000 72671 4386 The corresponding pvalue is Pt lt 438600000 The critical value for a 5 1076 significance level is t00571076 1646 The rejection region is tS 1646 Since the 1 value is less than the critical value and the pvalue is less than 005 we reject the null hypothesis We conclude that for a 20year old house with SQF T 2300 an extra year of age decreases the price by more than 1000 Remark A comparison of the results in parts g and h with those from the quadratic function with the interaction term in Exercise 513c shows that similar conclusions are reached although the interval estimate in g is narrower and the estimated marginal effect is smaller Similarly the marginal effect in h is smaller in absolute value and estimated more precisely than its counterpart in Exercise 513c Chapter 5 Exercise Solutions Principles of Econometrics 46 157 EXERCISE 515 8 b C The estimated regression model is 176 5216 O6434GROWTH O1721INFLATION se 146 01656 O4290 The hypothesis test results on the signi cance of the coef cients are H0 83 0 H3 83 gt 0 pvalue 00003 signi cant at 10 level H0 83 0 H3 83 lt 0 pvalue 03456 not signi cant at 10 level Onetail tests were used because more growth is considered favorable and more in ation is considered not favorable for reelection of the incumbent party i For INFLATION 4 and GROWTH 3 the predicted percentage vote is 176 521565 064342gtlt 3 0172076gtlt 4 4954 ii For INFLATION 4 and GROWTH 0 the predicted percentage vote is e 521565 064342gtlt 0 0172076gtlt 4 5147 iii For INFLATION 4 and GROWTH 3 the predicted percentage vote is 176 521565 064342gtlt3 0172076gtlt 4 5340 Ignoring the error term the incumbent party will get the majority of the vote when 8183GROWTH 83INFLATION gt 50 When INFLATION 4 this requirement becomes 8183GROWTH 483 gt 50 i When GROWTH 3 the hypotheses are H0281382483S50 H1281383483gt50 Given that I 2457 We reject H 0 when 09930 193319 4193 50 gt 2457 se193 3193 4193 Now var191 3193 4193 var191 32 var193 42 var193 2gtlt 3cov191193 2x4 2x3x4 2127815 9x 0027433 16gtlt 0184003 6x 0048748 8gtlt 0498011 24gtlt 0011860 134252 Chapter 5 Exercise Solutions Principles of Econometrics 46 158 Exercise 515ci continued The calculated Ivalue is 3 19l 3193 4193 50 49538 5O O399 se19l 3b3 4b3 134252 Since 0399 lt 2457 We do not reject H 0 There is no evidence to suggest that the incumbent part will get the majority of the vote when INFLATION 4 and GROWTH 3 ii When GROWTH 0 the hypotheses are HllBl4B3 5O HlBl4B3gt50 19l 4193 50 Seb1 4193 gt 2457 We reject H0 when t The standard error can be calculated from a similar expression to that given in ci Using computer software We nd se19l 4193 104296 The calculated Ivalue is 19l 4193 50 514682 50 1408 se19l 4b3 104296 Since 1408 lt 2457 We do not reject H0 There is insuf cient evidence to suggest that the incumbent part will get the majority of the vote when INFLATION 4 and GROWTH 0 iii When GROWTH 3 the hypotheses are HllBl3B34B3s5O HlBl3B34B3gt5O 19l 3193 4193 50 se19l 3193 4193 gt 245739 We reject H0 when t The standard error can be calculated from a similar expression to that given in ci Using computer software We nd se19l 3192 4193 115188 The calculated Ivalue is t 19l 3193 4193 50 533985 5O 2950 se19l 3193 4b3 115188 Since 2950 gt 2457 We reject H0 We conclude that the incumbent part will get the majority of the vote when INFLATION 4 and GROWTH 3 As a president seeking reelection you would not Want to conclude that you would be re elected Without strong evidence to support such a conclusion Setting up reelection as the alternative hypothesis with a 1 signi cance level re ects this scenario Chapter 5 Exercise Solutions Principles of Econometrics 46 159 EXERCISE 516 a The estimated regression is SAL1 22963 470845PR1 92990PR2 165113PR3 R2 0443 se 9806 79578 70o13 93670 b The estimate 93 470845 suggests that holding PR2 and PR3 constant a one cent increase in the price of brand 1 leads to a decrease in the sales of brand 1 by 471 units The estimate 93 92990 suggests that holding PR and PR3 constant a one cent increase in the price of brand 2 leads to an increase in the sales of brand 1 by 93 units The estimate 94 165113 suggests that holding PR1 and PR2 constant a one cent increase in the price of brand 3 leads to an increase in the sales of brand 1 by 165 units The estimates of B3 B3 and B4 have the expected signs The sign of B3 is negative re ecting the fact that quantity demanded will fall as price rises While the signs of the other two coef cients are positive re ecting the fact that brands 2 and 3 are substitutes Increases in their prices will increase the demand for brand 1 c The hypothesis test results on the signi cance of the coef cients are H0 82 0 H1 32 lt 0 pvalue 00000 signi cant at 5 level H0 83 0 H1 83 gt 0 pvalue 00952 not signi cant at 5 level H0 84 0 H1 34 gt 0 pvalue 00422 signi cant at 5 level d i The hypotheses are H0 B3 300 H1B3 300 Since I 2011 We reject H0 iftb3 300seb3gt2011 or tlt 2011 097548 The Ivalue is t 19 300 4708453OO seb 79578 2 2147 Since 2147 lt 2011 we reject H0 and conclude that a 1cent increase in the price of brand 1 does not reduce its sales by 300 cans Chapter 5 Exercise Solutions Principles of Econometrics 46 160 Exercise 516d continued ii iii W The hypotheses are H0B3300 H1B3 300 Since I 2011 we reject H0 iftb3 300seb3gt2011 or tlt2011 097548 The Ivalue is t 193 300 92990 300 2957 seb3 70013 Since 2957 lt2011 we reject H0 and conclude that a 1cent increase in the price of brand 2 does not increase sales of brand 1 by 300 cans The hypotheses are H0B4300 H1B4 300 Since I 2011 we reject H0 iftb4 300seb4gt2011 or tlt2011 097548 The Ivalue is t 194 300 165113 300 24440 seb4 93670 Since 2011lt 1440 lt 2011 we do not reject H 0 There is no evidence to suggest that the increase in sales of brand 1 from a 1cent increase in the price of brand 3 is different from 300 cans Price changes in brands 2 and 3 will have the same effect on sales of brand 1 if B32134 Thus we test H0 33 34 against the alternative H1 33 7134 and we reject H0 if t gt 2011 or tlt 2011 The Istatistic is calculated as follows 3 I94 92990165113 t 0586 seb3Z94 123118 The standard error seb3 b4123118 can be calculated using computer software or from the coefficient covariance matrix as follows seb3b4JxETET2 4901763 8774127 2 X 741048 123118 Since 2011lt 0586 lt 2011 we fail to reject H0 There is no evidence to suggest that price changes in brands 2 and 3 have different effects on sales of brand 1 Chapter 5 Exercise Solutions Principles of Econometrics 46 161 Exercise 516div continued V In part ii we concluded that the effect of a price increase in brand 2 was not 300 cans In part iii we concluded that the effect of a price increase in brand 3 could be 300 cans And in part iv we concluded that the effect of increases in prices for brands 2 and 3 could be equal On the surface this may seem like a contradiction the results from parts ii and iii suggest the effects are different and the part iv result suggests they are the same To appreciate that the hypothesistest conclusions are indeed compatible it must be appreciated that we never conclude null hypotheses are true only that we have insufficient evidence to reject them Thus in part iii the effect of a price increase in brand 3 could be 300 cans but it also could be something else And in part iv it could be true that B3 B4 but it could also be true that they are not equal Suppose that prices are set at PR10 PR20 and PR30 and that average sales are SAL10 That is SAL10 13 BPR10 B3PR20 B4PR30 Strickly speaking we are looking at no change in average sales so we can ignore the error term Now suppose that all prices go up by 1 cent and that average sales do not change That is SAL10 B1BPR10 1B3 PR20 1B4 PR30 1 13 BPR10 B3PR20 B4PR30 B 133 134 For SAL10 to be the same in these two equations we require B2 133 134 0 Thus we test H0 3B2B3 B4 0 The Ivalue is calculated as follows bb3b4 47O8459299O1651131724 sebb3b4 123416 39 H13B2B3B4 750 Since 2011 lt 1724 lt 2011 we fail to reject H0 The results are compatible with the hypothesis that sales remain unchanged if all 3 prices go up by 1 cent For calculation of seb2 193 194 123416 we can use computer software or sebb3 b4JE ExE 2 2 2 6332635 4901763 8774127 2gtlt1642598 2gtlt 4815 2gtlt 741048 1234l6 Chapter 5 Exercise Solutions Principles of Econometrics 46 162 EXERCISE 517 8 b C The estimated linear regression from Exercise 516 is 19ALI 22963 470845PR1 92990PR2 165113PR3 R2 0443 se 9806 79578 70013 93670 A point estimate for expected sales when PR1 90 PR2 75 and PR3 75 is Q31 2296343 4708447 x 90 929900 X 75 1651 129gtlt 75 5488 Using 16 10975 48 2011 a 95 interval estimate is given by SAL1 1 se L1 5488 2011gtlt1385523 2841 2731 with se1SquotAL1 se191 90192 75193 75194 1385523 found using computer software The interval estimate contains a wide range of negative values which are clearly infeasible Sales cannot be negative The values PR190PR275 and PR375 are unfavorable ones for sales of brand 1 but they are nevertheless within the ranges of the sample data Thus the linear model is not a good one for forecasting The estimated loglinear regression is 1nsQT 1045595 0062176PR1 0014174PR2 002147210123 se 103046 0008362 0007357 0009843 A point estimate for expected logsales when PR1 90 PR2 75 and PR3 75 is 1ns2 1045595 0062176gtlt900014174gtlt750021472gtlt75 753356 Using 16 t097548 2010635 a 95 interval estimate for expected logsales is given by lnSAL1 1 selt 753356 2010635gtlt 0145589 724083782629 Converting this interval into one for sales using the exponential function we have exp724083 exp782629 1395 2506 Comparing this interval with the one obtained from the linear function we find that the two upper bounds of the intervals are of similar magnitude but the lower bound for the interval from the loglinear model is positive and much larger than that from the linear model Also the width of the interval from the loglinear model is much narrower suggesting more accurate estimation of expected sales When SAL is the dependent variable the coefficients show the change in number of cans sold from a 1cent change in price When lnSAL1 is the dependent variable by multiplying the coefficients by 100 we get the the percentage change in number of cans sold from a 1cent change in price Chapter 5 Exercise Solutions Principles of Econometrics 46 163 EXERCISE 518 The estimated regression is LEAVE 3482 2433PRBARR 08077PRBCONV R2 0601 se 0351 0320 01110 0333 SPRBPRIS 2006POLPC 0002187WCON 04700 43 6 0000834 All ve variables are expected to have negative effects on the crime rate We expect each of them to act as a deterrent to crime In the estimated equation the probability of an arrest and the probability of conviction have negative signs as expected and both coef cients are signi cantly less than zero with pvalues of 00000 On the other hand the coef cients of the other three variables the probability of a prison sentence the number of police and the Weekly Wage in construction have positive signs which is contrary to our expectations Of these three variables the coef cient of PRBARR is not signi cantly different from zero but the other two POLPC and WCON are signi cantly different from zero and have unexpected positive signs Thus it appears that the variables PRBARR and PRBCON V are the most important for crime deterrence The positive sign for the coef cient of POLPC may have been caused by endogeneity a concept considered in Chapter 10 In the context of this example high crime rates may be more likely to exist in counties with greater numbers of police because more police are employed to counter high crime rates It is less clear Why WCON should have a positive sign Perhaps construction companies have to pay higher Wages to attract Workers to counties with higher crime rates Chapter 5 Exercise Solutions Principles of Econometrics 46 164 EXERCISE 519 a b C d The estimated regression is 1nW AFE 11005 009031EDUC 0005776EXPER 0008941HRSWK se 01095 000608 0001275 0001581 R2 02197 The estimate 92 00903 implies that holding other variables constant an additional year of education increases Wage by 903 on average The estimate 93 0005776 implies that holding other variables constant an extra year of related Work experience increases Wage on average by 058 The estimate 94 0008941 implies that holding other variables constant Working an extra hour per Week increases Wage by 089 on average All coef cient estimates are signi cantly different from zero vvithpvalues of 00000 The null and alternative hypotheses are H0B2201 H1B2lt01 The critical value for a 5 signi cance level is t005996 1646 We reject H0 when 1 6 01 se6 lt 1646 The value of the Istatistic is t 6 01 00903101 se6 000608 1595 The corresponding pvalue is 00555 Since 1565 gt 1646 We do not reject H 0 There is not suf cient evidence to show that the return to another year of education is less than 10 A 90 con dence interval for 100 B4 is given by 100 x 94 i t095996 se100 x 194 08941 i 1646 x 01581 0634 1154 We estimate with 90 con dence that the Wage return to Working an extra hour per Week lies between 063 and 115 The estimates with quadratic terms and interaction term for ED UC and EXPER are given in the table on page 165 The coef cient estimates for variables EDUC2 EXPER EXPER2 and HRSWK are signi cantly different from zero at a 5 level of signi cance That for EDUC XEXPER is signi cant at a 10 level The coef cient of the remaining variable EDUC is not signi cantly different from zero Chapter 5 Exercise Solutions Principles of Econometrics 46 165 Exercise 519d continued 6 f g Estimates of wage equation with quadratic and interaction terms included Variable Coef cient Estimate Std Error IValue pValue C 3 09266081 03404072 2722 00066 EDUC B2 00490281 00366258 1339 01810 EDUC2 B3 00023649 00011048 2141 00325 EXPER B4 00527446 00097493 5410 00000 EXPER2 B5 00006287 00000888 7080 00000 EDUCgtlt EXPER B6 00009238 00005054 1828 00679 HRSWK B7 00066930 00015681 4268 00000 De ning the coefficients as they appear in the above table the marginal effects on lnWAGE are 8lnWAGE 2 EDUC EXPER MUG B2 B3 B jf1 13 2B5EXPER B6EDUC F01 Jill a 92 32b3 lOb6 0049028 32 X 00023649 10 X 00009238 0115 For Wendy 92 9 t 0049028 24 X 00023649 10 x 00009238 0097 We estimate that Jill has a greater marginal effect of education than Wendy As education increases the marginal effect of education increases There are increasing returns to education J ill s marginal effect of education will be greater than that of Wendy if B 32B3 10B6 gt B 2413 10B6 Which Will be true if and only if 32B3 gt 24B3 Now the inequality 32B3 gt 24B3 holds if 33 gt 0 and does not hold if 33 S 0 Thus a suitable test is H0 B3 S 0 against H1 33 gt 0 From the above table the pValue for this test is 00325 2 00163 Thus We reject H0 and conclude that J ill s marginal effect of education is greater than that of Wendy Chapter 5 Exercise Solutions Principles of Econometrics 46 166 Exercise 519 continued h i For Chris c 0052745 40 x 00006287 16gtlt 00009238 00128 For Dave 121 0 94 60195 16196 8EXPER 0052745 60 X 00006287 16 x 00009238 00002 We estimate that Chris has a greater marginal effect of exerience than Dave As experience increases the marginal effect of experience decreases There are decreasing returns to experience For someone with 16 years of education the marginal effect of experience is 8lnWAGE aEXPER 34 ZBSEXPER 16B6 Assuming 35 lt 0 the marginal effect of experience will be negative when EXPER gt 4342131636 EXPER 5 A point estimate for EXPER is 94 16196 00527446 16 gtlt 00009238 2b5 2 X 00006287 EXPER quot 3019 The delta method is required to get the standard error se1 xPER se l916b6j 15163 5 A 95 interval estimate is given by EXPER t0975p993seEXPER 30191 1962 gtlt15163 2722 3317 We estimate with 95 con dence that the number of years of experience after which the marginal return to experience becomes negative is between 272 and 332 years Chapter 5 Exercise Solutions Principles of Econometrics 46 167 EXERCISE 520 8 b C ADVERT 0 175 will be optimal if 83 2 x 17584 1 Thus the null and alternative hypotheses are H0 83 3584 1 and H4 83 3584 at 1 The tvalue is b335b4 1 12151235gtlt 276796 1 1 2149 seb3 35194 068085 and the corresponding pvalue is 00350 Thus We reject H0 and conclude that AD VERT0 175 is not optimal ADVERT 0 19 will be optimal if 83 2 x 1984 1 Thus the null and alternative hypotheses are H0 83 3884 1 and H4 83 3884 at 1 The tvalue is 193 38b4 1 12151238gtlt 276796 1 seb3 38b4 065419 0968 and the corresponding pvalue is 03365 Thus We fail to reject H0 and conclude that AD VERT0 19 could be optimal AD VERT0 23 will be optimal if 83 2 x 2384 1 Thus the null and alternative hypotheses are H0 83 4684 1 and H4 83 4684 at 1 The tvalue is 193 46b4 1 12151246gtlt 276796 1 1500 seb3 46194 105435 and the corresponding pvalue is 01381 Thus We fail to reject H0 and conclude that AD VERT0 23 could be optimal Note that We have found that both 19 and 23 could be optimal values for advertising expenditure A null hypothesis that used any value for AD VERT0 in between these two values would also not be rejected This outcome illustrates Why We never accept null hypotheses as the truth The best We can do is to say there is insuf cient evidence to conclude a null hypothesis is not true You might be surprised by the fact that 23 lies outside the 95 interval estimate for AD VERT0 found on page 195 of the text To appreciate how the difference can arise note that for part c We could also have set up the hypothesis H0 ADVERT0 121333 23 4 which is identical algebraically to H 0 83 468 4 1 In this case the tvalue is Chapter 5 Exercise Solutions Principles of Econometrics 46 168 Exercise 520 continued 1 b3j23 1 121512 j23 I 2194 2gtlt 276796 2219 1193 012872 S6 2194 The pValue is 00297 and H0 is rejected The different outcome arises because the delta method used to nd se 1 b32194 is a large sample approximation needed for nonlinear functions of the b s Whereas seb3 46194 involves getting the standard error for a linear function of the b s something We can do exactly Without a large sample approximation Chapter 5 Exercise Solutions Principles of Econometrics 46 169 EXERCISE 521 8 b C d The estimated equation is T7ME 199166 O36923DEPART 13353REDS 27548TRA1Ns se 12548 001553 01390 03038 Interpretations of each of the coefficients are 31 The estimated time it takes Bill to get to Work when he leaves Carnegie at 630AM and encounters no red lights and no trains is 1992 minutes B2 If Bill leaves later than 630AM his traveling time increases by 37 minutes for every 10 minutes that his departure time is later than 630AM assuming the number of red lights and trains are constant B3 Each red light increases traveling time by 134 minutes B4 Each train increases traveling time by 275 minutes The 95 con dence intervals for the coef cients are 8 6 z07se6 199166 1970 x 12548 1744 2239 8 6 t0975D227seZ92 036923 1970 x 001553 0339 0400 0 6 z07se6 13353 1970 x 01390 106161 04 I94 z0se6 27548 1970 X 03038 216 335 In the context of driving time these intervals are relatively narrow ones We have obtained precise estimates of each of the coefficients The hypotheses are H0 33 2 2 and H1 33 lt2 The critical value is t005227 1652 We reject H 0 when the calculated tvalue is less than 1652 This Ivalue is t133532 478 01390 Since 478 lt 1652 We reject H0 We conclude that the delay from each red light is less than 2 minutes The hypotheses are H0 13 4 3 and H1 13 4 3 The critical values are t005p227 1652 and 60959227 1652 We reject H0 when the calculated tvalue is such that tlt 1652 or t gt1652 This Ivalue is t 275483 O8O7 03038 Since 1652 lt 0807 lt 1652 We do not reject H0 The data are consistent with the hypothesis that each train delays Bill by 3 minutes Chapter 5 Exercise Solutions Principles of Ec0n0meIrz39cs 4e 170 Exercise 521 continued 6 f Delaying the departure time by 30 minutes increases travel time by 30132 Thus the null hypothesis is H0 230132 210 or H0 2132 213 and the alternative is H 2132 lt13 We reject H 0 if I S I005227 1652 Where the calculated Ivalue is t 036923 033333 001553 231 Since 231gt 1652 We do not reject H0 The data are consistent with the hypothesis that delaying departure time by 30 minutes increases travel time by at least 10 minutes If We assume that 132133 and B4 are all nonnegative then the minimum time it takes Bill to travel to Work is B Thus the hypotheses are H0 13 s 20 and H1 13 gt 20 We reject H 0 if I 2 I 1652 Where the calculated Ivalue is 095227 t199166 20 12548 0066 Since 0066 lt 1652 We do not reject H0 The data support the null hypothesis that the minimum travel time is less than or equal to 20 minutes It was necessary to assume that 132133 and B4 are all positive or zero otherwise increasing one of the other variables will lower the travel time and the hypothesis would need to be framed in terms of more coef cients than B Chapter 5 Exercise Solutions Principles of Econometrics 4e 171 EXERCISE 522 The estimated equation is T7ME 199166 O36923DEPART 13353REDS 27548TRIINS se 12548 001553 01390 O3038 a The delay from a train is B4 and the delay from a red light is B3 Thus the null and alternative hypotheses are H03B3B4 and H13B3 B4 The critical Values for the ttest are 20973337 1970 and 0039733227 1970 The rejection region is tlt 1970 or t gt1970 The calculated Value of the Itest statistic is 3293 294 3gtlt13353 27548 2404 se3293 294 05205 Where the standard error is computed from se3b3 159xvE153vE15 2x3xc zE 9 X 0019311 0092298 6 x 000081 05205 The null hypothesis is rejected because 2404 gt 1970 The pValue is 0017 The delay from a train is not equal to three times the delay from a red light b This test is similar to that in part a but it is a onetail test rather than a twotail test The hypotheses are H0B423B3 and H1B4lt3B3 The rejection region for the Itest is tlt t0033227 1652 and the calculated tValue is Z 294 3293 275483gtlt13353 24O4 se294 3293 05205 Since 2404 lt 1652 We reject H0 The delay from a train is less than three times the delay from a red light c The delay from 3 trains is 3134 The extra time gained by leaving 5 minutes earlier is 5 5133 Thus the hypotheses are H03B4 55B3 and H03B4gt55B3 The rejection region for the Itest is tgt t0933227 1652 where the tValue is calculated as 3 3194 519 5 3gtlt275485gtlt0369235 1546 se3294 5293 09174 Chapter 5 Exercise Solutions Principles of Econometrics 4e 172 Exercise 522c continued d and the standard error is computed from se3b4 51a9xvE1o425xvE1o 30xcT391aT 9 X 0092298 25 x 0000241 30 x 0000165 09174 Since 1546 lt 1652 We do not reject H0 at a 5 signi cance level Alternatively We do not reject H0 because the pvalue 00617 which is greater than 005 There is insuf cient evidence to conclude that leaving 5 minutes earlier is not enough time The expected time taken when the departure time is 715AM and no red lights or trains are encountered is B 45 B2 Thus the null and alternative hypotheses are H0 B145B2 45 and H1B145B2 gt45 The rejection region for the Itest is tgt t095D227 1652 where the tvalue is calculated as t b1 45 45 199166 45gtlt 036923 45 744 seb1 4519 11377 and the standard error is computed from seb145b2xTa1 1452 xvEb90xcT bT 1574617 2025 gtlt 000024121 90 x 000854061 11377 Since 744 lt 1652 We do not reject H 0 at a 5 signi cance level Alternatively We do not reject H 0 because the pvalue 1000 which is greater than 005 There is insufficient evidence to conclude that Bill will not get to the University before 800AM Chapter 5 Exercise Solutions Principles of Econometrics 4e 173 EXERCISE 523 8 b The estimated model is WE 39594 47024gtlt AGE 20222 x AGE2 2749gtlt AGE3 se 28153 27810 8901 0925 The Within sample predictions with age expressed in terms of years not units of 10 years are graphed in the following figure They are also given in a table on page 176 5 I O I V O o SCORE 39 39 I IZr 39 SCOREHAT 5 l O I O 0 O I I I AGEUNTS Figure xr523 Fitted line and observations To test the hypothesis that a quadratic function is adequate We test H 0 84 0 The Ivalue is 2972 with corresponding pvalue 00035 We therefore reject H0 and conclude that the quadratic function is not adequate For suitable values of B2 B3 and B4 the cubic function can decrease at an increasing rate then go past a point of in ection after which it decreases at a decreasing rate and then it can reach a minimum and increase These are characteristics Worth considering for a golfer That is the golfer improves at an increasing rate then at a decreasing rate and then declines in ability These characteristics are displayed in Figure Xr5 23 i Using the predictions in the table on page 176 We find the predicted score is lowest 629 at the age of 30 Thus We predict that Lion was at the peak of his career at age 30 Mathematically We can nd the value for AGE at which E SCORE is a minimum by considering the derivative dESCORE CZAGE 13 2B3lGE 3B4lGE2 Setting this derivative equal to zero and solving for age yields AGE ZB3 i 4 12B2B4 654 Chapter 5 Exercise Solutions Principles of Econometrics 4e 174 Exercise 523bi continued 0 Replacing B2 B3 B4 by their estimates 92 I93 94 gives the two solutions 2 x 202222 J4 gtlt 2022222 12 x 4702386 gtlt 274934 AGTE 3008 6gtlt274934 A 2 2 2222 4 2 2222212 4 2 2 4 4 AGE2 gtlt 0 gtlt 0 x 70 386x 793 21895 6gtlt274934 The second derivative 2 LQRE 2b3AGE 6b4AGE dAGE is positive when AGEAG77 and negative when AGEAGYTZ Thus the expected score is a minimum when AGE 3008 which is equivalent to 3008 years ii Lion s game is improving at an increasing rate between the ages of 20 and 25 where the differences between the predictions are increasing iii Lion s game is improving at a decreasing rate between the ages of 25 and 30 where the differences between the predictions are declining We can consider ii and iii mathematically in the following way When Lion s game is improving the first derivative will be negative It can be veri ed that the estimated first derivative will be negative for values of AGE between 2 and 3 If Lion s game is improving at an increasing rate the second derivative will also be negative it will be positive when Lion s game is improving at a decreasing rate Thus to nd the age at which Lion s improvement changes from an increasing rate to a decreasing rate we find that AGE for which the second derivative is zero namely AGE 219 2 gtlt 202222 2452 6194 6 X 274934 which is equivalent to 2452 years iv At the age of 20 Lion s predicted score is 44403 His predicted score then declines and rises again reaching 41 145 at age 36 Thus our estimates suggest that when he reaches the age of 36 Lion will play worse than he did at age 20 v At the age of 40 Lion s predicted score becomes positive implying that he can no longer score less than par At the age of 70 the predicted score relative to par for Lion Forrest is 24171 To break 100 it would need to be less than 28 100 72 Thus he will not be able to break 100 when he is 70 Chapter 5 Exercise Solutions Principles of Econometrics 4e 175 Exercise 523 continued Predicted scores at different ages age predicted scores 20 44403 21 45621 22 47420 23 49633 24 52097 25 54646 26 57116 27 59341 28 61157 29 62398 30 62900 31 62497 32 61025 33 58319 34 54213 35 48544 36 41145 37 31852 38 20500 39 06923 40 09042 41 27561 42 48799 43 72921 44 100092 Chapter 5 Exercise Solutions Principles of Econometrics 4e 176 EXERCISE 524 8 b C d The coefficient estimates standard errors Ivalues and pvalues are in the following table Dependent Variable lnPROD Coeff Std Error Ivalue pvalue C 15468 02557 60503 00000 lnAREA 03617 00640 56550 00000 lnLABOR 04328 00669 64718 00000 lnFERT 02095 00383 54750 00000 All estimates have elasticity interpretations For example a 1 increase in labor will lead to a 04328 increase in rice output A 1 increase in fertilizer will lead to a 02095 increase in rice output All pvalues are less than 00001 implying all estimates are signi cantly different from zero at conventional significance levels The null and alternative hypotheses are H0 B2 05 and H1 B2 7 05 The 1 critical values are t0995D348 259 and t0005D348 259 Thus the rejection region is t2 259 or rs 259 The calculated value of the test statistic is t 03617 05 0064 Since 259 lt 216 lt 259 We do not reject H0 The data are compatible with the hypothesis that the elasticity of production with respect to land is 05 216 A 95 interval estimate of the elasticity of production with respect to fertilizer is given by 94 it x seI94 02095 i 1967 x 003826 0134 0285 0975348 This relatively narrow interval implies the fertilizer elasticity has been precisely measured This hypothesis test is a test of H0 33 S 03 against H1 33 gt 03 The rejection region is t2 t095D348 1649 The calculated value of the test statistic is t 0433 03 0067 199 We reject H0 because 199 gt1649 There is evidence to conclude that the elasticity of production with respect to labor is greater than 03 Reversing the hypotheses and testing H0 33 203 against H1B3 lt03 leads to a rejection region of zs 1649 The calculated Ivalue is t 199 The null hypothesis is not rejected because 199 gt 1649 Chapter 5 Exercise Solutions Principles of Econometrics 4e 177 EXERCISE 525 a Taking logarithms yields the equation lnYB1B2lnKB3lnLB4lnEB5lnM e where B lnoL This form of the production function is linear in the coef cients B1 B2 B3 B4 and B5 and hence is suitable for least squares estimation b Coefficient estimates and their standard errors are given in the following table Estimated Standard coefficient error 32 005607 025927 33 022631 044269 34 004358 038989 35 066962 036106 c The estimated coefficients show the proportional change in output that results from proportional changes in K L E and M All these estimated coefficients have positive signs and lie between zero and one as is required for profit maximization to be realistic Furthermore they sum to approximately one indicating that the production function has constant returns to scale However from a statistical point of view all the estimated coefficients are not signi cantly different from zero the large standard errors suggest the estimates are not reliable CHAPTER 6 Exercise Solutions 178 Chapter 6 Exercise Solutions Principles of Econometrics 46 179 EXERCISE 61 3 b To compute R2 We need SSE and SST We are given SSE We can nd SST from the equation 2 M 0 1345222 N l N l Solving this equation for SST yields SST 6 x N 1 13452222 X 39 70575267 Thus SSE 1 979830 T 08612 SST 70575267 The Fstatistic for testing H 0 82 53 0 is de ned as F SST SSEK 1 70575267 9798302 SSEN K 979830403 ll475 At OL 005 the critical value is 17095D237 325 Since the calculated F is greater than the critical F We reject H 0 There is evidence from the data to suggest that 32 0 andor 53 0 Chapter 6 Exercise Solutions Principles of Econometrics 4e 180 EXERCISE 62 The model from Exercise 61 is y31B2x33ze The SSE from estimating this model is 979830 The model after augmenting with the squares and the cubes of predictions f2 and j3 is yB1B2xB3zy1A2 y2A3 e The SSE from estimating this model is 6965375 To use the RESET We set the null hypothesis H0 y y O The F Value for testing this hypothesis is F SSER SSEUJ 97983O 69653752 71175 SSEU N K 696537340 5 The critical Value for signi cance level oc005 is 170952353267 Since the calculated F is greater than the critical F We reject H0 and conclude that the model is misspeci ed Chapter 6 Exercise Solutions Principles of Econometrics 46 181 EXERCISE 63 3 b C d Let the total variation unexplained variation and explained variation be denoted by SST SSE and SSR respectively Then We have SSE Z N 1ltx82 20 3gtlt25l93 428281 Also R2 l l 09466 SST and hence the total variation is SST SSE 428281 1122 1 O9466 8020243 and the explained variation is SSR SST SSE 8020243 428281 7591962 A 95 con dence interval for 32 is 6 zseb 069914 2110 X x0048526 O234311639 A 95 con dence interval for 33 is 6 zseb 17769 2110 x 0037120 13704 21834 To test H0 32 2 1 against the alternative H1 B2 lt 1 We calculate t 92 B2 O69914 1 seb x0048526 At a 5 signi cance level We reject H0 if I lt I l3658 00 1740 Since 13658 gt 1740 We fail to reject H 0 There is insuf cient evidence to conclude 32 lt 1 To test H 0 52 33 0 against the alternative H1 32 0 andor 53 0 We calculate explained variationK 1 7591962 2 unexplained variation N K 428281 17 The critical value for a 5 level of signi cance is 11109572717 359 Since 151 gt 359 We reject H0 and conclude that the hypothesis B2 33 O is not compatible with the data Chapter 6 Exercise Solutions Principles of Econometrics 4e 182 Exercise 63 continued e The Istatistic for testing H 0 2B2 53 against the alternative H1 2B2 33 is 2b2 b3 se2b2 b3 For a 5 signi cance level We reject H0 if tltt0025D17 211 or tgtt097517211 The standard error is given by se2b2 b3J22xEEbQ 2x2x 4gtlt 0048526 oo3712 2gtlt 2gtlt 0031223 059675 The numerator of the Istatistic is 2192 3 2 x 069914 17769 O37862 leading to a Ivalue of t O37862 0634 059675 Since 211 lt 0634 lt 211 We do not reject H0 There is no evidence to suggest that 2B2 B3 Chapter 6 Exercise Solutions Principles of Econometrics 46 183 EXERCISE 64 a b The value of the 1 statistic for the signi cance tests is calculated from 19 1 quot 8609 We reject the null hypothesis H0 5k 0 if ltlgttC 2 The Ivalues for each of the coef cients are given in the following table Those which are signi cantly different from zero at an approximate 5 level are marked When EDUC and EDUC2 both appear in an equation their coef cients are not signi cantly different from zero with the exception of eqn B where EDUC2 is signi cant In addition the interaction term between EXPER and EDUC is not signi cant in eqn A Variable tvaluesa Eqn A Eqn B Eqn C Eqn D Eqn E C B 397 659 838 2382 942 EDUC B 126 084 104 1590 EDUC2 B 189 212 173 EXPER B4 458 628 517 611 EXPER2 B 538 531 490 513 EXPEREDUC B6 106 HRSWK B7 834 843 987 1011 871 a Note These 2 values were obtained from the computer output Some of them do not agree exactly with the 2 ratios obtained using the coef cients and standard errors in Table 64 Rounding error discrepancies arise because of rounding in the reporting of values in Table 64 Using the labeling of coef cients in the above table we see that the restriction imposed on eqn A that gives eqn B is 36 0 The Ftest value for testing H0 B6 0 against H1 2136 73 0 can be calculated from restricted and unrestricted sums of squared errors as follows F SSER SSEUJ 2226674 22241661 1120 SSEU N K 2224166993 The corresponding pvalue is 0290 The critical value at the 5 signi cance level is 7095L993 3851 Since the F value is less than the critical value or the pvalue is greater than 005 we fail to reject the null hypothesis and conclude that the interaction term ED UC gtlt EXPER is not signi cant in determining the wage The Ivalue for testing H0 2136 0 against H1 2136 7amp0 is 1058 At the 5 level its absolute value is less than the critical value t0975p993 1962 Thus the Itest gives the same result The two tests are equivalent because x1120 1058 and 3851 1962 Chapter 6 Exercise Solutions Principles of Econometrics 4e 184 Exercise 64 continued 0 d The restrictions imposed on eqn A that give eqn C are 34 0 B5 0 and B6 0 Thus We test H0B40 B5 0andB60 H1 At least one of B4 or B5 or B6 is nonzero The F value is calculated from F SSER SSEUJ 23383l7 2224l663 16988 ssE N K 2224166993 The corresponding pvalue is 00000 The critical value at a 5 signi cance level is 11195939993 2614 Since the F value is greater than the critical value or the pvalue is less than 005 We reject the null hypothesis and conclude at least one of B4 or B5 or B6 is I1OI1ZCI39O By performing this test We are asking Whether experience is relevant for determining the Wage level All three coef cients relate to variables that include EXPER The test outcome suggests that experience is indeed a relevant variable The restrictions imposed on eqn B that give eqn D are 32 0 and B3 0 Thus We test H0 2B 0 B3 0 H1 At least one of B2 or B3 is nonzero The F value is calculated from SSE SSE J 2805061 2226674 2 F R U 1291 ssE N K 2226674994 The corresponding pvalue is 00000 The critical value at a 5 signi cance level is 11195929994 3005 Since the F value is greater than the critical value or the pvalue is less than 005 We reject the null hypothesis and conclude at least one of B2 or B3 is nonzero By performing this test We are asking Whether education is relevant for determining the Wage level Both coef cients relate to variables that include EDUC The test outcome suggests that education is indeed a relevant variable Chapter 6 Exercise Solutions Principles of Econometrics 46 185 Exercise 64 continued 6 f g The restrictions imposed on eqn A that give eqn E are 33 0 and B6 0 Thus We test H0 B3 0 B6 0 H1 At least one of B3 or B6 is nonzero The F value is calculated from SSE SSE J 2236716 2224166 2 F R U 2802 SSEU N K 2224166993 The corresponding pvalue is 00612 The critical value at a 5 signi cance level is 11195929993 3005 Since the F value is less than the critical value or the pvalue is greater than 005 We do not reject the null hypothesis The assumption B3 0 B6 0 is compatible with the data By performing this test We are asking Whether it is suf cient to include education as a linear term or Whether We should also include it as a quadratic and or interaction term The test outcome suggests that including it as a linear term is adequate Eqn E is the preferred model All its estimated coef cients are signi cantly different from zero It includes both EXPER and EXPER2 which were shown to be jointly signi cant and it excludes the interaction term and EDUC2 Which jointly were not signi cant The AIC for eqn D AICD ln 51 2 K ln 2805061 8 1263 N N 1000 1000 The SC for eqn A SCA 1namp9E1W1nJJ 1455 N N 1000 1000 Eqn B is favored by the AIC criterion Eqn E is favored by the SC criterion Chapter 6 Exercise Solutions Principles of Econometrics 46 186 EXERCISE 65 3 b 0 Education and experience will have the same effects on lnWAGE if 32 B4 and B3 B5 The null and alternative hypotheses are H0 3B2 ZB4 and B3 B5 H1B2 B4 or B3 B5 orboth The restricted model assuming the null hypothesis is true is ln WAGE 13 13 EDUCEXPER 3 EDUC2 EXPER2B HRSWKe 1 4 5 6 The F Value is calculated from SSE SSE J 2541726 2226674 2 F R U 7032 SSEU N K 2226674994 The corresponding pValue is 00000 Also the critical Value at a 5 signi cance level is 7095p2p994 3005 Since the F Value is greater than the critical Value or the pValue is less than 005 We reject the null hypothesis and conclude that education and experience have different effects on lnWAGE Chapter 6 Exercise Solutions Principles of Econometrics 4e 187 EXERCISE 66 Consider for example the model y5152x53ze If We augment the model with the predictions 3 the model becomes yB1BxB3zyj2e However 3 bl bzx 932 is perfectly collinear with x and z This perfect collinearity means that leastsquares estimation of the augmented model will fail Chapter 6 Exercise Solutions Principles of Econometrics 46 188 EXERCISE 67 8 b 0 Least squares estimation of y 31 Bzx B3w e gives 93 04979 seb3 01174 and t O4979 01174 424 This result suggests that 93 is signi cantly different from zero and therefore w should be included in the model Additionally the RESET based on the equation y B1B2xe gives Fvalues of 1798 and 872 which are much higher than the 5 critical values of 170951D32415 and 14109532331 33O respectively Thus the model omitting w is inadequate Let I be the least squares estimator for 32 in the model that omits w The omitted variable bias is given by E093B2 my varx Now W gt 0 because rm gt O Thus the omitted variable bias will be positive This result is consistent with What We observe The estimated coefficient for 32 changes from 09985 to 41072 When w is omitted from the equation The high correlation between x and w suggests the existence of collinearity The observed outcomes that are likely to be a consequence of the collinearity are the sensitivity of the estimates to omitting w the large omitted variable bias and the insignificance of 92 when both variables are included in the equation Chapter 6 Exercise Solutions Principles of Econometrics 46 189 EXERCISE 68 There are a number of Ways in which the restrictions can be substituted into the model with each one resulting in a different restricted model We have chosen to substitute out 84 and 83 With this in mind We rewrite the restrictions as 83 1 3884 84 80 682 1983 36184 Substituting the first restriction into the second yields 84 80 68 191 3884 36184 Substituting this restriction and the rst one 83 1 3884 into the equation SALES 5 82PRICE 83LDVERT 84LDVERT2 e yields SALES 80 682 l91 3884 36184 82PRICE 1 3884IDVERT 84IDVERT2 e Rearranging this equation into a form suitable for estimation yields SALES ADVERT 781 3 PRICE 6 34 361 38IDVERT ADVERT2 e Chapter 6 Exercise Solutions Principles of Econometrics 46 190 EXERCISE 69 f The results of the tests in parts a to e appear in the following table Note that in all cases there is insufficient evidence to reject the null hypothesis at the 5 level of signi cance Part H0 F value df F6 5 pvalue a B2 0 0047 120 435 0831 b 32 33 0 0150 220 349 0862 c 32 34 0 0127 220 349 0881 d 32 33 34 0 0181 320 310 0908 e 32 33 34 35 1 0001 120 435 0980 The auxiliary R2 s and the explanatoryvariable correlations that are exhibited in the following table suggest a high degree of collinearity in the model Correlation with Variables Variable Auxiliary R2 lnL lnE lnM lnIQ 0969 0947 0984 0959 lnL 0973 0972 0986 lnE 0987 0983 lnM 0984 To examine the effect of collinearity on the reliability of estimation we examine the estimated equation with 1 values in parentheses 177 0035 0056lnK 0226lnL 0044lnE 0670lnM 1 08000216 0511 0112 1855 R2 0952 The very small tvalues for all variables except lnM our inability to reject any of the null hypotheses in parts a through e and the high R2 are indicative of high collinearity Collectively all the variables produce a model with a high level of explanation and a good predictive ability Furthermore our economic theory tells us that all the variables are important ones in a production function However we have not been able to estimate the effects of the individual explanatory variables with any reasonable degree of precision Chapter 6 Exercise Solutions Principles of Econometrics 46 191 EXERCISE 610 8 b C The restricted and unrestricted least squares estimates and their standard errors appear in the following table The two sets of estimates are similar except for the noticeable difference in sign for lnPL The positive restricted estimate 0187 is more in line with our a priori views about the crossprice elasticity with respect to liquor than the negative estimate 0583 Most standard errors for the restricted estimates are less than their counterparts for the unrestricted estimates supporting the theoretical result that restricted least squares estimates have lower variances CONST lnPB lnPL lnPR lnI Unrestricted 3243 1020 0583 0210 0923 3743 0239 0560 0080 0416 Restricted 4798 1299 0187 0167 0946 3714 0166 0284 0077 0427 The high auxiliary R25 and sample correlations between the explanatory variables that appear in the following table suggest that collinearity could be a problem The relatively large standard error and the wrong sign for lnPL are a likely consequence of this correlation Sample Correlation With Variable Auxiliary R2 lnPL lnPR 161 lnPB 0955 0967 0774 0971 lnPL 0955 0809 0971 lnPR 0694 0821 lnD 0964 We use the Ftest to test the restriction H0 02 53 34 55 0 against the alternative hypothesis H1 02 53 34 55 0 The value of the test statistic is F 250 with ap value of 0127 The critical value is 17109571725 424 Since 250 lt 424 we do not reject H 0 The evidence from the data is consistent with the notion that if prices and income go up in the same proportion demand will not change This idea is consistent with economic theory The Fvalue can be calculated from restricted and unrestricted sums of squared errors as follows F SSER SSEUJ 0098901 0089921 2 50 ssE N K 00899225 Chapter 6 Exercise Solutions Principles of Econometrics 4e 192 Exercise 610 continued de The results for parts d and e appear in the following table The tvalues used to construct the interval estimates are t097525 2060 for the unrestricted model and t0975D26 2056 for the restricted model The two 95 prediction intervals are 706 1279 and 596 1167 The effect of the nonsample restriction has been to increase both endpoints of the interval by approximately 10 litres 1I1Q Q 175 se f tc lower upper lower upper d Restricted 45541 014446 2056 4257 4851 706 1279 e Unrestricted 44239 016285 2060 4088 4759 596 1167 Chapter 6 Exercise Solutions Principles of Econometrics 46 193 EXERCISE 611 3 b The estimated CobbDouglas production function with standard errors in parentheses is lnQ 01290559lnL0488lnK R2 0688 se 0546 0816 0704 The magnitudes of the elasticities of production coefficients of lnL and lnK seem reasonable but their standard errors are very large implying the estimates are unreliable The sample correlation between lnL and lnK is 0986 It seems that labor and capital are used in a relatively fixed proportion leading to a collinearity problem which has produced the unreliable estimates After imposing constant returns to scale the estimated function is 175 0020 0398lnL 0602lnK se 0053 0559 0559 We note that the relative magnitude of the elasticities of production with respect to capital and labor has changed and the standard errors have declined However the standard errors are still relatively large implying that estimation is still imprecise Chapter 6 Exercise Solutions Principles of Econometrics 4e 194 EXERCISE 612 The RESET results for the loglog and the linear demand function are reported in the table below Test F value df 5 Critical F pvalue Loglog 1 term 00075 124 4260 09319 2 terms 03581 223 3422 07028 Linear 1 term 88377 124 4260 00066 2 terms 47618 223 3422 00186 Because the RESET returns pvalues less than 005 00066 and 00186 for one and two terms respectively at a 5 level of signi cance we conclude that the linear model is not an adequate functional form for the beer data On the other hand the loglog model appears to suit the data well with relatively high pvalues of 09319 and 07028 for one and two terms respectively Thus based on the RESET we conclude that the loglog model better re ects the demand for beer Chapter 6 Exercise Solutions Principles of Econometrics 46 195 EXERCISE 613 3 b C The estimated model is Y 06254 00302 00794 RG 00005111 03387 RF R2 06889 se 02582 00034 00817 00918 01654 7 2422 8785 0972 0005 2047 We expect the signs for 323354 and 35 to be all positive We expect the wheat yield to increase as technology improves and additional rainfall in each period should increase yield The signs of 22 and 25 are as expected but those for 23 and 24 are not However the I statistics for testing signi cance of I23 and 24 are very small indicating that both of them are not signi cantly different from zero Interval estimates for 33 and 34 would include positive ranges Thus although 23 and 24 are negative positive values of B3 and 54 are not in con ict with the data We want to test H0 83 54 33 35 against the alternative H1283 34 and B5 are not all equal The value of the F test statistic is F SSER SSEUJ 4863664 43035042 SSEU T K 430350448 5 27985 The corresponding pvalue is 0072 Also the critical value for a 5 signi cance level is 100957743 3214 Since the Fvalue is less than the critical value and the pvalue is greater than 005 we do not reject H 0 The data do not reject the notion that the response of yield is the same irrespective of whether the rain falls during germination development or owering The estimated model under the restriction is Y0651500314t00138RG00138RD00138RF se 0267900035 00567 00567 00567 1 2432 889 02443 02443 02443 With the restrictions imposed the signs of all the estimates are as expected However the response estimates for rainfall in all periods are not signi cantly different from zero One possibility for improving the model is the inclusion of quadratic effects of rainfall in each period That is the squared terms RG2RD2 and RF 2 could be included in the model These terms could capture a declining marginal effect of rainfall Chapter 6 Exercise Solutions Principles of Econometrics 46 196 EXERCISE 614 a The estimated model is 1 8123621933HE O1997HA R2 01655 se 41583 01801 00675 1 1954 12182 2958 An increase of one year of a husband s education leads to a 219 increase in wages Also older husbands earn 20 cents more on average per year of age other things equal b A RESET with one term yields F 9528 with pvalue 00021 and with two terms F 4788 and pvalue 00086 Both pvalues are smaller than a signi cance level of 005 leading us to conclude that the linear model suggested in part a is not adequate c The estimated equation is 17 455675 14580HE 01511HE2 28895HA 003O1H12 R2 01918 se 175436 11228 00458 07329 00081 1 2597 1298 3298 3943 3703 Wages are now quadratic functions of age and education The effects of changes in education and in age on wages are given by the partial derivatives 1 IT M 14580 03022HE M 28895 00602HA 8HE 8H1 The first of these two derivatives suggests that the wage rate declines with education up to an education level of HEM 1458 030522 48 years and then increases at an increasing rate A negative value of 8HW8HE for low values of HE is not realistic Only 7 of the 753 observations have education levels less than 48 so the estimated relationship might not be reliable in this region The derivative with respect to age suggests the wage rate increases with age but at a decreasing rate reaching a maximum at the age HAmaX 2889500602248 years d A RESET with one term yields F 0326 with pvalue 0568 and with two terms F 0882 and pvalue 0414 Both pvalues are much larger than a significance level of 005 Thus there is no evidence from the RESET test to suggest the model in part c is inadequate Chapter 6 Exercise Solutions Principles of Econometrics 4e 197 Exercise 614 continued e The estimated model is 17 370540 22076HE 01688HE2 26213HA se 170160 10914 00444 07101 1 2178 2023 3800 3691 00278H12 79379CIT R2 02443 00079 11012 3525 7208 The wage rate in large cities is on average 794 higher than it is outside those cities f The pvalue for 96 the coefficient associated with CIT is 00000 This suggests that 26 is signi cantly different from zero and CIT should be included in the equation Note that when CIT was excluded from the equation in part c its omission was not picked up by RESET The RESET test does not always pick up misspecifications g From part c we have ml 14580 03022HE M1 28895 00602HA 8HE 8Hl and from part f Ml 22076 03376HE Ml 26213 00556HA 8HE 8Hl Evaluating these expressions for HE 6 HE 15 HA 35 and HA 50 leads to the following results aHWaHE aHWaHA IIE6 IIEl5 IIA35 IIA50 Partc 0356 3076 0781 0123 Parte 0182 2855 0678 0156 The omitted variable bias from omission of CIT does not appear to be severe The remaining coefficients have similar signs and magnitudes for both parts c and e and the marginal effects presented in the above table are similar for both parts with the exception of 6HW8HE for HE 6 where the sign has changed The likely reason for the absence of strong omitted variable bias is the low correlations between CIT and the included variables HE and HA These correlations are given by corrCITHE 02333 and corrCITPL4 00676 Chapter 6 Exercise Solutions Principles of Econometrics 46 198 EXERCISE 615 3 b C The estimated model is E17275 111543106800LIVAREA 11334AGE 155524BEDS 701930BATHS se 65551 2731 80502 19700 290382 All coef cients are signi cantly different from zero with the exception of that for AGE The negative signs on BEDS and BAT HS might be puzzling Recall however that their coef cients measure the effects on price of adding more bedrooms or more bathrooms while keeping LITAREA constant Taking space from elsewhere to add bedrooms or bathrooms might reduce the price An estimate of the expected difference in prices is SPRICEAGH SPRICEAGE10 19 x 2 19 X 10 22668 11334 90672 Holding other variables constant on average the price of a 2year old house is 9067 dollars more than the price of a 10year old house A 95 interval is given by ltSPRCEAGE2 SPRCEAGE10 i t09751495 X Se8b3 90672 i 1962 X 8 X 80502 1173 1354 With 95 confidence we estimate that the average price difference between houses that are 2 and 10 years old lies between 1173 and 1354 This interval is a relatively narrow one but it is uninformative in the sense that the difference could be negative or positive Given that the living area is measured in hundreds of square feet the expected increase in price is estimated as LIVAREA22 L1VAMA20 92 X 22 92 X 20 10680 x 2 21360 Holding other variables constant we estimate that extending the living area by 200 square feet will increase the price of the house by 21360 The null and altemative hypotheses are H11 21 320000 and H1 21 gt20000 that we write alternatively as H11 13 310000 and H1 13 gt10000 Note In the rst printing of the text the wording of the question suggested the altemative hypothesis should be H1 13 210000 Since a null hypothesis should always include an equality we have change the hypotheses accordingly Chapter 6 Exercise Solutions Principles of Econometrics 46 199 Exercise 615c continued C1 6 At a 5 signi cance level we reject H 0 if t gt t0951495 1646 The calculated Ivalue is t 1 10000 2 2489 seb2 The corresponding pvalue is 00065 Since the tvalue is greater than the critical value of 1646 or because the pvalue is less than 005 we reject the null hypothesis and conclude that an increase in the price of the house is more than 20000 dollars Adding a bedroom of size 200 square feet will change the expected price by 2B2 B4 Thus an estimate of the price change is 2192 94 2x 10680 155524 5808 A 95 interval estimate of the price change is 219 194 i t09751495se2b 194 58076 i 1962 x 18699 2139 9476 With 95 con dence we estimate the price increase will be between 2139 and 9476 The standard error can be found from computer software or from se2bb4J22v5b42x2 4 gtlt 7461043 3880922 4 gtlt1706802 18699 A RESET with one term yields F 11780 with pvalue 00000 and with two terms F 73985 and pvalue 00000 Both pvalues are smaller than a signi cance level of 005 leading us to conclude that the linear model suggested in part a is not reasonable Chapter 6 Exercise Solutions Principles of Econometrics 46 200 EXERCISE 616 a The estimated regression is i1 1FE 797557 299465LIVAREA 83038lGE 119219BEDS 497106BATHS se 87443 77230 19778 19721 279737 16909LIVIREA2 142326AGE2 1613 33559 b To see if LIVAREA2 and AGE2 are relevant variables we test the hypotheses H0 36 0B7 0 H1B6 0andor 37 0 The restricted SSE is that from Exercise 6l5a SSER21111419gtlt1012 The unrestricted SSE is that from part a with LIVAREA2 and AGE 2 included The F value is calculated as follow g5ER SSEUJ 21111419x10 19434999x10 2 SSEUN K 19434999gtltl0121500 7 644 The corresponding pvalue is 00000 The critical value at a 5 signi cance level is 300 Since the F value is larger than the critical value or because the pvalue is smaller than 005 we reject the null hypothesis and conclude that including LIVAREA2 and AGE2 has improved the model c b An estimate of the expected difference in prices is SPRICEAGEZ2 13139e AG 29 gtlt2b7 gtlt22 b3 gtlt10b7 x102 8b3 9619 8gtlt 8303785 96gtlt1423261 52767 Holding other variables constant we estimate that the average price difference between a 2year old house and a 10year old house is 5277 Using se 8b3 96197 129195 from computer software a 95 interval is AGE2 AGE10 i t09751495 X Se8b3 96177 52767 i 1962gtlt 129195 27419 78115 With 95 confidence we estimate that the average price difference between houses that are 2 and 10 years old lies between 2742 and 7812 This interval is a relatively wide one but a more realistic one than that obtained using the speci cation in Exercise 615 Chapter 6 Exercise Solutions Principles of Econometrics 46 201 Exercise 616 continued 0 C c An estimate of the expected increase in price is d SPRICE6VAREA SPR1CE6VAREA0 2219 222 196 2019 202 196 219 84196 2 x 2994652 84 x 1690916 20193 Holding other variables constant we estimate that extending the living area by 200 square feet will increase the price of the house by 20193 The null and alternative hypotheses are H6 213 8406 s 20000 H 213 84136 gt 20000 Note In the rst printing of the text the wording of the question suggested the alternative hypothesis should be H1220 8406 220000 Since a null hypothesis should always include an equality we have change the hypotheses accordingly At a 5 signi cance level we reject H0 if tgtt60951493 1646 The calculated 1 value is t 219 84194 20000 19300 2 0361 se219 84194 53455 The corresponding pvalue is 03591 Since the tvalue is less than the critical value of 1646 or because the pvalue is greater than 005 we fail to reject the null hypothesis and conclude that there is not suf cient evidence to show that the increase in the price of the house will be more than 20000 dollars This test outcome is opposite to the conclusion reached in Exercise 615 It shows that test conclusions can be sensitive to the model speci cation Adding a bedroom of size 200 square feet will change the expected price by 208 202 86 8 BEDS 1 188 18286 B4BEDS 28 7606 86 Thus an estimate of the price change is 219 76196 194 2x 2994652 76x 1690916 1192192 69183 A 95 interval estimate of the price change is 219 76196 194 2 166676 493se219 76196 194 69183 2 1962 x 1802468 338210455 With 95 con dence the estimated price increase is between 3382 and 10455 Chapter 6 Exercise Solutions Principles of Econometrics 46 202 Exercise 616 continued 0 6 A RESET with one term yields F 990 with pValue 00017 with two terms it yields F 3256 with pValue 00000 Both pValues are smaller than a signi cance level of 005 leading us to conclude that the model with LIVAREA2 and AGE2 included is not adequate despite being an improvement over the model in Exercise 615 Chapter 6 Exercise Solutions Principles of Econometrics 46 203 EXERCISE 617 a The estimated regression is ln 107453 0082609LIIAREA 000050364LIVARE12 00079785lGE se 00505 0004477 000009629 00011799 000014110AGE 0075423BEDS 000002001 0011316 b The null and alternative hypotheses are H0 2B2 0B3 0 H1 2B 75 0 or 33 0 or both are nonzero The F value can be calculated as SSE SSE J l779768 694625 2 F R U 116696 SSEU N K 6946251494 The corresponding pvalue is 00000 Also the critical value is 1709521494 3002 Since the F value is greater than the critical value or because the pvalue is less than 005 We reject the null hypothesis and conclude that living area helps explain the selling price c The null and alternative hypotheses are H0 2B4 0B5 0 H1204 73 0 or B5 75 0 or both are nonzero The F value can be calculated as SSE SSE J 717908 694625 2 2504 SSEU N K 6946251494 The corresponding pvalue is 00000 The relevant critical value is 3002 Since the F value is greater than the critical value or because the pvalue is less than 005 we reject the null hypothesis and conclude that age of house helps explain the selling price Chapter 6 Exercise Solutions Principles of Econometrics 46 204 Exercise 617 continued d The predicted price using the natural predictor is EFIEE eXpl 074528 0082609LIIAREA 0000503644LIVARE12 00079785AGE 000014110AGE 0075423BEDS exp1074528 0082609 x 20 0000503644 x 202 00079785gtlt10000014110gtlt102 0075423gtlt3 147865 The predicted price using the corrected predictor is 1 9391 1E 31715175 eXp622 147865 gtlt eXp0046494 12 151343 e To nd a 95 prediction interval for SPRICE we rst nd such an interval for lnSPRICE h11S 1 z IFl Jrt0975494 sef 11904057 i 196155 gtlt 0215938 11480484 12327630 which yields the following prediction interval for SPRICE eXp11480484 eXp12327630 96808 225851 With 95 con dence we predict that the selling price of a house with the speci ed characteristics will lie between 96808 and 225851 The standard error of the forecast error for lnSPRICE se f 0215938 was found using computer software f Using the natural predictor the estimated price of Wanling s house after the extension is SEEIFE eXpl074528 0082609 x 22 0000503644 x 222 00079785 x 10 000014110 gtlt102 0075423 x 3 167204 Chapter 6 Exercise Solutions Principles of Econometrics 46 205 Exercise 617 continued 13 h Ignoring the error term the increase in price of the house is given by SPRCELIVAREA22 SPRCELIVAREA20 exp13 2213 22213 1013 10213 313 exp13 2013 20213 1013 10213 313 exp13 1013 100135 3136exp2213 48413 exp2013 40013 Let g13 exp13 1013 10013 313exp2213 48413 exp2013 40013 Then the null and alternative hypotheses are H g13 3 20000 H g13 gt 20000 Note In the rst printing of the text the wording of the question suggested the alternative hypothesis should be H g13 2 20000 Since a null hypothesis should always include an equality we have change the hypotheses accordingly At a 10 signi cance level we reject H0 if tgtt 1282 The calculated Ivalue 0901494 is gb 20000 661464 segb 580951 The corresponding pvalue is 08725 Since the tvalue is less than the critical value of 1282 or because the pvalue is greater than 005 we fail to reject the null hypothesis and conclude that there is not sufficient evidence to show that the increase in the price of the house will be more than 20000 1139 The standard error segb 580951 was found using computer software that utilized the delta method since gb is a nonlinear function A comparison of this test result to that from similar tests in Exercises 615 and 616 illustrates the sensitivity of test results to model speci cation In Exercises 615 and 616 the tvalues were 2489 and 0361 respectively A RESET with one tenn yields F 0968 with pvalue 03254 using two terms yields F 0495 with pvalue 06094 Both pvalues are larger than a significance level of 005 leading us to conclude that the model suggested in part a is a reasonable specification This conclusion is in contrast to those from similar tests in Exercises 615 and 616 It appears that the log specification is a better model than the linear and quadratic ones considered earlier Chapter 6 Exercise Solutions Principles of Econometrics 46 206 EXERCISE 618 a The estimated regression is l 103149 012680LIVAREA 00012677LIVARE12 0016916lGE se 02408 002125 00005148 0007373 000029391AGE2 0062799BEDS 0013812LIVAREA gtlt BEDS 000012498 0071877 0005 844 00002401 1LIVAREA2 gtlt BEDS 00026419AGE gtlt BEDS 000013163 00021610 0000045123AGE2 gtlt BEDS 000003 6997 The estimated relationships for 2 3 and 4 bedroom houses are as follows BEDS 2 BEDS 3 BEDS 4 C 104405 105033 105661 LIIAREA 0099175 0085363 0071550 LIVAREA2 000078751 000054740 000030730 AGE 001 16321 00089902 00063483 AGE2 000020366 000015854 000011342 b The null and altemative hypotheses are H0B6O9 B3 09 B9209 B1020 H1 At least one of B6 B8 B9 and B10 is nonzero The Value of F statistic is SSE SSE J 6924671 6902920 4 1174 SSEU N K 69029201490 The corresponding pValue is 03205 Also the critical Value is 17095741490 2378 Since the F Value is less than the critical Value or because the pValue is greater than 005 we do not reject the null hypothesis at the 5 level and conclude that 56 B8 39 and 510 are jointly not signi cantly different from zero This results suggests that the number of bedrooms effects the price only through its interaction with the living area Chapter 6 Exercise Solutions Principles of Econometrics 46 207 Exercise 618continued C The estimated regression is l 105518 0090116LIVAREA 000034819LIVAREA2 00080479lGE se 00479 0004903 000009426 00011784 000014243AGE2 00039957LVAREA gtlt BEDS 000001998 00005695 The estimated relationships for 2 3 and 4 bedroom houses are as follows BEDS 2 BEDS 3 BEDS 4 C 105518 105518 105518 LIVIRE1 0082125 0078129 0074133 LIVAREA2 000034819 000034819 000034819 AGE 00080479 00080479 00080479 AGE2 000014243 000014243 000014243 In this case only the eoef cient of LI IAREA changes with the number of bedrooms d The AIC and SC Values for the two models are Model in part a AIC 3065 SC 3030 Model in part C AIC 3068 SC 3046 Thus the model in part C is favored by both the AIC and the SC Chapter 6 Exercise Solutions Principles of Econometrics 46 208 EXERCISE 619 81 b The predicted time it takes Bill to reach the University if he leaves at 700AM is g 1 gtlt30b3 gtlt6b4 gtltl 1991660369227gtlt30133532gtlt 6275483 41760 Using suitable computer software the standard error of the forecast error can be calculated as se f 40704 Thus a 95 interval estimate for the travel time is 77173 zse f 41760 1970 x 40704 3374 4978 Rounding this interval to 34 50 minutes a 95 interval estimate for Bill s arrival time is from 734AM to 750AM The predicted time it takes Bill to reach the University if he leaves at 745AM is TI EZb1bgtlt75b3gtlt10b4gtlt4 1991660369227gtlt75l33532gtlt10275483gtlt4 71981 Using suitable computer software the standard error of the forecast error can be calculated as se f 42396 Thus a 95 interval estimate for the travel time is r7ME zse f 719814 1970 x 42396 63638033 Rounding this interval to 64 80 minutes a 95 interval estimate for Bill s arrival time is from 849AM to 905AM Chapter 6 Exercise Solutions Principles of Ec0n0meIrics 46 209 EXERCISE 620 8 b C We are testing the null hypothesis H 0 B2 33 against the alternative H1 82 at B3 The test can be performed with an F or a I statistic Using an Ftest We reject H0 when F gt 70957L348 Where I7095pL348 3868 The calculated Fvalue is 0342 Thus We do not reject H0 because 0342 lt 3868 Also the pvalue of the test is 0559 con rming non rejection of H0 The hypothesis that the land and labor elasticities are equal cannot be rejected at a 5 signi cance level Using a Itest We reject H0 when IgtI0975348 or IltI0025348 Where I0975348 1967 and I0025348 1967 The calculated Ivalue is 192 Z93 036174 043285 I 0585 seb2 193 012165 In this case H 0 is not rejected because 1967 lt 0585 lt 1967 The pvalue of the test is 0559 The hypothesis that the land and labor elasticities are equal cannot be rejected at a 5 signi cance level We are testing the null hypothesis against the alternative H0 5B3B4 1 H1 82 33 34 1 using a 10 signi cance level The test can be performed with an F or a I statistic Using an F test We reject H 0 when F gt 70907L348 272 The calculated Fvalue is 00295 Thus We do not reject H0 because 00295 lt 272 Also the pvalue of the test is 0864 con rming nonrej ection of H 0 The hypothesis of constant returns to scale cannot be rejected at a 10 signi cance level Using a Itest We reject H 0 when I gt I or I lt I Where I 1649 and 095348 1649 The calculated Ivalue is 005348 095348 t005348 Z t 192 93 194 1 036174 043285 0209502 1 0172 seb2 93 194 0023797 In this case H0 is not rejected because 1649 lt 0172 lt 1649 The pvalue of the test is 0864 The hypothesis of constant returns to scale is not rejected at a 10 signi cance level In this case the null and alternative hypotheses are B2B320 H B2B3 0 andor 0 B2B3B41 1 B2B3B4 1 We reject H0 when F gt I70952348 302 The calculated Fvalue is 0183 Thus We do not reject H0 because 0183lt 302 Also the pvalue of the test is 0833 con rming nonrejection of H 0 The joint null hypothesis of constant returns to scale and equality of land and labor elasticities cannot be rejected at a 5 signi cance level Chapter 6 Exercise Solutions Principles of Econometrics 46 Exercise 620 continued d The restricted model for part a where 32 33 is 210 lnPROD 31 32 lnAREA gtlt LABOR 34 lnFERT e The restricted model for part b where 32 133 134 1 is lnPROD 3 13 1nAREA 1 B B4lnLABOR 134 lnFERT e or In PR OD LABOR PROD FERT jB1B21n AREA LABOR jB1B21n The restricted model for part c where 32 33 and 32 133 134 1 is AREA gtlt LABOR Wt FER T2 FERT LABOR The estimates and standard errors from these restricted models and the unrestricted model are given in the following table Because the unrestricted estimates almost satisfy the restriction 32 133 134 1 imposing this restriction changes the unrestricted estimates and their standard errors very little Imposing the restriction B2 33 has an impact changing the estimates for both 32 and B3 and reducing their standard errors considerably Adding 32 133 134 1 to this restriction reduces the standard errors even further leaving the coef cient estimates essentially unchanged B2 B3 U d 1 nrestr1cte B2 B3 32 133 134 B2 B3 B4 1 C 15468 14095 15381 14030 O2557 01011 02502 00913 1nIREA 03617 03964 03595 03941 00640 00241 00625 00188 1nLIBOR 04328 03964 04299 03941 00669 00241 00646 00188 1nFERT 02095 02109 02106 02118 003 83 003 82 00377 00376 SSE 405654 406052 405688 406079 Chapter 6 Exercise Solutions Principles of Econometrics 4e EXERCISE 621 i ii iii The results are summarized in the following table Full FERT LABOR AREA model omitted omitted omitted 92 AREA 03617 04567 06633 93 LABOR 04328 05689 07084 94 FERT 02095 03015 02682 RESET1pvalue 05688 08771 04281 01140 RESET2 pvalue 02761 04598 05721 00083 211 With FERT omitted the elasticity for AREA changes from 03617 to 04567 and the elasticity for LABOR changes from 04328 to 05689 The RESET Fvalues pvalues for 1 and 2 extra terms are 0024 0877 and 0779 0460 respectively Omitting FERT appears to bias the other elasticities upwards but the omitted variable is not picked up by the RESET With LABOR omitted the elasticity for AREA changes from 03617 to 06633 and the elasticity for FERT changes from 02095 to 03015 The RESET Fvalues pvalues for 1 and 2 extra terms are 0629 0428 and 0559 0572 respectively Omitting LABOR also appears to bias the other elasticities upwards but again the omitted variable is not picked up by the RESET With AREA omitted the elasticity for FERT changes from 02095 to 02682 and the elasticity for LABOR changes from 04328 to 07084 The RESET Fvalues pvalues for 1 and 2 extra terms are 2511 0114 and 4863 0008 respectively Omitting AREA appears to bias the other elasticities upwards particularly that for LABOR In this case the omitted variable misspeci cation has been picked up by the RESET with two extra terms Chapter 6 Exercise Solutions Principles of Econometrics 4e 212 EXERCISE 622 8 b The model for parts a and b is PIZZA 31 B2lGE B3INC0ME 34 AGEgtlt INCOME e The hypotheses are H0 32 34 0 and H1 32 0 andor 34 7 0 The value of the F statistic under the assumption that H0 is true is F SSER SSEUJ 819286 580609 2 7 40 39 SSEUNK 39 58060936 The 5 critical value for 2 36 degrees of freedom is F6 326 and the pvalue of the test is 0002 Thus we reject H0 and conclude that age does affect pizza expenditure The marginal propensity to spend on pizza is given by 5E PIZZA B3 P 5INCOME Point estimates standard errors and 95 interval estimates for this quantity for different ages are given in the following table Age Point Standard Con dence Interval Est1mate Error Lower Upper 20 4515 1520 1432 7598 30 3283 0905 1448 4731 40 2050 0465 1107 2993 50 0818 0710 0622 2258 55 0202 0991 1808 2212 The interval estimates were calculated using tc 60975736 20281 The point estimates for the marginal propensity to spend on pizza decline as age increases as we would expect However the con dence intervals are relatively wide indicating that our information on the marginal propensities is not very reliable Indeed all the con dence intervals do overlap Chapter 6 Exercise Solutions Principles of Econometrics 46 Exercise 622 continued 213 c This model is given by PIZZA B1B2AGE B3INC B4IGE gtlt INC BSAGE2 gtlt INC e The marginal effect of income is now given by 8EPIZZA T AGE AGE 2 8INCOME B3B B5 If this marginal effect is to increase with age up to a point and then decline then 35 lt 0 The results are given in the table below The sign of the estimated coefficient 95 00042 did not agree with our expectation but with a pvalue of 0401 it was not significantly different from zero Variable Coefficient Std Error Ivalue pvalue C 10972 13557 0809 04238 AGE 20383 35419 0575 05687 INCOME 140962 88399 1595 01198 AGEgtlt INCOME 04704 04139 1 136 02635 AGE2gtlt INCOME 0004205 0004948 0850 04012 d The marginal propensity to spend on pizza in this case is given by LWIZZA 5 5 AGE 5 AGE2 8INCOME 3 4 5 Point estimates standard errors and 95 interval estimates for this quantity for different ages are given in the following table Age Point Standard Con dence Interval Est1mate Error Lower Upper 20 6371 2664 0963 11779 30 3769 1074 1589 5949 40 2009 0469 1056 2962 50 1090 0781 0496 2675 55 0945 1325 1744 3634 The interval estimates were calculated using I6 60975335 20301 Chapter 6 Exercise Solutions Principles of Econometrics 4e 214 Exercise 622d continued 6 0 As in part b the point estimates for the marginal propensity to spend on pizza decline as age increases There is no lifecycle effect where the marginal propensity increases up to a point and then declines Again the con dence intervals are relatively wide indicating that our information on the marginal propensities is not very reliable The range of ages in the sample is 1855 The quadratic function reaches a minimum at AcEm J 5593 2gtlt 0004205 Thus for the range of ages in the sample the relevant section of the quadratic function is that where the marginal propensity to spend on pizza is declining It is decreasing at a decreasing rate The pvalues for separate t tests of signi cance for the coef cients of AGE AGEgtlt INCOME and AGE2 gtlt INCOME are 05687 02635 and 04012 respectively Thus each of these coef cients is not signi cantly different from zero To perform a joint test of the signi cance of all three coef cients we set up the hypotheses H0 3B2 ZB4 B5 0 H1 At least one of B2 B4 and B5 is nonzero The Fvalue is calculated as follows SSE SSE J 8192858 5688692 3 F R U 5136 SSEUN K 568869235 The corresponding pvalue is 00048 Also the critical value at the 5 signi cance level is 1119573735 2874 Since the Fvalue is greater than the critical value or because the p value is less than 005 we reject the null hypothesis and conclude at least one of B2 B4 and B5 is nonzero This result suggests that age is indeed an important variable for explaining pizza consumption despite the fact each of the three coef cients was insigni cant when considered separately Collinearity is the likely reason for this outcome We investigate it in part f Two ways to check for collinearity are i to examine the simple correlations between each pair of variables in the regression and ii to examine the R2 values from auxiliary regressions where each explanatory variable is regressed on all other explanatory variables in the equation In the tables below there are 3 simple correlations greater than 094 for the regression in part c and 5 when AGE3 gtltINC is included The number of auxiliary regressions with R2s greater than 099 is 3 for the regression in part c and 4 when AGE3 gtltINC is included Thus collinearity is potentially a problem Examining the estimates and their standard errors con rms this fact In both cases there are no Ivalues which are greater than 2 and hence no coef cients are signi cantly different from zero None of the coef cients are reliably estimated In general including squared and cubed variables can lead to collinearity if there is inadequate variation in a variable Chapter 6 Exercise Solutions Principles of Econometrics 4e 215 Exercise 622f continued Simple Correlations AGE AGEgtltINC AGE2 gtltINC AGE3 gtltINC INC 04685 09812 09436 08975 AGE 05862 06504 06887 AGEgtlt INC 09893 09636 AGE2 gtlt INC 09921 R2 Values from Auxiliary Regressions LHS Variable R2 in part C R2 in part f INC 099796 099983 AGE 068400 082598 AGEgtlt INC 099956 099999 AGE2 gtlt INC 099859 099999 AGE3 gtlt INC 099994 Chapter 6 Exercise Solutions Principles of Econometrics 4e 216 EXERCISE 623 Coef cient estimates standard errors tvalues and pvalues obtained for this model are given in the following table Variable Coefficient Std Error Ivalue pvalue C 113408 033982 3337 00009 EDUC 0046418 0036936 1257 02092 EDUC2 00026509 00011122 2383 00173 EXPER 0057775 0009761 5919 00000 EXPER2 00006946 00000882 7875 00000 EDUCgtltEXPER 00010256 00005092 2014 00442 a The percentage change in WAGE from an extra year of education is calculated from 8lnWA GE 8ED UC The percentage change in WAGE from an extra year of experience is calculated from 8lnWA GE 8EXPER gtlt100 3 23EDUC B6EXPERgtlt100 X100 34 ZBSEXPER B6EDUCgtlt100 1 When EDUC 10 and EXPER 10 W0 8ED UC 0046418 2x 00026509 x 10 00010256 x 10 008918 Se z 8EDUC 2 0014685 Using t0975994 19624 a 95 interval estimate for 100gtlt 8lnWAGE8EDUC is 8918 i19624 x 14685 604 1180 ii When EDUC 10 and EXPER 10 W0 0057775 2 x 00006946 x 10 00010256 x 10 003363 8EXPER Se z 8EXPER j 0004262 A 95 interval estimate for 100gtlt 8lnWAGE8EXPER is 3363 i19624gtlt 04262 253 420 Chapter 6 Exercise Solutions Principles of Econometrics 4e 217 Exercise 623a continued iii When ED UC 20 and EXPER 20 am UC 0046418 2 x 00026509gtlt 20 00010256 x 20 013194 0014807 8EDUC Using t0975994 19624 a 95 interval estimate for 100gtlt 8lnWAGE8EDUC is 13194i19624gtlt14807 10291610 iv When ED UC 20 and EXPER 20 075 0057775 2 x 00006946 x 20 00010256 x 20 0009478 8EXPER se 4 0003324 8EXPER A 95 interval estimate for 100gtlt 8lnWAGE8EXPER is 09478 i 19624 gtlt 03324 030 160 These results suggest that the return to an extra year of education is greater than the return to an extra year of experience Furthermore the return to education increases with further education Whereas the return to experience decreases with further experience b The null and alternative hypotheses are H0 2B2 20B3 10B6 01 and 34 20B5 10B6 004 H1202 20B3 10B6 at 01 andor 34 20B5 10B6 73 004 Using econometric software the Fvalue and the pvalue are computed as 1118 and 03273 respectively Since the pvalue is larger than 005 We do not reject the null hypothesis We conclude that for 10 years of experience and 10 years of education the data are compatible with the hypothesis that the return to an extra year of education is 10 and the return to an extra year of experience is 4 c The null and alternative hypotheses are H0 2B2 40B3 20B6 012 and 34 40B5 2006 001 H1 2B2 40B3 20B6 75 012 andor 34 40B5 20B6 75 001 Using econometric software the Fvalue and the pvalue are computed as 0335 and 07154 respectively Since the pvalue is larger than 005 We do not reject the null hypothesis We conclude that for 20 years of experience and 20 years of education the data are compatible with the hypothesis that the return to an extra year of education is 12 and the return to an extra year of experience is 1 Chapter 6 Exercise Solutions Principles of Econometrics 4e 218 Exercise 623 continued d 6 The null and alternative hypotheses are H6 2133 20133 10136 01 134 20133 10136 004 133 40133 20136 012 and 134 40133 20136 001 H1 At lease one of the above equations does not hold Using econometric software the Fvalue and the pvalue are computed as 07695 and 05452 respectively Since the pvalue is larger than 005 We do not reject the null hypothesis We conclude that the data are compatible with the hypothesis that for 10 years of experience and 10 years of education the return to an extra year of education is 10 and the return to an extra year of experience is 4 and for 20 years of experience and 20 years of education the return to an extra year of education is 12 and the return to an extra year of experience is 1 From the joint hypotheses in part c We have 132 012 40133 20136 134 001 40133 20136 Substituting these expressions into the original equation yields lnWAGE133 012 40133 20136 EDUC 133EDUC2 001 40133 20136 EXPER 133EXPER2 136 EDUC gtlt EXPER e lnWAGE 012EDUC 001EXPER 13 133 EDUC2 40EDUC 133 EXPER2 40EXPER 136 ED UC gtlt EXPER 20ED UC 20EXPER 6 Estimating the above model and substituting into the restrictions to nd estimates for 132 and 134 yields Variable Coefficient Std Error Ivalue pvalue C 104522 024712 4230 00000 EDUC 0063536 0021249 2990 00029 EDUC2 00018907 00004659 4058 00001 EXPER 00570590 00083390 6842 00000 EXPER2 00006974 00000879 7934 00000 EDUCgtlt EXPER 00009582 00002697 3553 00004 To con rm the result in c We can manually calculate the F value SSER SSEU J 2531464 25297592 O 335 SSEU N K 2529759994 Chapter 6 Exercise Solutions Principles of Econometrics 4e 219 EXERCISE 624 8 b C C1 B2 is the direct price elasticity of sales of brand 1 with respect to changes in the price of brand 1 The expected sign of B2 is negative Holding other variables constant a 1 increase in price per can of brand 1 changes brand 1 s sales by 32 B3 is the cross price elasticity of sales of brand 1 with respect to changes in the price of brand 2 The expected sign of B3 is positive Holding other variables constant a 1 increase in price per can of brand 2 changes brand 1 s sales by 33 B4 is the cross price elasticity of sales of brand 1 with respect to changes in the price of brand 3 The expected sign of B4 is positive Holding other variables constant a 1 increase in price per can of brand 3 changes brand 1 s sales by 34 The regression results are Variable Coef cient Std Error Ivalue pvalue C 78894 02514 31376 00000 lnAPR 46246 06383 7245 00000 lnAPR2 09904 05338 1855 00697 lnAPR3 16871 07460 2262 00283 All coef cients have the expected signs and all are signi cantly different from zero at a 5 level of significance with the exception of 93 which is the coef cient of lnAPR2 If 32 33 34 0 We can rewrite the regression equation as lnSAL B 33 B4lnAPR 33 lnAPR2 34 lnAPR3 e 3 3 lnAPR2 lnAPR 34 lnAPR3 lnAPR e B1B3ln B4lne APR APR APR APR B1B31niF22jB41nAPR3ie APR APR OL1OL2lI1 4 OL3lI1 j e APR2 APR3 Where We have set ocl 31 OL2 33 and OL3 34 The null and alternative hypotheses are H03B2B3B4O H1B2B3B4 O Using econometric software We find the Fvalue for this hypothesis to be 3841 with corresponding pvalue of 00588 Since 00588 lt 010 We reject H0 at a 10 significance level The data do not support the marketing manager s claim Chapter 6 Exercise Solutions Principles of Econometrics 4e 220 Exercise 624 continued e The estimated regression is 83567 13177ln APR 27001ln APR APR2 APR3 se 00820 05215 05534 a2 1318 implies that holding other variables constant a 1 increase in the price ratio of brand 1 to brand 2 tuna decreases the sales of brand 1 tuna by 1318 a3 270 implies that holding other variables constant a 1 increase in the price ratio of brand 1 to brand 3 tuna decreases the sales of brand 1 tuna by 270 The Ivalues for a2 and a3 are 2527 and 4879 respectively indicating that both these estimated coef cients are signi cantly different from zero The Ftest result in part d can be con rmed using the sums of squared errors from the restricted and unrestricted models SSER SSEUJ 166956 154585 1 F 39 SSEUNK 15458548 3398 f Both estimated models in parts b and e suggest that brand 3 is the stronger competitor to brand 1 because 94 gt 193 and a3 lt a2 A price change in brand 3 has a greater effect on sales of brand 1 than a price change in brand 2 g To con rm that brand 3 is the stronger competitor We set up an alternative hypothesis that brand 3 is a stronger competitor than brand 2 For the model in part a H0 2B4 S133 against H1B4 gt 33 The value of the Istatistic is 4 I93 16871 09904 seb4 b3 09507 20733 The corresponding pvalue is 0234 Also the critical value at a 5 level of signi cance is t09548 1677 Since tlt1677 We do not reject the null hypothesis At a 5 level of signi cance the evidence is not suf ciently strong to con rm that brand 3 is a stronger competitor than brand 2 The standard error can be calculated as follows seb4b3JvEWv Q2x 0556547 O284986 2gtlt0031110 09507 Chapter 6 Exercise Solutions Principles of Econometrics 46 221 Exercise 624g continued For the model in part c H0 ZOL3 20 against H12Ot3 ltOt2 The value of the Istatistic is a a 27001 13177 t sea3 a 09092 1520 The corresponding pvalue is 00674 Also the critical value at a 5 level of signi cance is t00549 1677 Since tgt 1677 We do not reject the null hypothesis At a 5 level of signi cance the evidence is not suf ciently strong to con rm that brand 3 is a stronger competitor than brand 2 The opposite conclusion is reached if We use a 10 signi cance level In this case t010749 1299 gt 1520 and the evidence is suf ciently strong to con nn that brand 3 is a stronger competitor The standard error can be calculated as follows sea3 a2Jv Qva c52x 0306213 0271995 2gtlt 0124246 09092 Chapter 6 Exercise Solutions Principles of Econometrics 46 222 EXERCISE 625 81 To appreciate the relationship between the 3 equations we begin by rewriting the rst equation as follows SAL B1BAPR1 B3APR2 B4APR3 e PR1 PR2 PR3 l31l32Wjl33Wjl34Wj OL1 ot2PRI OL3PR2 oc4PR3 e where OL1B1 oz B2100 013 B3100 OL4 B4100 Thus the coefficients ofPRI PR2 and PR3 in the second equation will be 100 times smaller than the coefficients of APRI APR2 and APR3 in the first equation The intercept coef cient remains unchanged For the third equation we write SAL OL1 oc2PR1 oc3PR2 oc4PR3 e 1000 gtlt SALES on oc2PR1 oc3PR2 OL4PR3 e SALES iiPR1 3 PR2 4 PR3 6 1000 1000 1000 1000 1000 yl y2PR1 y3PR2 y4PR3 e where y1 O611000 y oz1000 13 0131000 14 OL41000 Thus all coefficients in the third equation including the intercept will be 1000 times smaller than those in the second equation The estimated regressions are STE 2296343 4708447APRI 929900PR2 1651129PR3 SALI 22963 43 4708447PRI 9299OOPR2 1651 12910123 SALES 22963 O47084PRI 009299PR2 O16511PR3 The relationships between the estimated coefficients in these three equations agree with the conclusions we reached by algebraically manipulating the equations Chapter 6 Exercise Solutions Principles of Econometrics 4e 223 Exercise 625 continued b To obtain the relationship between the coef cients of the rst two equations we write 1nSAL1 31 BZAPRI B3lPR2 B4lPR3 e PR1 PR2 PR3 l31l32Wjl33Wjl34Wj 011 oL2PRI OL3PR2 oc4PR3 e where 011 31 OL2 32100 013 B3100 OL4 B4100 The relationships between the coef cients are the same as those in part a The coefficients of PR PR2 and PR3 in the second equation will be 100 times smaller than the coef cients of APR APR2 and APR3 in the rst equation The intercept coef cient remains unchanged To obtain the third equation from the second we write lnSALI 011 oc2PRI oc3PR2 oc4PR3 e lnSALES X 1000 011 OL2PR1 OL3PR2 oc4PR3 e lnSALES 011 lnl 000 OL2PR1 OL3PR2 oc4PR3 e y1 y2PRI y3PR2 y4PR3 e where V OL1 ln1000 y 012 v3 063 v4 014 The coefficients of the third equation are identical to those of the second equation with the exception of the intercept which differs by the amount ln1000 6907755 The estimated regressions are 1ns39F 1045595 62176lPR1 14174APR2 21472IPR3 1ns F 1045595 0062176PR1 0014174PR2 0021472PR3 1111F13 354819 0062176PR1 0014174PR2 0021472PR3 These estimates agree with the relationships established algebraically Note that al ln1000 1045595 690776 354819 371 Chapter 6 Exercise Solutions Principles of Econometrics 4e 224 Exercise 625 continued c To obtain the relationship between the coef cients of the rst two equations We Write lnSAL1 31 132 lnAPR1 B3 lnAPR2 34 lnAPR3 e B1 132 ln O1jB3 lnfT12fj B4 ln2O3j e 13 132 lnPR1B3 lnPR2B4 lnPR3 B2 133 34 ln100 e OL1 OL2 lnPR1 OL3lI1PR2 OL4lI1PR3 e where ocl 13 32 133 B4ln100 on B2 oc3 33 OL4 B4 Thus all coef cients of the second equation are identical to those of the first equation with the exception of the intercept which differs by the amount 32 133 B4ln100 To obtain the third equation from the second We Write lnSALI on OL2lI1PR1 013 lnPR2 014 lnPR3 e lnSALES X 1000 011 012 lnPRI OL3 lnPR2 oc4 lnPR3 e lnSALES 011 ln1000 012 lnPRI OL3 lnPR2 014 lnPR3 e y1 y lnPRI y3 lnPR2 y4 lnPR3 e where y OL1 lI1l00O y 012 13 013 14 014 This result is the same as that obtained in part b The coef cients of the third equation are identical to those of the second equation with the exception of the intercept which differs by the amount ln1000 6907755 In all three cases only the intercept changes This is a general result Changing the units of measurement of Variables in a loglog model does not change the Values of the coefficients which are elasticities The estimated regressions are 1ns39F 788938 46246lnAPR1 09904lnAPR2 16871lnAPR3 1ns7F 1685591 46246lnPR1 09904lnPR2 16871lnPR3 ms1F13 994816 46246lnPR1 09904lnPR216871lnPR3 As expected the elasticity estimates are the same in all three equations To reconcile the three different intercepts first note that al ln1000 16855913 6907755 9948158 171 Comparing equations 1 and 2 We note that b1 b 93 194 ln100 7889381 4624576 0990379 1687140 gtlt 460517 1685591 a1 CHAPTER 7 Exercise Solutions 225 Chapter 7 Exercise Solutions Principles of Econometrics 46 226 EXERCISE 71 8 b C When a GPA is increased by one unit and other variables are held constant We estimate that the average starting salary is estimated to increase by the amount 1643 1 466 and the coef cient is signi cant at on 0001 Students who take econometrics are estimated to have a starting salary which is 5033 higher on average than the starting salary of those who did not take econometrics t 1103 and the coef cient is signi cant at or 0001 The intercept suggests the starting salary for someone with a zero GPA and who did not take econometrics is 24200 However this figure is likely to be unreliable since there would be no one with a zero GPA The R2 074 implies 74 of the variation of starting salary is explained by GPA and MET RICS A suitably modi ed equation is SAL B BZGPA B3METRICS B4FEMALE e The parameter B4 is an intercept indicator variable that captures the effect of gender on starting salary all else held constant B B2GPA B3METRICS ifFEMALE 0 B B BZGPA B3METRICS ifFEMALE 1 ESAL To see if the value of econometrics is the same for men and Women We change the model to SAL B B2GPA B3METRICS B4FEMALE BSMETRICS gtlt FEMALE e The parameter B4 is an intercept indicator variable that captures the effect of gender on starting salary all else held constant The parameter B5 is a slopeindicator variable that captures any change in the slope for females relative to males 3 BZGPA B3METRICS ifFEMALE 0 B1B4B2GPA B B METRICS ifFEMALE 1 EsAL Chapter 7 Exercise Solutions Principles of Econometrics 4e 227 EXERCISE 72 3 b Considering each of the coef cients in turn we have the following interpretations Intercept At the beginning of the time period over which observations were taken on a day which is not Friday Saturday or a holiday and a day which has neither a full moon nor a half moon the estimated average number of emergency room cases was 9369 T We estimate that the average number of emergency room cases has been increasing by 00338 per day other factors held constant This time trend has a Ivalue of 306 and a p value 0003 lt 001 HOLIDAY The average number of emergency room cases is estimated to go up by 1386 on holidays holding all else constant The holiday effect is signi cant at the 005 level of signi cance FRI and SA T The average number of emergency room cases is estimated to go up by 69 and 106 on Fridays and Saturdays respectively holding all else constant These estimated coefficients are both signi cant at the 001 level F ULLMOON The average number of emergency room cases is estimated to go up by 245 on days when there is a full moon all else constant However a null hypothesis stating that a full moon has no in uence on the number of emergency room cases would not be rejected at any reasonable level of signi cance NE WMOON The average number of emergency room cases is estimated to go up by 64 on days when there is a new moon all else held constant However a null hypothesis stating that a new moon has no in uence on the number of emergency room cases would not be rejected at the usual 10 level or smaller Therefore hospitals should expect more calls on holidays Fridays and Saturdays and also should expect a steady increase over time There are very small changes in the remaining coef cients and their standard errors when F ULLMOON and NE WMOON are omitted The equation goodnessof t statistic decreases slightly as expected when variables are omitted Based on these casual observations the consequences of omitting F ULLMOON and NE WMOON are negligible Chapter 7 Exercise Solutions Principles of Econometrics 46 228 Exercise 72 continued c The null and alternative hypotheses are H0 B6 B7 0 H1206 or B7 is nonzero The test statistic is F SSER SSEU2 SSEU 229 7 Where SSER 2742419 is the sum of squared errors from the estimated equation with F ULLMOON and NE WMOON omitted and SSEU 2710882 is the sum of squared errors from the estimated equation with these Variables included The calculated Value of the F statistic is 129 The 05 critical Value is 1410957 27 222 3307 and corresponding pValue is 0277 Thus We do not reject the null hypothesis that new and full moons have no impact on the number of emergency room cases Chapter 7 Exercise Solutions Principles of Econometrics 46 229 EXERCISE 73 3 b C d 6 The estimated coefficient of the price of alcohol suggests that if the price of pure alcohol goes up by 1 per liter the average number of days out of 31 that alcohol is consumed will fall by 0045 The price elasticity at the means is given by Q3 2478 p q 0045 x 0320 3 49 We estimate that a 1 increase in the price of alcohol will reduce the number of days of alcohol usage by 032 holding all else fixed To compute this elasticity We need g for married black males in the 2130 age range It is given by g 4099 0045 x 2478 0000057 x 12425 1637 0807 0035 0580 397713 Thus the price elasticity is g 2478 51 61 We estimate that a 1 increase in the price of alcohol will reduce the number of days of alcohol usage by a married black male by 028 holding all else fixed The coef cient of income suggests that a 1 increase in income will increase the average number of days on which alcohol is consumed by 0000057 If income was measured in terms of thousanddollar units which would be a sensible thing to do the estimated coef cient would change to 0057 The magnitude of the estimated effect is small but based on the t statistic the estimate is statistically significant at the on 001 level The effect of GENDER suggests that on average males consume alcohol on 1637 more days than Women On average married people consume alcohol on 0807 less days than single people Those in the 1220 age range consume alcohol on 1531 less days than those who are over 30 Those in the 2130 age range consume alcohol on 0035 more days than those who are over 30 This last estimate is not significantly different from zero however Thus two age ranges instead of three 1220 and an omitted category of more than 20 are likely to be adequate Black and Hispanic individuals consume alcohol on 05 80 and 0564 less days respectively than individuals from other races Keeping in mind that the critical Ivalue is 1960 all coef cients are signi cantly different from zero except that for the indicator variable for the 2130 age range Chapter 7 Exercise Solutions Principles of Econometrics 46 230 EXERCISE 74 8 b C The estimated coef cient for SQF T suggests that an additional square foot of oor space will increase the price of the house by 7279 holding all other factors xed The positive sign is as expected and the estimated coef cient is signi cantly different from zero The estimated coefficient for AGE implies the house price is 179 less for each year the house is older The negative sign implies older houses cost less other things being equal The coefficient is signi cantly different from zero The estimated coef cients for the indicator variables are all negative and they become increasingly negative as We move from D92 to D96 Thus house prices have been steadily declining in Stockton over the period 199196 holding constant both the size and age of the house Including a indicator variable for 1991 would have introduced exact collinearity unless the intercept was omitted Exact collinearity would cause least squares estimation to fail The collinearity arises between the dummy variables and the constant term because the sum of the dummy variables equals 1 the value of the constant term Chapter 7 Exercise Solutions Principles of Econometrics 46 231 EXERCISE 75 8 b The model to estimate is lnPRICEB1 5UT0mv BsQET ySQFT gtlt UTOIWV B3AGE BZPOOL 53FPLACE e The estimated equation with standard errors in parentheses is 1W9Y21cT 44638 03334UTOWN 003596SQFT 0003428SQFT gtlt UTOWN se 0026400359 oo01o4 ooo1414 0000904AGE 00l899POOL 0006556FPLACE R2 08619 0000218 000510 0004140 In the loglinear functional form ln y B Bzx e We have Q1 dry 32 or d yB2dx y Thus a 1 unit change in x leads to approximately a percentage change in 2 equal to 1OOgtlt32 In this case aPR1CE 1 asQET PRICE aPR1CE 1 a1 GE PRICE B yUTOIWV B3 Using this result for the coef cients of SQF T and AGE We estimate that an additional 100 square feet of oor space is estimated to increase price by 36 for a house not in University town and 325 for a house in University town holding all else fixed A house which is a year older is estimated to sell for 00904 less holding all else constant The estimated coef cients of UT OWN AGE and the slopeindicator variable SQF T UT OWN are significantly different from zero at the 5 level of signi cance Chapter 7 Exercise Solutions Principles of Econometrics 46 232 Exercise 75 continued 0 d 6 Using the results in Section 731 lnPRICEp00 1nPR1CEWx100 5 x 100 z APRICE An approximation of the percentage change in price due to the presence of a pool is 190 Using the results in Section 732 nopool PRICEP00 PRICEW0 x 100 e52 1 X 100 PRICE The exact percentage change in price due to the presence of a pool is estimated to be 192 From Section 731 lnPRICE replace 1nPR1CE0 pm X 100 5 X 100 z APRICE An approximation of the percentage change in price due to the presence of a fireplace is 066 From Section 732 PR replace PRn0 replace gtlt100e53 1gtlt100 PRICEn0 replace The exact percentage change in price due to the presence of a fireplace is also 066 In this case the difference in logprices is given by 1 SQFT25 l SQFT25 03334UTOWN 0003428gtlt 25gtlt UTOVWV 03334 0003428 X 25 02477 and the percentage change in price attributable to being near the university for a 2500 squarefeet home is e 2 77 1gtlt100 2811 Chapter 7 Exercise Solutions Principles of Econometrics 46 233 EXERCISE 76 3 b C The estimated equation is 1J3E 89848 37463APR1 11495118122 1288APR3 0423 7DISP se O6464 O5765 04486 O6053 01052 14313DISPAD R2 08428 O1562 The estimates of B2 B3 and B4 are all significant and have the expected signs The sign of B2 is negative While the signs of the other two coef cients are positive These signs imply that Brands 2 and 3 are substitutes for Brand 1 If the price of Brand 1 rises then sales of Brand 1 will fall but a price rise for Brand 2 or 3 will increase sales of Brand 1 Furthermore with the loglinear function the coef cients are interpreted as proportional changes in quantity from a 1unit change in price For example holding all else xed a oneunit increase in the price of Brand 1 is estimated to lead to a 375 decline in sales a oneunit increase in the price of Brand 2 is estimated to lead to a 115 increase in sales These percentages are large because prices are measured in dollar units If We Wish to consider a 1 cent change in price a change more realistic than a 1dollar change then the percentages 375 and 115 become 375 and 115 respectively There are three situations that are of interest i No display and no advertisement SALI1 expB1 BZAPRI B3lPR2 B4IPR3 Q ii A display but no advertisement SAL exp8 BZAPRI B3lPR2 B4IPR3 8 Qexp 5 iii A display and an advertisement SAL expB1 BZAPRI B3IPR2 B4IPR3 06 QexpB6 The estimated percentage increase in sales from a display but no advertisement is SAL12 L11 SALI1 The estimated percentage increase in sales from a display and an advertisement is x 100 TQeXpZ 5 quot Q x 100 e 4237 lgtlt 100 528 SAL13 SAL11 gtlt100 TQeXpb639Q gtltlOO 614313 1gtlt100 318 SAL11 Q The signs and relative magnitudes of I95 and 96 lead to results consistent with economic logic A display increases sales a display and an advertisement increase sales by an even larger amount Chapter 7 Exercise Solutions Principles of Econometrics 46 234 Exercise 76 continued d The results of these tests appear in the table below s Ho 333 g f W i 35 0 t 403 46 201 Reject H0 ii 36 0 t 917 46 201 Reject H0 iii 35 36 0 F 420 246 320 Reject H0 iv 36 3 B5 t 686 46 168 Reject H0 e The test results suggest that both a store display and a newspaper advertisement will increase sales and that both forms of advertising will increase sales by more than a store display by itself Chapter 7 Exercise Solutions Principles of Econometrics 46 235 EXERCISE 77 3 The estimated regression is 06885 000162LVR 00593REF O4816INSUR 00344RATE se 02115 000078 00238 002364 00086 0023 SAMOUNT 000044CREDIT 001262TERM O1283ARM 00127 000020 000354 00319 The explanatory variables with the positive signs are LVR RATE AMOUNT and ARM and these signs are as expected because LVR A higher ratio of the amount of loan to the value of the property will lead to a higher probability of delinquency The higher the ratio the less the borrower has put as a down payment perhaps indicating nancial stress RATE A higher interest rate of the mortgage will result in a higher probability of delinquency Lenders target higher risk borrowers and charge a higher rate as a risk premium AMOUNT As the amount of mortgage gets larger holding all else xed it is more likely that the borrower will face delinquency ARM With the adjustable rate the interest rate may rise above what the borrower is able to repay which leads to a higher probability of delinquency On the other hand the explanatory variables with the negative signs are REF INSUR CREDIT and TERM and these signs are also as expected because REF Re nancing the loan is usually done to make repayments easier to manage which has a negative impacts upon the loan delinquency INSUR Taking insurance is an indication that borrower is more reliable reducing the probability of delinquency However the magnitude of the estimated coef cient is unreasonably large CREDIT A borrower with a higher credit rate will have a lower probability of delinquency After all the higher credit rate is earned by borrowers who have a good track record of paying pack loans and debts in a timely fashion TERM As the term of the mortgage gets longer it is less likely that the borrower faces delinquency A longer term means lower monthly payments which are easier to t into a budget Chapter 7 Exercise Solutions Principles of Econometrics 46 236 Exercise 77 continued b C d The coef cient estimate for INS UR is 04816 If a borrower is insured We estimate that the probability of their having a delinquent payment falls by 04816 This is an extremely large effect We Wonder if INSUR has captured some omitted explanatory variable and thus has an in ated coefficient The estimated coef cient of CREDIT is 000044 suggesting an increase in the credit score by one point decreases the probability of missing at least three payments by 000044 Thus if CREDIT increases by 50 points the estimated probability of delinquency decreases by 0022 The predicted value of DELINQ UEN T at the 1000th observation is O6885 000162 X 882 00593 x 1 04816gtlt 0 00344 x 7650 00238gtlt2910 000044gtlt624 001262gtlt3001283gtlt1 05785 the exact calculation using software This suggests that the probability that the last observation an individual misses at least three payments is 05785 Despite the fact that this predicted probability is greater than 05 the 1000th borrower was not in fact delinquent Out of the 1000 observations the predicted values of 135 observations were less than zero but none of the observations had its predicted value greater than 1 This is problematic because We cannot have a negative probability Chapter 7 Exercise Solutions Principles of Econometrics 46 237 EXERCISE 78 a The line plots of variables against TIME The reference lines are a TIME 17 and TIME 23 8 3 0 5 10 15 20 25 b month 1march 2003 25march 2005 1 0 percentage motel occupancy percentage competitors occuparlcy The graphical evidence suggests that the damaged motel had the higher occupancy rate before the repair period During the repair period the damaged motel and the competitor had similar occupancy rates The average occupancy rates during the nonrepair period MOTELo 7935 COMPo 6249 The difference is MOTEL1 COMP1 7935 6249 1686 The average occupancy rates during the repair period M1 6611 1 6337 The difference is MOT EL1 COMP1 6611 6337 274 The estimate of lost occupancy is computed as follows MOTEL 6337 1686 8023 MOTEL M0TEL1 8023 6611 1412 Therefore the estimated amount of revenue lost is based on lost revenue from 1412 X 100 1412 rooms 215gtlt1412gtlt5661 17183539 Chapter 7 Exercise Solutions Principles of Econometrics 46 238 Exercise 78 continued 0 d In the gure below We observe Points A and B D and E Point C is inferred under the common trend assumption Occupancy damaged B ntfIIni IVLOSS Zjzjjj D A E H A competitor Before During repair repair Point A COMPo 6249 B MOTELO 7935 C MOTEL 8023 is an estimate of What occupancy rate would have been in the absence of the damage D MOTEL1 6611 E COMP1 6337 Loss 8023 6611 1412 The estimated model is A 1207561 06326COMP PCT 1069659RELPRICE 181441REPAIR se 45735 0194 49378 4192 92 O6326 This implies that holding other variables constant on average a one percentage increase in the competitor s occupancy rate is estimated to increase the damaged motel s occupancy rate by 063 percent The signi cance test suggests that the estimate is signi cant both at the one and ve percent levels 93 10697 Holding other variables constant on average a one unit increase in the relative price of the damaged motel and its competitor decreases the occupancy rate of the damaged motel by 107 A oneunit change is a change in relative price of 100 which is too large to be relevant If the relative price increases by only 10 the estimated reduction in the occupancy rate is 107 The signi cance test suggests that the estimate is signi cant at the ve percent level but not at the one percent level Chapter 7 Exercise Solutions Principles of Econometrics 46 239 Exercise 78d continued 94 18144 Holding other variables constant on average the occupancy rate of the damaged motel when it is under repair is 1814 percent less than when it is not under repair The signi cance test suggests that the estimate is signi cant at the one percent level e The expected revenue loss is computed as 215gtlt 5661gtlt 1814 2208344 This calculation is based on the 1814 decline in the occupancy of a 100 unit motel or 1814 rooms per day The simple estimate of the revenue loss calculated in part b is 17183539 The 95 interval estimate for the estimated loss is calculated as follows 215 x 5661 X 2 i t097521 se215 gtlt 5661gtlt I94 22078466 i 208 x 5102504 326947 114722 The simple estimate from part b is Within this interval estimate f The RESET value with three terms is 054 with a p value of 06601 There is no evidence from this RESET to suggest the model in part c is misspeci ed g The graph below depicts the least square residuals over time Residuals AM zU0 V O 10 5 10 month 1march 2003 25march 2005 The residuals trend down a little over time Testing for serial correlation is delayed until Chapter 9 Chapter 7 Exercise Solutions Principles of Econometrics 4e EXERCISE 79 a The estimated average test scores are regular sized class with no aide 9180429 regular sized class with aide 9183568 small class 9319419 240 From the above gures the average scores are higher with the small class than the regular class The effect of having a teacher aide is negligible The results of the estimated models for parts bg are summarized in the following table Exercise 79 1 2 3 4 5 b C d e 9 C 918043 904721 923250 931755 918272 1641 2228 3121 3940 4357 SMALL 13899 14006 13896 13980 15746 2409 2395 2294 2302 2096 AIDE 0314 0601 0698 1002 1782 2310 2306 2209 2217 2025 TCHEXPER 1469 1114 1156 0720 0167 0161 0166 0167 BOY 14045 14008 12121 1846 1843 1662 FREELUNCH 34117 32532 34481 2064 2126 2011 WHITEASIAN 11837 16233 25315 2211 2780 3510 TCIWHITE 7668 1538 2842 3284 TCIZMASTERS 3560 2621 2019 2184 SCIURBAN 5 750 2858 SCIRURAL 7 OO6 2559 N 5786 5766 5766 5766 5766 adj Rsq 0007 0020 0101 0104 0280 BIC 66169500 65884807 65407272 65418626 64062970 SSE 31232400314 30777099287 28203498965 28089837947 22271314955 Standard errors in parentheses pltO10 plt005 plt001 Chapter 7 Exercise Solutions Principles of Econometrics 4e 241 Exercise 79 continued b C d 6 f g The estimated regression results are in column 1 of the Table above The coef cient of SMALL is the difference between the average of the scores in the regular sized classes 91836 and the average of the scores in small classes 93194 That is 92 9319419 9180429 13899 Similarly the coef cient of AIDE is the difference between the average score in classes with an aide and regular classes The Itest of signi cance of B3 is 93 0314 0136 seb3 2310 The critical value at the 5 signi cance level is 196 We cannot conclude that there is a signi cant difference between test scores in a regular class and a class with an aide The estimated regression after including T CHEXPER is in column 2 above The t statistic for its signi cance is 878 and we reject the null hypothesis that a teacher s experience has no effect on total test scores The inclusion of this variable has a small impact on the coef cient of SMALL and the coef cient of AIDE has gone from positive to negative However AIDE s coef cient is not signi cantly different from zero and this change is of negligible magnitude so the sign change is not important The estimated regression after including BOY FREELUNCH and WHITEASIAN is in column 3 of the Table above The inclusion of these variables has little impact on the coef cients of SMALL and AIDE The variables themselves are statistically signi cant at the on 001 level of signi cance We estimate that holding all of the factors constant boys score 1405 points lower than girls that students receiving a free lunch score 3411 points lower than those who do not and that white andor Asian students score 1184 points higher The estimated regression after including the additional four variables is in column 4 of the Table above The regression result suggests that T CH WHITE SCHRURAL and SCHURBAN are signi cant at the 5 level and T CHMASTERS is signi cant at the 10 level The inclusion of these variables has only a very small and negligible effect on the estimated coef cients of AIDE and SMALL The results found in parts c d and e suggest that while some additional variables were found to have a signi cant impact on total scores the estimated advantage of being in small classes and the insigni cance of the presence of a teacher aide is unaffected The fact that the estimates of the key coef cients did not change is support for the randomization of student assignments to the different class sizes The addition or deletion of uncorrelated factors does not affect the estimated effect of the key variables The estimated model including school xed effects is in column 5 of the Table above The estimates of the school effects themselves are suppressed We nd that inclusion of the school effects increases the estimates of the bene ts of small classes and the presence of a teacher aide although the latter effect is still insigni cant statistically The F test of the joint signi cance of the school indicators is 1915 The 5 Fcritical value for 78 numerator and 5679 denominator degrees of freedom is 128 thus we reject the null hypothesis that all the school effects are zero and conclude that at least some are not zero The variables SCHURBAN and SCHRURAL drop out of this model because they are exactly collinear with the included 78 indicator variables Chapter 7 Exercise Solutions Principles of Econometrics 4e 242 EXERCISE 710 8 b The table below displays the sample means of LNPRICE and LN UNIT S as well as the percentage differences using only the data for 2000 IZLA W 1 IZLAW 0 Pct Diff LNPRICE 128914 122851 6063 LNUNITS 99950 95449 4501 The approximate percentage differences in the price and units for cities with and without the law are 6063 and 4501 respectively using the approximation 100lny1 ln0 E Ay Since the average price is higher under the law it suggests that the law failed to achieve its objective of making housing more affordable There are however more units available in cities with the law The sample means of LNPRICE and LN UNITS before the year 1990 are IZLAW 1 IZLAW 0 LNPRICE 123383 120646 LNUNITS 98992 94176 The diagrams for LN UNIT S and LNPRICE are on the following page For LN UNIT S the diagram follows The line segment AD represents what happens in cities without the law The line segment BC represents what happened in cities with the law The line segment BE represents what would have happened to LN UNIT S in the absence of the law assuming that the common trend assumption is valid We see that in the absence of the law we estimate that the number of units would have actually been larger For LNPRICE the line segment AD represents what happens in cities without the law The line segment BC represents what happened in cities with the law The line segment BE represents what would have happened to LNPRICE in the absence of the law assuming that the common trend assumption is valid We see that in the absence of the law we estimate that the average price of units would have been smaller Chapter 7 Exercise Solutions Principles of Econometrics 416 243 Exercise 710b continued Exercise 710 LNUNITS E100255 1 J B98992 quot39 e9 C99950 04451 D939M49 A94176 Before After Exercise 710 LNPRICE C128914 E125588 06063 A120646 B efore A her Chapter 7 Exercise Solutions Principles of Econometrics 48 Exercise 710 continued The regressions for parts ee are summarized in the following tables Discussion follows Exercise 710 LNPRICE 1610 0398 0150 0029 0182 0059 0238 0083 1300 0038 5518 0790 0147 0032 0058 0050 0194 0070 0589 0074 1940 0126 0515 0296 0039 0011 1 C C 12065 0033 D 0221 0046 IZLAW 0274 0100 IZLAWD 0333 0141 LMEDHHINC EDUCATTAIN PROPPOVERTY LPOP N 622 ad R sq 0 109 BIC 1026124 SSE 181891 Standard errors in parentheses plt010 plt005 88 plt001 244 Exercise 710 continued 0 Chapter 7 Exercise Solutions Principles of Econometrics 46 Exercise 710 LNUNITS IZLAW IZLAWD LMEDHHINC EDUCATTAIN PROPPOVERTY LPOP 9418 0057 0127 0081 0482 0176 0031 0249 9005 1199 0116 0087 0479 0176 0034 0249 0039 0114 14023 0404 0077 0016 0007 0026 0027 0036 0764 0038 1343 0064 2620 0151 0998 0006 0021 1732039 565846 Standard errors in parentheses plt010 plt005 plt001 245 See column 1 in each of the above tables The treatment effect is estimated by the coef cient of DgtltIZLAW which is represented in the table as IZLlWD LNPRICE equation We estimate that the result of the law was to increase prices by about 333 395 using the exact calculation of Chapter 732 and this effect is statistically signi cant at the 5 level t 235 For the LNUNITS equation the effect carries a negative sign which is opposite the direction We expect but the coef cient is not statistically different from zero so that its sign should not be interpreted t 013 To summarize these models suggest that the policy effect is to increase prices but not to increase the number of housing units contrary to the intention of the policy In the Chapter 7 Exercise Solutions Principles of Econometrics 46 246 Exercise 710 continued d 6 See column 2 in each of the above tables In the LNPRICE equation holding other variables constant We estimate that a one percent increase in the households median income increases the price of housing by 13 percent This effect is statistically signi cant with a t value of 3436 The inclusion of this control variable reduces the magnitude of the estimated treatment effect to approximately 283 The treatment effect is statistically signi cant at the 1 level with a t value of 287 In the LN UNIT S equation the median income variable is not statistically signi cant and the estimate of the treatment effect remains statistically insigni cant See column 3 in the above tables In the LNPRICE equation the effects are ED UCA T T AIN Holding all else constant We estimate that an increase in the proportion of the population holding a college degree will increase prices by a statistically signi cant amount A oneunit change of a proportion is very large If there is an increase in the proportion by 001 or 1 the estimated increase in house prices is 194 PROPOVERT Y Holding all else constant an increase in the proportion of the population in poverty decreases house prices by a statistically signi cant amount If the poverty rate increases by 001 or 1 We estimate that house prices will fall by 0515 LPOP Holding all else constant an increase in the population of 1 is estimated to increase house prices by 0039 percent This effect is statistically signi cant at the 1 level The addition of these additional controls slightly reduces the estimated treatment effect to 194 The treatment remains statistically signi cant at the 1 level In the LN UNITS equation the effects are ED UCA T T AIN We estimate that holding other factors xed an increase in the percent of the population with a college degree increases by 001 or 1 the number of housing units will increase by 1343 percent which is signi cant at the 1 level PROPOVERT Y We estimate that holding other factors constant an increase of the proportion living in poverty of 001 or 1 is associated with a decrease of housing units of 262 and this effect is signi cant at the 1 level LPOP Holding all else constant We estimate that a 1 increase in population is associated with a 0998 or about 1 increase in housing units Again this effect is strongly signi cant The inclusion of these control variables does not alter the insigni cance of the treatment effect There is no evidence that the policy increased the number of housing units Chapter 7 Exercise Solutions Principles of Econometrics 4e 247 Exercise 710 continued 0 California s Inclusionary Zoning policies are designed to increase the supply of affordable housing The policy which is implemented in some California cities requires developers to provide a percentage of homes in new developments at below market price That is if the average price of homes in a development is 900000 the developer is required to provide some at a much lower price The policy has a noble intention but it has failed based on an analysis of the data Comparing housing in cities across California in 2000 after the policy change was implemented in some cities to housing in cities before the policy change we find that there has been no significant increase in the number of housing units attributable to the policy change Indeed the data show that the number of housing units in cities in which the policy was implemented has increased less than in cities in which the policy was not implemented However there does in fact appear that there has been an increase in average price resulting from the policy change Using an array of models which control for median income the level of educational attainment the percent of the population living in poverty and the population size we estimate the increase in average house price due to the law change to be between 333 the high estimate and 194 the low estimate A 95 interval estimate of the effect on prices from the model providing the low estimate is 56 to 332 One conjecture is that the law reduces the pro tability of builders and thus actually may reduce the supply of homes Chapter 7 Exercise Solutions Principles of Econometrics 46 248 EXERCISE 711 Note In the following question the interpretation of coef cient estimates is based on the 8 characteristics of changes in logarithms of Variables In Appendix A equation A3 We note that 1OOlny1 lny0 1OOAln y E percentage change in y Thus in a regression equation Alny 0 0A1nx gt 100Alny 1000 0100A1nx A percentage change in x is associated with a 32 percent change in 2 approximately If there is an indicator Variable D on the righthand side then Alny 0 51 gt 1OOAlny 1000 1005D The effect of the indicator Variable is 1008 change in 2 approximately The estimated regression for price is 1 02205 03326323IZLA W se 00152 00466 The estimated differencesindifferences regression is LNPRICE 120646 022051 02737 IZLAW 03326323 IZLAW x D se 00325 04602 00999 01413 Note that the estimate of the treatment effect is the same in both equations though standard errors are different due to estimation with different numbers of observations The estimated regression for changes in LN UNITS is z W9 01273 00314075IZLAW se 00119 00366 And for LN UNITS LNUNITS 9417601273D 04815IZLAW 00314075IZLAWgtltD se 00574 00812 01762 02492 The estimate of treatment effects are the same as the treatment effects from the differencesindifferences regression though the standard errors are different Chapter 7 Exercise Solutions Principles of Econometrics 4e 249 Exercise 711 continued b From equation 718 We see that the differencesindifferences estimator of the treatment effect is 8 fm Em ll fcb abbreviating Treatment Control Before and After Using the differenced data the regression 724 is Ayl 33 5d error i 1 N Where Ayl ym yib with a denoting After and b denoting Before and with 0 being the treatment Variable The least squares estimator of 8 is 11ltA1 Edz 67 Z A1dz c72 1 Where Ay NZl1Ay 8 From Appendix 7B the denominator is NON N Where M is the number receiving treatment and N 0 is the number in the control group Working then with the numerator of the expression We have 253W Ardi 67 Z ZA A rdi X5iA W07 Z1AEdi67Z1Arz5 Zf1Ay A ydt Zf1Aridt Zifdi ZZ1Aydi AyZ1d Where We have used the fact that Z1Ay 0 We can simplify the rst term in the last line of 1 as Zf1Ayt d Zf1va Md Zf1yiadi Zfimbdt N N 02 02 N1Z1yza lN1Z11Jlb I N N N 1 J N12 N E 2 The last line arises from the fact that for example yiadl is the sum of the outcome Variable only for the treated group where 01 1 The second term in the last line of 1 is dz N15 and NA N N N1 N d 1 d 1 yWZi1yiayibWZj1 z39yz39ayz39b z39yz39ayz39b N N F1iN1ya Nlytb Noyca Noycbi F1N1ya Jb N0 yea yd 8 Chapter 7 Exercise Solutions Principles of Econometrics 46 250 Exercise 711b continued 0 Then expression 1 becomes N Zi1Ayidi A3Z11dz39 2 N1Jta ytbF1N1ta Jb N0 yca ycb 7 NN N2 NN V Jtaytb71ytaytb V ycayd 2 7ta 7tbNN 7ca 7cb N NN ta ytb N N1yca cb N1N0 N 2 tb ca where in the last line we have used the fact that N N1 NO The last line of 3 is the numerator of The denominator is already noted NON1N so that 5la ytbyca ycb This is exactly the differencesindifferences estimator The estimated regression for price is 1 01439 02397IZLA W 12801DLMEDHHINC se 003 84 00415 O1268 The interpretation of the coefficient estimate for DLMEDHHINC is Holding other factors constant we estimate that one percent growth in the median household income between 1990 and 2000 increases housing price by 128 percent This estimate is statistically very signi cant with a Ivalue of 1009 The estimate of the treatment effect falls from 3326 to 2397 but the estimate remains statistically signi cant with a t value of 577 The estimated regression for units is zfvfW9 00480 00761IZLA W 06157DLMEDHHINC se 00331 00358 01094 The interpretation of the coefficient estimate for DLMEDHHINC is Holding other factors constant one percent growth in median household income between 1990 and 2000 is associated with an increase of 062 percent increase in the number of housing units The coef cient of IZLAW is negative and now statistically significant at the 5 level We estimate that holding all else constant the presence of the law is associated with 76 fewer housing units being available Chapter 7 Exercise Solutions Principles of Econometrics 46 251 Exercise 711 continued d The estimated regression for price is 1 01494 O1896IZLA W 10372DLMEDHHINC se 00481 00371 O1478 11841DEDUCITTAIN 0323 8DPROPPOVERTY 02448DLPOP 01828 05609 00528 Interpretation of new variables DED UCA T T AIN DPROPPOVERT Y and DLPOP DED UCA T ION Holding other factors constant a 1 increase in the proportion of people with a college education between 1990 and 2000 is associated with an increase in the housing price by 118 This estimate is signi cantly different from zero at the 1 level with a Ivalue of 648 DPROPPOVERT Y Holding other factors constant a 1 increase in the proportion of people below the poverty level between 1990 and 2000 is associated with a decrease in housing prices by 032 This estimate is not statistically signi cant from zero DLPOP Holding other variables constant a 1 increase in the size of population between 1990 and 2000 is associated with a decrease in housing prices by 024 This estimate is statistically signi cant with a Ivalue of 463 but the sign is difficult to rationalize The estimated regression for units is 9 0064O 00223IZLA W 00424DLMEDHHINC se 00148 00115 00456 03251DEDUCITTAIN 01873DPROPPOVERTY 08489DLPOP 00564 01731 00163 First note that the effect of the law passage is associated with a numerically smaller fall in the number of housing units available of 22 but the effect is still statistically signi cant at close to the 5 level We now estimate that a 1 increase in median income is associated with a 00424 increase in the number of housing units but this estimate is not statistically signi cant Interpretation of new variables DED UCA T T AIN DPROPPOVERT Y and DLPOP DEDUCATION Holding other factors constant we estimate that a 1 increase in the proportion of people with a college education between 1990 and 2000 is associated with an increase in the housing supply by 0325 This estimate is signi cant at the 1 level DPROPPOVERT Y Holding other factors constant we estimate that a 1 increase in the proportion of people below the poverty level between 1990 and 2000 is associated with a decrease in the housing supply by 0187 This estimate is not statistically signi cant DLPOP Holding other factors constant we estimate that a 1 increase in the size of the population between 1990 and 2000 is associated with an increase in the housing supply by 085 This estimate is very signi cant with a Ivalue of 5205 Chapter 7 Exercise Solutions Principles of Econometrics 46 252 EXERCISE 712 a The estimated regression is 1nV FE 09561 00905EDUC 00331EXPER 0000497EXPER2 02014FEMALE gtlltgtlltgtllt gtlltgtlltgtllt gtlltgtlltgtllt gtlltgtlltgtllt se 01039 00059 00048 00000835 00318 O1191BLACK 00301MIRRIED 00158SOUTH 00512 00331 00346 02044FULL TIME 01713METRO 00460 003 77 gtlltgtlltgtllt The 5 critical Ivalue for testing the signi cance of the coef cients and for other hypothesis tests is tc t09757990 1962 Considering the variables individually The intercept estimate cannot be reliably interpreted in this equation Its presence facilitates predictions and is present for mathematical completeness and it is the base from which all our indicator variables are measured EDUC We estimate that an increase in education by one year is associated with an approximate 905 increase in hourly Wages holding all else constant This estimate is signi cantly different from zero at a 1 level of signi cance That more educated Workers earn signi cantly higher salaries may occur because of their accumulated human capital or perhaps because smarter people stay in school longer and smarter Workers earn higher salaries EXPER and EXPER2 The marginal effect of another year of experience is estimated to be 003315 2gtlt 00004973gtlt EXPER For Workers with 1 5 25 and 50 years of experience these marginal effects are estimated to be approximately 32 28 083 and 17 respectively These estimated changes are all statistically different from zero The turning point in the relationship occurs at EXPER bEXPER2bEXPER 003312 0000497 323 The lifecycle effect of experience on earnings re ects the additional productivity that less experienced Workers receive from additional experience compared to a Worker with long years of experience Whose productivity changes little as experience is accumulated FEMALE We estimate that holding all else constant females earn approximately 2014 less than their male counterparts Using the exact calculation the difference is 1824 This estimate is statistically different from 0 at the 1 level Discrimination in the Workplace is re ected in these lower Wages Chapter 7 Exercise Solutions Principles of Econometrics 46 253 Exercise 712a continued BLACK We estimate that Wages for black Workers are approximately 119 lower than they are for nonblack Workers holding all else constant This estimate is statistically different from 0 at the 5 level Discrimination in the Workplace is re ected in these lower Wages MARRIED We estimate that Wages for married Workers are 301 higher than those who are not married This estimate is not statistically different from zero so using these data there is no significant evidence that married Workers earn more SOUTH We estimate that Wages for southerners are 158 less than their nonsouthern counterparts holding all else equal This estimate is not statistically signi cant We cannot reject the hypothesis that southern Workers do not earn less than nonsouthern Workers This outcome is different from results in many model estimations using data from earlier periods These data are from the 2008 CPS see Exercise 215 The current sample is only 1000 observations so the effect may not be estimated precisely F ULLT ME We estimate that the hourly Wage for full time Workers is approximately 2044 2268 using the exact calculation higher than it is for those who do not Work full time The estimate is statistically different from zero at the 1 level That Wages are higher for fulltime Workers than parttime Workers is not surprising Full time Workers tend to have more specialized training and more education as Well MET R0 We estimate that the hourly Wage for someone who lives in a metropolitan area is approximately 1713 higher 1869 using the exact calculation than nonmetro Workers This estimate is signi cant at the 1 level Workers in metropolitan areas have a Wider variety of Work opportunities resulting in higher average Wages Exercise 712 continued b Exercise 712 Chapter 7 Exercise Solutions Principles of Econometrics 46 To facilitate comparison from using the alternative data sets We have tabled them 1 2 CPS5 CPS4 C 0956 0906 0104 0047 EDUC 009l 0092 0006 0003 EXPER 0033 0029 0005 0002 EXPERAZ 0 497E3 0430E 3 0000 0000 FEMALE 020l 0l90 0032 0014 BLACK 0ll9 0l45 0051 0023 MARRIED 0030 0083 0033 0015 SOUTH 0016 0042 0035 0015 FULLTIME 0204 0266 0046 0020 METRO 0l7l 0l46 0038 0017 N 1000 4838 ad Rsq 0306 0336 SSE 231666 1057723 254 Standard errors in parentheses plt0l0 plt005 plt00l There are only slight differences in the estimated coef cient values and the signs of the coef cients are the same What is evident is that the t values are all much larger in magnitude for estimation from the cps4dat data This re ects the use of a larger sample size of 4838 observations in cps4dat relative to the 1000 observations in cps5dat Using a larger sample size improves the reliability of our estimated coef cients because We have more information about our regression function The larger tvalues also mean that the estimates have smaller pvalues and will therefore be significantly different from zero at a smaller level of significance We now nd for example that the effects of being married and being a southern Worker are statistically significant using cps4dat Whereas they were not using cps5dat Chapter 7 Exercise Solutions Principles of Econometrics 46 255 EXERCISE 713 The regressions for parts a d are summarized in the following table Exercise 713 1 2 3 4 a b C 01 C 45431 16894 5 8691 16400 0893 0041 1010 0046 EDUC 20315 00950 21053 00977 0058 0003 0071 0003 BLACK 51386 02463 59040 03000 0790 0036 1153 0052 FEMALE 53191 02589 54824 02642 0333 0015 0388 0018 BLACKFEM 45892 02147 61055 02800 1048 0048 1555 0071 SOUTH 08266 00460 21615 00612 0451 0020 1768 0080 MIDWEST 16721 00724 0465 0021 WEST 05658 00254 0465 0021 EDUCSOUTH 0 2077 0 0075 0 123 0 006 BLACKSOUTH 1 2764 0 0934 1 597 0 073 FEMALESOUTH 0 6517 0 0212 0 755 0 034 BLACKFEMALESOUTII 2 8406 0 1203 2 145 0 097 N 4838 4838 4838 4838 adj Rsq 0239 0253 0236 0249 BIC 369312299 70119091 369699545 70496046 SSE 5771884128 11899787 5797898271 11950878 Standard errors in parentheses plt001 plt010 plt005 Chapter 7 Exercise Solutions Principles of Econometrics 46 256 Exercise 713 continued 3 The estimated regression with standard errors in parentheses is i ii 1 4543 1 203l5EDUC 5l386BLACK 53 l9lFEMALE se 0892500578 07903 03325 45892BLACK gtlt FEMALE 08266SOUTH 1 672 IMIDWEST 10475 04510 O4653 O5658WEST R2 02404 O4648 To test Whether there is interaction between BLACK and FEMALE We test the null hypothesis that the coefficient of BLACK gtlt FEMALE is zero against the alternative that it is not zero The t statistic given by the computer output is 438 with a pvalue of 0000 Since this value is less than 001 We reject the null at a 1 level of signi cance and We conclude that there is a significant interaction between BLACK and FEMALE To test the hypothesis that there is no regional effect We test that the coef cients of SOUTH MIDWEST and WEST are jointly zero against the alternative that at least one of the indicator variables coef cients is not zero The F value can be calculated from the restricted regression Without regional variables and the unrestricted models SSER SSEUJ 5805445 57718843 9 3615 39 SSEUN K 39 57718844838 8 39 39 The corresponding pvalue is 0000 Also the critical value at the 5 significance level is 2607 Since the F value is larger than the critical value or the pvalue is less than 005 We reject the null hypothesis at the 5 level and conclude the regional effect is signi cant in determining the Wage level Chapter 7 Exercise Solutions Principles of Econometrics 46 257 Exercise 713 continued b The estimated regression using ln WAGE as a dependent variable i ii iii l 16894 00950EDUC 02463BLACK 02589FEMALE se 00405 00026 00359 00151 02147BLA CK gtlt FEMALE 0046OS0 UTH 00724MID WEST 00476 00204 00211 00254WEST R2 02540 00211 Comparing the results with the estimated equation in part a We find the signs of all the coef cient estimates are exactly the same The major difference lies in the value of coef cient estimates and their respective standard errors This is due to the nature of the linear versus the loglinear model In part a the estimated coefficients measure an impact on WAGE In part b they measure an impact on ln WAGE For example in model a We estimate that each additional year of education holding all else constant is associated with an increase in the hourly Wage of 203 In part b We estimate that the effect of an extra year of education holding all else constant is associated with approximately a 95 increase in the hourly Wage The loglinear model suggests that the variable SOUTH is significant at the 5 level While in the linear model in part a it is significant at only the 10 level To test Whether there is interaction between BLACK and FEMALE We test the null hypothesis that the coefficient of BLACK gtlt FEMALE is zero against the alternative that it is not zero The t statistic given by the computer output is 451 with a pvalue of 0000 Since this value is less than 001 We reject the null at a 1 level of significance and We conclude that there is a significant interaction between BLACK and FEMALE To test the hypothesis that there is no regional effect We test that the coefficients of SOUTH MIDWEST and WEST are jointly zero against the alternative that at least one of the indicator variable s coefficients is not zero The Fvalue can be calculated from the restricted regression Without regional variables and the unrestricted models SSER SSEUJ 1196854 11899793 39 SSEUN K 39 11899794838 8 39 9302 The corresponding pvalue is 0000 Also the critical value at the 5 significance level is 2607 Since the F value is larger than the critical value or the pvalue is less than 005 We reject the null hypothesis at the 5 level and conclude the regional effect is signi cant in determining the lnWAGE level Chapter 7 Exercise Solutions Principles Of Econometrics 46 258 Exercise 713 continued c The estimated regression is WAc 58691 21053EDUC 59040BLACK 54824FEMALE se 10099 00708 11535 O3885 61055BLACK gtlt FEMALE 21615SOUTH 02077EDUC gtlt SOUTH 11535 17682 01229 12764BLACK gtlt SOUTH 06517FEMALE gtlt SOUTH 15969 07554 28406BLA CK gtlt FEMALE gtlt SOUTH 21450 To test the null hypothesis that the Wage equation in the south is the same as the Wage equation for nonsoutherners We test the joint hypothesis that the coef cients of SOUTH and all the interaction variables with SOUTH are zero The alternative is that at least one these coefficients is not zero which would indicate a difference between south and non south Wage equations The F statistic is calculated from the sum of squared residuals of restricted and unrestricted models and is given by SSER SSEUJ 5805445 57978985 1257 SSEUN K 57978984838 10 The corresponding pvalue is 02798 Also the critical value at the 5 signi cant level is 2216 Since the Fstatistic is less than the critical value or the pvalue is greater than 005 We do not reject the null hypothesis at the 5 level and conclude that there is no signi cant difference between Wage equations for southern and nonsouthern Workers Chapter 7 Exercise Solutions Principles of Econometrics 46 259 Exercise 713 continued d The estimated regression for the loglinear model is ii 1W4 16400 00977ED UC 03 OOOBLACK 02642FEMALE se 011459 011032 011524 00177 02800BLACK gtlt FEMALE 00612SOUTH 00075EDUC gtlt SOUTH 00706 00803 00056 00934BLACK gtlt SOUTH 00212FEMALE gtlt SOUTH 00725 00343 01203BLACK gtlt FEMALE gtlt SOUTH 00974 Comparing the results with the estimated equation in part a we nd the signs of all the coef cient estimates are exactly the same The major difference lies in the value of the coef cient estimates and their respective standard errors This is due to the nature of the linear versus the loglinear model In part a the estimated coef cients measure an impact on WAGE In part b they measure an impact on ln WAGE For example in model a we estimate that each additional year of education holding all else constant is associated with an increase in the hourly wage of 211 In part b we estimate that an extra year of education holding all else constant is associated with approximately a 977 increase in the hourly wage In the loglinear model the interaction between EDUC and SOUTH is not signi cant at even the 10 level while in the linear relationship it is Otherwise SOUTH and its interactions are not signi cantly different from zero in both models To test the null hypothesis that the wage equation in the south is the same as the wage equation in the nonsouth we test the joint hypothesis that the coef cients of SOUTH and all the interaction variables with SOUTH are zero The alternative is that at least one these coef cients is not zero which would indicate a difference between south and nonsouth wage equations The Fstatistic is calculated from the sum of squared residuals of restricted and unrestricted models and is given by SSER SSEUJ 1196854 11950885 1427 SSEUN K 11950884838 10 The corresponding pvalue is 02110 Also the critical value at the 5 signi cance level is 2216 Since the Fvalue is less than the critical value or the pvalue is greater than 005 we do reject the null hypothesis at the 5 level and conclude that there is no signi cant difference between wage equations for southern and non southern workers Chapter 7 Exercise Solutions Principles of Econometrics 46 260 EXERCISE 714 8 b C We expect the parameter estimate for the dummy variable PERSON to be positive because of reputation and knowledge of the incumbent However it could be negative if the incumbent was on average unpopular andor ineffective We expect the parameter estimate for WAR to be positive re ecting national feeling during and immediately after first and second world wars The regression functions for each value of PART Y are E VOTE PARTY 1 0 0 BZGROWTH B3INFLA TION B4GOODNEWS BSPERSON B6DURA TION B8WAR EVOTE PARTY 1 0 0 BZGROWTH B3INFLA TION B4GOODNEWS BSPERSON B6DURA TION B8WAR The intercept when there is a Democrat incumbent is 31 137 When there is a Republican incumbent it is 31 137 Thus the effect of PARTY on the vote is 2137 with the sign of 37 indicating whether incumbency favors Democrats 37 gt 0 or Republicans 37 lt 0 The estimated regression using observations for 19162004 is WJFE 472628 06797GROWTH 06572INFLA TION 10749GOODNEWS se 253 84 01 107 02914 02493 32983PERSON 33300DURATION 26763PARTY 56149WAR 14081 12124 06264 26879 The signs are as expected We expect the coef cient of GROWTH to be positive because society rewards good economic growth For the same reason we expect the coef cient of GOODNEWS to be positive We expect a negative sign for the coef cient of INFLA T ION because increased prices impact negatively on society We expect the coef cient for PERSON to be positive because a party is usually in power for more than one term we expect the incumbent to get the majority vote for most of the elections We expect that for each subsequent term it is more likely that the presidency will change hands therefore we expect the parameter for DURATION to be negative The sign for PARTY is as expected if one knows that the Democratic Party was in power for most of the period 19162004 We expect the parameter for WAR to be positive because voters were more likely to stay with the incumbent party during the World Wars All the estimates are statistically signi cant at a 1 level of signi cance except for INFLATION PERSON DURATION and WAR The coef cients of INFLATION DURATION and PERSON are statistically signi cant at a 5 level of signi cance however The coef cient of WAR is statistically insigni cant at a level of 5 Lastly an R2 of 09052 suggests that the model ts the data very well Chapter 7 Exercise Solutions Principles of Econometrics 46 261 Exercise 714 continued d 6 0 Using the data for 2008 and based on the estimates from part c we summarize the actual and predicted vote as follows along with a listing of the values of the explanatory variables votehat 4809079 vote 466 22 growth inflation goodnews person duration party war 288 3 0 1 1 0 Thus we predict that the Republicans as the incumbent party will lose the 2008 election with 48091 of the vote This prediction was correct with Democrat Barack Obama defeating Republican John McCain with 529 of the popular vote to 457 A 95 confidence interval for the vote in the 2008 election is 176352012 it gtltse f 48091i21315gtlt2815 4209 5409 o97515 For the 2012 election the Democratic party will have been in power for one term and so we set DURATION 1 and PARTY 1 Also the incumbent Barack Obama is running for election and so we set PERSON 1 WAR 0 We use the value of in ation 30 anticipating higher rates of in ation after the policy stimulus We consider 3 scenarios for GROWTH and GOODNEWS representing good economic outcomes moderate and poor if there is a doubledip recession The values and the prediction intervals based on regression estimates with data from 19162008 are GROWTH INFLATION GOODNEWS lb Vote ub 35 3 6 456 515 573 1 3 3 404 465 525 3 3 1 350 415 480 We see that if there is good economic performance then President Obama can expect to be reelected If there is poor economic performance then we predict he will lose the election with the upper bound of the 95 prediction interval for a vote in his favor being only 48 In the intermediate case with only modest growth and less good news then we predict he will lose the election though the interval estimate upper bound is greater than 50 meaning that anything could happen Readers can keep up with Professor Fair s model httpfairmodeleconyaleeduvote2012index2htm and predictions at Chapter 7 Exercise Solutions Principles of Econometrics 4e EXERCISE 715 262 a A table of selected summary statistics Variable Mean Median Std Dev Skewness Kurtosis AGE 1957407 18 1719425 093851 3561539 BATHS 1973148 2 0612067 0912199 655344 BEDROOMS 317963 3 0709496 0537512 5751031 FIREPLACE 0562963 1 049625 0253 87 1064451 OWNER 0488889 0 0500108 0044455 1001976 POOL 007963 0 0270844 3105585 1064466 PRICE 1548632 130000 1229128 6291909 6094976 SQFT 2325938 21865 1008098 1599577 7542671 TRADITIONAL 0538889 1 0498716 015603 1024345 40 30 Percent 20 10 1 II 0 500000 1000000 sale price dollars 1500000 Figure xr715 Histogram of PRICE We can see from Figure Xr715 that the distribution of PRICE is positively skewed In fact the measure of skewness is 6292 We can see that the median price 130000 is Very different from the maximum price of 1580000 Exercise 715 continued b The results from estimating the regression model are below Chapter 7 Exercise Solutions Principles of Econometrics 46 263 I Coef Std Err t Pgtt 95 Conf Interval C 3980833 0458947 8674 0000 3890779 4070886 SQFTS 0299011 0014059 2127 0000 0271425 0326597 BEDROOMS 031506 0166109 1 90 0058 0640996 0010875 BATHS 190119 0205579 925 0000 1497807 2304573 AGE 0062145 0005179 12 00 0000 0072308 0051982 OWNER 0674655 017746 380 0000 0326445 1022864 POOL 0042748 0315812 014 0892 0662429 0576933 TRADITIONAL 0560925 0170267 329 0001 0895021 022683 FIREPLACE 0842748 019015 443 0000 0469639 1215857 WATERFRONT 10997 033355 30 0001 0445213 1754186 The estimated model ts the data well with R2 0737 though we should recall that the dependent variable is logarithmic The generalized R2 value calculated as the squared correlation between price and its predictor is corr1 IEPRICE2 08092 The estimated coef cient of SQF T is positive and signi cant indicating that an additional 100 square feet of living space holding all else xed will increase the price of the house by approximately 3 The estimated effect of an increase in the number of BEDROOMS is to reduce the house price by 315 This is consistent with the notion that more bedrooms holding all else xed results in smaller bedrooms which is less desirable This estimate is signi cant at the 10 level The estimated effect of an increase in the number of BA T HS is positive and signi cant with additional baths increasing the value of the house by approximately 19 holding all else constant This estimate is signi cant at the 1 level The estimated coef cient of AGE suggests that depreciation reduces the value of the home by 062 per year Again this estimate is signi cant at the 1 level Homes that are occupied rather than vacant are estimated to sell for 67 more holding all else constant It is reasonable that a livedin looking home is more attractive than a vacant one Empty houses may also indicate sellers are more anxious for a sale because they have moved on The presence of a POOL is statistically insigni cant One would think that an amenity such as a pool would carry a positive value so this result is somewhat surprising However the presence of a pool does increase maintenance costs and thus it is not a totally positive factor TRADITIONAL style homes are estimated to sell for 56 less other things being equal Since style is a matter of taste it is dif cult to form an a priori expectation about the sign of this factor Chapter 7 Exercise Solutions Principles of Econometrics 46 264 Exercise 715b continued 0 A FIREPLACE is a nice amenity for a home and the positive and signi cant estimate is as We would expect The estimated 84 increase in the house Value is perhaps a bit high The coefficient of WATERFRONT can be used to tell us the percentage increase or decrease associated with a Waterfront house On average a Waterfront house sells for 100 x eXp01 100 1 1162 higher than a house that is not Waterfront After including the Variable TRADITIONAL gtlt WATERFRONT the results from estimating the two regression models are summarized below 1 2 b C C 39808 39711 0046 0046 SQFTS 00299 00300 0001 0001 BEDROOMS 00315 00313 0017 0017 BATHS 01901 01883 0021 0021 AGE 00062 00061 0001 0001 OWNER 00675 00684 0018 0018 POOL 00043 00024 0032 0032 TRADITIONAL 00561 00449 0017 0018 FIREPLACE 00843 00873 0019 0019 WATERFRONT O1100 O1654 0033 0040 WFTRAD 01722 0069 N 1080 1080 ad R sq 0 735 0736 SSE 779809 775256 Chapter 7 Exercise Solutions Principles of Econometrics 46 265 Exercise 715c continued Let lnP0 be the mean logprice for a nontraditional house that is not on the Waterfront and let 39 B10 and 311 be the coef cients of TRADITIONAL WATERFRONT and TRADITIONAL gtlt WATERFRONT respectively Then the mean logprice for a traditional house not on the Waterfront is 1nPT 1HPo 139 The mean logprice for a nontraditional house on the Waterfront is 1nPW 1HPoB1o The mean logprice for a traditional house on the Waterfront is 1nPTW1nBB9 1310 1311 The approximate percentage difference in price for traditional houses not on the Waterfront is lnPT lnP0 x 100 39 gtlt100 45 The approximate percentage difference in price for nontraditional houses on the Waterfront is lnPW lnP0gtlt1OO 310 gtlt100 165 The approximate percentage difference in price for traditional houses on the Waterfront is lnPTW lnP0gtlt1OO 39 310 B11gtlt1OO 517 Thus traditional houses on the Waterfront sell for less than traditional houses elsewhere The price advantage from being on the Waterfront is lost if the house is a traditional style The approximate proportional difference in price for houses which are both traditional and on the Waterfront cannot be obtained by simply summing the traditional and Waterfront effects 39 and B10 The extra effect from both characteristics B11 must also be added Its estimate is signi cant at a 5 level of signi cance The corresponding exact percentage price differences are as follows For traditional houses not on the Waterfront 100 x exp 00449 1 439 For nontraditional houses on the Waterfront 100 gtltexpO1654 1 1798 For traditional houses on the Waterfront 100 X exp 00449 01654 01722 1 504 Chapter 7 Exercise Solutions Principles of Econometrics 46 266 Exercise 715 continued d 6 The Chow test requires the original model plus an interaction variable of TRADITIONAL with every other variable We want to test the joint null hypotheses that the coefficients of TRADITIONAL and all its interactions are zero against the alternative that at least one is not zero Rejecting the null indicates that the equations for traditional and nontraditional home prices are not the same On the following page four models are summarized The restricted model is the one in which it is assumed that there is no difference between TRADITIONAL and nontraditional houses Rest Two models are for the subsets of the data for which the variable TRADITIONAL is 1 or O and the last model is the fully interacted model The F value for this test is F SSER SSEUJ 7877197579959 46272 SSEUN K 7579951080 18 Since 4627gtI7095971062l889 the null hypothesis is rejected at a 5 level of signi cance We conclude that there are different regression functions for traditional and nontraditional styles Note that SSEU 757995 is equal to the sum of the SSE from traditional houses 3 1 05 82 and the SSE from nontraditional houses 447413 Using the model from part c we find that the prediction for lnPRICE1000 is 4992 The natural predictor is PRICE exp1 x1000 eXp4992 x 1000 147 265 The corrected predictor is PRICEC PRICE gtlt expe2 2147265gtlt 00725 2 152703 Chapter 7 Exercise Solutions Principles of Econometrics 46 267 Exercise 715d continued Rest Trad1 Trad0 Unrest sqfts 00302 00271 00324 00324 0001 0002 0002 0002 bedrooms 00405 00275 00714 00714 0016 0021 0027 0024 baths 01894 02142 01831 01831 0021 0026 0033 0029 age 00062 00068 00055 00055 0001 0001 0001 0001 owner 00650 00975 00388 00388 0018 0021 0029 0026 pool 00008 00216 00021 00021 0032 0041 0047 0042 fireplace 00912 01228 00578 00578 0019 0022 0034 0030 waterfront 01226 00340 01730 01730 0033 0051 0046 0041 traditional 03351 0094 sqfttr 00053 0003 bedstr 00989 0034 bathtr 00311 0041 agetr 00013 0001 owntr 00587 0035 poo1tr 00238 0063 fptr 00650 0039 wftr 02070 0071 cons 39701 37322 40673 40673 0046 0065 0065 0058 N 1080 582 498 1080 ad R sq 0733 0752 0730 0741 SSE 787719 310582 447413 757995 Standard errors in parentheses plt010 plt005 plt00l Chapter 7 Exercise Solutions Principles of Econometrics 4e 268 EXERCISE 716 a The histogram for PRICE is positively skewed On the other hand the logarithm of PRICE is much less skewed and is more symmetrical Thus the histogram of the logarithm of PRICE is closer in shape to a normal distribution than the histogram of PRICE Percent I I I 0 200000 400000 600000 800000 selling price of home dollars Figure xr716a Histogram of PRICE 15 Percent 12 ogseing price Figure xr716b Histogram of lnPRICE Chapter 7 Exercise Solutions Principles of Econometrics 46 269 Exercise 716 continued b C d 6 The estimated equation is 1 39860 00539LIVAREA 00382BEDS 00103BATHS se 00373 00017 00114 00165 O2531LGELOT 00013AGE 00787PO0L 00255 00005 00231 All coefficients are signi cant with the exception of that for BATHS All signs are reasonable increases in living area larger lot sizes and the presence of a pool are associated with higher selling prices Older homes depreciate and have lower prices Increases in the number of bedrooms holding all else fixed implies smaller bedrooms which are less valued by the market The number of baths is statistically insigni cant so its negative sign cannot be reliably interpreted The price of houses on lot sizes greater than 05 acres is approximately 100exp 02531 1 288 larger than the price of houses on lot sizes less than 05 acres The estimated regression after including the interaction term is 1 39649 00589LIIAREA 0048OBEDS 0020131 THS se 00370 00019 00113 00164 O6134LGELOT 00016IGE 00853PO0L 00632 00005 00228 00161LGELOT gtlt LIVAREA 00026 Interpretation of the coef cient of LGELOTXLIVAREAZ The estimated marginal effect of an increase in living area of 100 square feet in a house on a lot of less than 05 acres is 589 holding other factors constant The same increase for a house on a large lot is estimated to increase the house selling price by 161 less or 427 However note that by adding this interaction variable into the model the coefficient of LGELOT increases dramatically The inclusion of the interaction variable separates the effect of the larger lot from the fact that larger lots usually contain larger homes To carry out a Chow test we use the sum of squared errors from the restricted model that does not distinguish between houses on large lots and houses that are not on large lots SSER 720633 and the sum of squared errors from the unrestricted model that includes LGELOT and its interactions with the other variables which is SSEU 654712 Then the value of the F statistic is Chapter 7 Exercise Solutions Principles of Econometrics 48 Exercise 716 continued SSER SSEUJ 720633 6547126 SSEUN K 6547121488 2497 270 The 5 critical F Value is 17095671488 210 Thus We conclude that the pricing structure for houses on large lots is not the same as that on smaller lots A summary of the alternative model estimations follows Exercise 716 9794 039 0607 002 0594 012 0262 017 0008 000 0989 024 9828 038 O604 002 0522 012 0334 017 OO16 000 O697 025 4293 141 O266 004 0434 037 1161 052 0002 001 0562 060 1 2 LGELOT1 LGELOT0 C 44121 39828 0183 0037 LIVAREA 00337 00604 0005 0002 BEDS 00088 00522 0048 0012 BATHS 00827 00334 0066 0017 AGE 00018 00016 0002 0000 POOL 01259 00697 0074 0024 LGELOT LorAREA LOIBEDS LorBATHs LorAGE LorPooL N 95 1405 adj R sq 0676 0608 BIC 508699 4392028 SSE 71268 583445 0667 252 72 8181 0633 0696 352 65 8402 4712 Standard errors in parentheses pltO10 plt005 plt001 LOIX indicates interaction between LGELOT and X CHAPTER 8 Exercise Solutions 271 Chapter 8 Exercise Solutions Principles of Econometrics 4e 272 EXERCISE 81 When 512 62 Chapter 8 Exercise Solutions Principles of Econometrics 46 273 EXERCISE 82 a Multiplying the first normal equation by zofxf and the second one by 26172 yields 2 Z6 xZ Z62 1Z6 x B2 Z61xfZ61yf Zc2Zo x 1Zc2Zx 2 Zltr2Zx yf Subtracting the rst of these two equations from the second yields o2x2o1x2 32 o2xy o1xo y Thus A2 Zlts2Zx y Zo xfZo yf ZltrZx 2 Zltr xi 2 Zofxf iofxj Z6 26 In this last expression the second line is obtained from the first by making the substitutions yf o 1y and x o 1x and by dividing numerator and denominator by 26172 2 Solving the first normal equation Zo2 1Zo1x 2 Zo 1y for 81 and making the substitutions y ojlyl and x ofxi yields 3 Zcfyi ZG2xi 3 l 2 2 2 X61 20 b When 612 62 for all i Zofyixi 522ix Zofy 622I Zofx cfzzxl and ZG2 N 62 Making these substitutions into the expression for 82 yields 622yixi 52234 2239 Zyixi A No No No T 2 cfzzxiz o2Zxj2 C2 No 2 NG2 and that for 81 becomes rs W1 r 92 ass No 2 No 2 These formulas are equal to those for the least squares estimators I91 and 192 See pages 52 and 8384 of the text Chapter 8 Exercise Solutions Principles of Econometrics 46 274 Exercise 82 continued c The least squares estimators I91 and 92 are functions of the following averages L 1 L L 2 Nzxi 3 Nzyi N292Vi N292 gtlt For the generalized least squares estimator for A3 and 32 these unweighted averages are replaced by the Weighted averages Z013 zcizyzj zcizyixij ZGIWJ Z6 Z6 Z6 26 In these Weighted averages each observation is Weighted by the inverse of the error variance Reliable observations with small error variances are weighted more heavily than those with higher error variances that make them more unreliable Chapter 8 Exercise Solutions Principles of Econometrics 46 275 EXERCISE 83 For the model yl B Bzxl el where varel ozxf the transformed model that gives a constant error variance is J B1xB2e Where yf yl xi xi 1 xi and ej el xi This model can be estimated by least squares with the usual simple regression formulas but with 31 and B2 reversed Thus the generalized least squares estimators for 31 and B2 are 31 NEW ZxZy2 and 32 I 7 m NZltxgt2 Zx Using observations on the transformed variables We nd Zf7 Zxf3712 Zxgjf478 Zxf2349144 With N 5 the generalized least squares estimates are 3 5478 37127 2984 5349144 37122 and 3 7 f31c 75 29848712 044 Chapter 8 Exercise Solutions Principles of Econometrics 46 276 EXERCISE 84 3 b C In the plot of the residuals against income the absolute value of the residuals increases as income increases but the same effect is not apparent in the plot of the residuals against age In this latter case there is no apparent relationship between the magnitude of the residuals and age Thus the graphs suggest that the error variance depends on income but not age Since the residual plot shows that the error variance may increase when income increases and this is a reasonable outcome since greater income implies greater exibility in travel We set up the null and alternative hypotheses as the one tail test H0 2612 6 versus H1 2612 gt 6 where of and 6 are arti cial variance parameters for high and low income households The value of the test statistic is of 29471x1o7100 4 A 7 28124 cs lO479gtltlO 100 4 The 5 critical value for 96 96 degrees of freedom is I709579696 1401 Thus we reject H 0 and conclude that the error variance depends on income Remark An inspection of the le vacation dat after the observations have been ordered according to INCOME reveals 7 middle observations with the same value for INCOME namely 62 Thus when the data are ordered only on the basis of INCOME there is not one unique ordering and the values for SSE and SSE will depend on the ordering chosen Those speci ed in the question were obtained by ordering rst by INCOME and then by AGE i All three sets of estimates suggest that vacation miles travelled are directly related to household income and average age of all adults members but inversely related to the number of kids in the household ii The White standard errors are slightly larger but very similar in magnitude to the conventional ones from least squares Thus using White s standard errors leads one to conclude estimation is less precise but it does not have a big impact on assessment of the precision of estimation iii The generalized least squares standard errors are less than the White standard errors for least squares suggesting that generalized least squares is a better estimation technique Chapter 8 Exercise Solutions Principles of Econometrics 46 277 EXERCISE 85 3 b C The table below displays the 95 confidence intervals obtained using the critical Ivalue t0975497 1965 and both the least squares standard errors and the White s standard errors After recognizing heteroskedasticity and using White s standard errors the confidence intervals for CRIME AGE and TAX are narrower while the con dence interval for ROOMS is wider However in terms of the magnitudes of the intervals there is very little difference and the inferences that would be drawn from each case are similar In particular none of the intervals contain zero and so all of the variables have coefficients that would be judged to be significant no matter what procedure is used 95 confidence intervals White s standard errors Least squares standard errors Lower Upper Lower Upper CRIME 0255 0112 0252 0114 ROOMS 5600 7143 5065 7679 AGE 0076 0020 0070 0026 TAX 0020 0005 0019 0007 Most of the standard errors did not change dramatically when White s procedure was used Those which changed the most were for the variables ROOMS TAX and PT RA T IO Thus heteroskedasticity does not appear to present major problems but it could lead to slightly misleading information on the reliability of the estimates for ROOMS TAX and PT RA T IO As mentioned in parts a and b the inferences drawn from use of the two sets of standard errors are likely to be similar However keeping in mind that the differences are not great we can say that after recognizing heteroskedasticity and using White s standard errors the standard errors for CRIME AGE DIST TAX and PT RA T IO decrease while the others increase Therefore using incorrect standard errors least squares understates the reliability of the estimates for CRIME AGE DIST TAX and PT RAT IO and overstates the reliability of the estimates for the other variables Remark Because the estimates and standard errors are reported to 4 decimal places in Exercise 55 Table 57 but only 3 in this exercise Table 82 there will be some rounding error differences in the interval estimates in the above table These differences when they occur are no greater than 0001 Chapter 8 Exercise Solutions Principles of Econometrics 46 278 EXERCISE 86 81 b ROOMS significantly effects the variance of house prices through a relationship that is quadratic in nature The coefficients for ROOMS and ROOMS2 are both signi cantly different from zero at a 1 level of signi cance Because the coefficient of ROOMS2 is positive the quadratic function has a minimum which occurs at the number of rooms for which A2 ii 0L 2ot3ROOMS 0 8ROOMS Using the estimated equation this number of rooms is ROOMSM OL2 305311 64 2amp3 quot 2gtlt23822 39 Thus for houses of 6 rooms or less the variance of house prices decreases as the number of rooms increases and for houses of 7 rooms or more the variance of house prices increases as the number of rooms increases The variance of house prices is also a quadratic function of CRIME but this time the quadratic function has a maximum The crime rate for which it is a maximum is OL4 2285 2293 CRIMEmaX A 20 2gtlt0039 Thus the variance of house prices increases with the crime rate up to crime rates of around 30 and then declines There are very few observations for which CRIME 2 30 and so We can say that generally the variance increases as the crime rate increases but at a decreasing rate The variance of house prices is negatively related to DIST suggesting that the further the house is from the employment centre the smaller the variation in house prices We can test for heteroskedasticity using the White test The null and alternative hypotheses are H0 oc2 oc3 oc6 0 H1 not all ocs in H0 are zero The test statistic is X2 NgtltR2 We reject H0 if X2 gt x2095 5 Where X39575 1107 The test value is X2 Ngtlt R2 506gtlt 008467 4284 Since 4284 gt 1107 We reject H 0 and conclude that heteroskedasticity exists Chapter 8 Exercise Solutions Principles 0fEc0n0metrics 46 279 EXERCISE 87 a Hand calculations yield Xx 0 Zy 311 Zxy 8935 Zxf 5234 7 O 7 38875 The least squares estimates are given by b2 Nzxiyl ZxlZy 8gtlt8935 Ogtlt31117O71 Nzxf Zxl2 8gtlt5234 O2 and 19 7 19 38875 l7071gtlt 0 38875 b The least squares residuals l yl 2 and other information useful for part c follow observation ln 2 z x ln 2 1 1933946 1319125 4353113 2 0733822 0618977 0185693 3 9549756 4513031 31591219 4 1714707 1078484 5068875 5 3291665 2382787 4527295 6 3887376 2715469 18465187 7 3484558 2496682 5742369 8 3746079 2641419 16905082 c To estimate oc we begin by taking logs of both sides of of CXpOtZl that yields lI1Gl2 oczi Then we replace the unknown 612 with f to give the estimating equation ln l2 otzl V Using least squares to estimate on from this model is equivalent to a simple linear regression Without a constant term See for example Exercise 24 The least squares estimate for on is 8 2 1 ei2 1 864674 zz 17817 6c 04853 Chapter 8 Exercise Solutions Principles 0fEc0n0metrics 46 280 Exercise 87 continued d Variance estimates are given by the predictions 62 CXp tZl eXp04853gtltzl These l values and those for the transformed variables y sj G1quot are given in the following table observation 62 yj xi l l 4960560 0493887 0224494 1156725 0464895 2789371 29879147 3457624 0585418 9785981 0287700 0575401 2514531 4036003 2144126 27115325 0345673 0672141 3053260 2575316 1373502 22330994 0042323 0042323 OOlOUlgtUJ e From Exercise 82 the generalized least squares estimate for B is 3 Z6 Z6 Z6 Ex Zofx 2 Z6 Z6 2193812x 0383851 03838512 2008623 8477148 7540580 11242 The generalized least squares estimate for B1 is 2 2 5 Z61 2 Z9 3 2193812 0383851gtlt1124226253 Z6 Z6 Chapter 8 Exercise Solutions Principles of Econometrics 46 281 EXERCISE 88 81 b 0 C1 The regression results with standard errors in parenthesis are l ah519315683907SQFT 2178433AGE se 358664 21687 350976 These results tell us that an increase in the house size by one square foot leads to an increase in house price of 6339 Also relative to new houses of the same size each year of age of a house reduces its price by 21784 For SQFT 1400 and AGE 20 z 519315 683907gtlt1400 2178433gtlt 20 96583 The estimated price for a 1400 square foot house which is 20 years old is 96583 For SQFT 1800 and AGE 20 z 519315 683907 gtlt1800 2178433 x 20 123940 The estimated price for a 1800 square foot house which is 20 years old is 123940 For the White test we estimate the equation f 011 o1sQFT 0131GE o14SQFT2 O15IGE2 o16SQFT gtlt AGE V and test the null hypothesis H 0 01 013 016 0 The Value of the test statistic is X2 Ngtlt R2 940gtlt 00375 3525 Since xfogsqs 1107 the calculated Value is larger than the critical Value That is X2 gtx2095 5 Thus We reject the null hypothesis and conclude that heteroskedasticity exists Estimating the regression log l2 011 o12SQF T V gives the results 6112163786 612 0001414 With these results We can estimate of as 63 eXp163786 0001414SQFT Chapter 8 Exercise Solutions Principles of Econometrics 46 282 Exercise 88 continued e Generalized least squares requires us to estimate the equation we14131418114 e119 6 6 6 6 6 When estimating this model we replace the unknown 61 with the estimated standard deviations 61 The regression results with standard errors in parenthesis are z 849114 653269SQFT 1876587AGE se 310943 20825 292844 These results tell us that an increase in the house size by one square foot leads to an increase in house price of 6533 Also relative to new houses of the same size each year of age of a house reduces its price by 18766 1 For SQF T 1400 and AGE 20 z 849114 653269gtlt1400 1876587gtlt 20 96196 The estimated price for a 1400 square foot house which is 20 years old is 96196 For SQFT 1800 and AGE 20 z 849114 653269 X1800 1876587 X 20 122326 The estimated price for a 1800 square foot house which is 20 years old is 122326 Chapter 8 Exercise Solutions Principles of Econometrics 416 283 EXERCISE 89 3 ii i Under the assumptions of Exercise 88 part a the mean and Variance of house prices for houses of size SQF T 1400 and AGE 20 are EPRICE 0 1400 0 20 0 VarPRICE G2 Replacing the parameters with their estimates gives EPRICE 96583 VarPRICE 22539632 Assuming the errors are normally distributed gt 115000 96583 225396 PZ gt 08171 PPRICE gt115000 PKZ 0207 Where Z is the standard normal random Variable Z N 01 The probability is depicted as an area under the standard normal density in the following diagram 5 J 3 FDF 6439latEitii fai s Z The probability that your 1400 square feet house sells for more than 115000 is 0207 For houses of size SQF T 1800 and AGE 20 the mean and Variance of house prices from Exercise 88a are EPRICE 123940 VarPRICE 22539632 The required probability is PPRICE lt110000 PEZ lt 110000423940 225396 PZ lt O6185 0268 The probability that your 1800 square feet house sells for less than 110000 is 0268 Chapter 8 Exercise Solutions Principles of Econometrics 46 284 Exercise 89 continued b i Using the generalized least squares estimates as the Values for 131132 and B3 the mean of house prices for houses of size SQFT 1400 and AGE 20 is from Exercise 88f EPRICE96196 Using estimates of ocl and 0c from Exercise 88d the Variance of these house types is VarPRICE expoL1 12704 0c x 1400 exp16378549 12704 000141417691gtlt 1400 3347172gtlt108 1829532 Thus PPRICE gt 115000 PZ gt L PZ gt 10278 0152 The probability that your 1400 square feet house sells for more than 115000 is 0152 ii For your larger house Where SQF T 1800 We nd that E PRICE 122326 and VarPRICE expoL1 12704 0c x 1800 exp16378549 12704 000141417691gtlt1800 5893127 X108 2427582 Thus PPRICE lt 110000 P Z lt 110000 122326 242758 PZ lt 05077 0306 The probability that your 1800 square feet house sells for less than 110000 is 0306 c In part a Where the heteroskedastic nature of the error term was not recognized the same standard deviation of prices was used to compute the probabilities for both house types In part b recognition of the heteroskedasticity has led to a standard deviation of prices that is smaller than that in part a for the case of the smaller house and larger than that in part a for the case of the larger house These differences have in turn led to a smaller probability for part i Where the distribution is less spread out and a larger probability for part ii Where the distribution has more spread Chapter 8 Exercise Solutions Principles of Econometrics 46 285 EXERCISE 810 a The transformed model corresponding to the Variance assumption of ozxl is L 9 2 9 xii 31xl52el where el P We obtain the residuals from this model square them and regress the squares on x to obtain 2 12379 2335x R2 013977 To test for heteroskedasticity We compute a Value of the X2 test statistic as X2 NxR2 40gtlt013977 559 A null hypothesis of no heteroskedasticity is rejected because 559 is greater than the 5 critical Value X39571 384 Thus the Variance assumption of ozxl was not adequate to eliminate heteroskedasticity b The transformed model used to obtain the estimates in 827 is 1 x e 4517 l52Tl97 Where 97 T 5 o G lt5 I I I I and 6 eXp093779596 232923 872 x lnx We obtain the residuals from this model square them and regress the squares on x to obtain 2 1117 005896x R2 002724 To test for heteroskedasticity We compute a Value of the X2 test statistic as X2 Ngtlt R2 40gtlt 002724 109 A null hypothesis of no heteroskedasticity is not rejected because 109 is less than the 5 critical Value X39571 384 Thus the Variance assumption of 62x17 is adequate to eliminate heteroskedasticity Chapter 8 Exercise Solutions Principles of Econometrics 46 286 EXERCISE 811 The results are summarized in the following table and discussed below part a part b part c 1 81000 76270 81009 seA31 32822 12004 33806 1 10328 10612 10323 sef3 1706 1024 1733 X2 Ngtlt R2 6641 2665 6955 The transformed models used to obtain the generalized estimates are as follows yi 1 xi 1 8139 3 X1025 J 2 B1 X1025 1 B2 X1025 1 8239 Where 8239 X1025 b p 0 1 0 e where e i xi xi xi xi c Tyquot 31 L 32 if ej where ej 18quot lnxl lnxl lnxl lnxl In each case the residuals from the transformed model were squared and regressed on income and income squared to obtain the R2 values used to compute the X2 values These equations were of the form 2 01 oL2xoL3x2 12 For the White test we are testing the hypothesis H0 01 013 0 against the alternative hypothesis H1 01 7 O andor 013 7 O The critical chisquared value for the White test at a 5 level of significance is X89572 5991 After comparing the critical value with our test statistic values we reject the null hypothesis for parts a and c because in these cases X2 gtX2095 2 The assumptions vareo2xi and varelo2 lnxl do not eliminate heteroskedasticity in the food expenditure model On the other hand we do not reject the null hypothesis in part b because X2 lt X39572 Heteroskedasticity has been eliminated with the assumption that varel ozxf In the two cases where heteroskedasticity has not been eliminated parts a and c the coef cient estimates and their standard errors are almost identical The two transformations have similar effects The results are substantially different for part b however particularly the standard errors Thus the results can be sensitive to the assumption made about the heteroskedasticity and importantly whether that assumption is adequate to eliminate heteroskedasticity Chapter 8 Exercise Solutions Principles of Econometrics 46 287 EXERCISE 812 3 b C This suspicion might be reasonable because richer countries countries with a higher GDP per capita have more money to distribute and thus they have greater exibility in terms of how much they can spend on education In comparison a country with a smaller GDP will have fewer budget options and therefore the amount they spend on education is likely to vary less The regression results with the standard errors in parentheses are EEI O1246 00732 LE P P 139 139 se 00485 00052 The tted regression line and data points appear in the following figure There is evidence of heteroskedasticity The plotted values are more dispersed about the fitted regression line for larger values of GDP per capita This suggests that heteroskedasticity exists and that the variance of the error terms is increasing with GDP per capita 14 12 10 08 06 04 02 00 390392 I I I I I I I I GDP per capita For the White test We estimate the equation GDB GDB 2 el OL1OL2 OL3 vl P P l I This regression returns an R2 value of 029298 For the White test We are testing the hypothesis H0 oc2 OL3 0 against the alternative hypothesis H1 oc2 7 O andor OL3 7 O The White test statistic is X2 Ngtlt R2 34 X 029298 9961 The critical chisquared value for the White test at a 5 level of signi cance is X89572 5991 Since 9961 is greater than 5991 We reject the null hypothesis and conclude that heteroskedasticity exists Chapter 8 Exercise Solutions Principles of Econometrics 46 288 Exercise 812 continued d 6 Using White s formula seb1 0040414 seb 0006212 The 95 con dence interval for B using the conventional least squares standard errors is Z i t09753seb 0073173 i 20369 x 000517947 00626 00837 The 95 con dence interval for 32 using White s standard errors is Z i t09753seb 0073173 i 20369 x 000621162 00605 00858 In this case ignoring heteroskedasticity tends to overstate the precision of least squares estimation The con dence interval from White s standard errors is Wider Reestimating the equation under the assumption that varel ozxl We obtain EEI39 00929 00693 LE P P 139 139 se 00289 00044 Using these estimates the 95 con dence interval for B2 is 92 it seb2 0069321 i 20369 x 000441 171 00603 00783 097532 The Width of this con dence interval is less than both confidence intervals calculated in part d Given the assumption varelo2xl is true We expect the generalized least squares con dence interval to be narrower than that obtained from White s standard errors re ecting that generalized least squares is more precise than least squares when heteroskedasticity is present A direct comparison of the generalized least squares interval with that obtained using the conventional least squares standard errors is not meaningful however because the least squares standard errors are biased in the presence of heteroskedasticity Chapter 8 Exercise Solutions Principles of Econometrics 46 289 EXERCISE 813 3 b C d For the model Ch 13 BZQU B3Q12l B4Q13l eh Where vare1 o2Q1 the generalized least squares estimates of B1 B2 B3 and B4 are estimated standard coef cient error 31 93595 23422 32 68592 17484 33 10744 3774 34 10086 02425 The calculated F value for testing the hypothesis that 31 34 0 is 1084 The 5 critical value from the F 2324 distribution is 340 Since the calculated F is greater than the critical F We reject the null hypothesis that 31 34 0 The F value can be calculated from SSER SSEU2 6131765 61111342 F 2 SSEU24 611113424 21084 The average cost function is given by Cu 2 L 2 ez an B1Q1lj B2 B3Q1t B4Q1t Q1 Thus if B 34 0 average cost is a linear function of output The average cost function is an appropriate transformed model for estimation when heteroskedasticity is of the form vare1 o2Q12 Chapter 8 Exercise Solutions Principles of Econometrics 46 290 EXERCISE 814 3 b C The least squares estimated equations are C 72774 83659Q 13796Qf 11911Qf sf 32485 se 23655 4597 02721 SSE1 779649 6 5118510829Q 20015Q 16131Q 6 84766 se 28933 6156 03802 ssE2034383 To see Whether the estimated coef cients have the expected signs consider the marginal cost function MC 0 2039 30492 We expect MC gt 0 when Q 0 thus We expect 32 gt 0 Also We expect the quadratic MC function to have a minimum for which We require 34 gt O The slope of the MC function is dMCdQ 2B3 6B4Q For this slope to be negative for small Q decreasing MC and positive for large Q increasing MC We require 33 lt 0 Both our leastsquares estimated equations have these expected signs Furthermore the standard errors of all the coef cients except the constants are quite small indicating reliable estimates Comparing the two estimated equations We see that the estimated coefficients and their standard errors are of similar magnitudes but the estimated error variances are quite different Testing H0 6126 against H1 612746 is a twotail test The critical values for performing a twotail test at the 10 signi cance level are 700524p24 00504 and 70952424 1984 The value of the F statistic is A2 F 2 6 2 84766 Z 261 61 32485 Since Fgt170952424 We reject H0 and conclude that the data do not support the proposition that of 6 Since the test outcome in b suggests 612 6 but We are assuming both rms have the same coef cients We apply generalized least squares to the combined set of data with the observations transformed using 61 and 62 The estimated equation is C 67270 89920Q 15408 Q2 13026Q3 se 16973 3415 02065 Remark Some automatic software commands will produce slightly different results if the transformed error variance is restricted to be unity or if the variables are transformed using variance estimates from a pooled regression instead of those from part a Chapter 8 Exercise Solutions Principles of Econometrics 46 291 Exercise 814 continued d Although We have established that of 75 G it is instructive to rst carry out the test for B2 52 B3 53 B4 54 under the assumption that of 6 and then under the assumption that of 75 6 H0B151 Assuming that of G the test is equivalent to the Chow test discussed on pages 268270 of the text The test statistic is SSER SSEUJ 39 SSEU N K Where SSEU is the sum of squared errors from the full dummy variable model The dummy variable model does not have to be estimated however We can also calculate SSEU as the sum of the SSE from separate least squares estimation of each equation In this case SSEU SSE SSE 779649 2034383 2814032 The restricted model has not yet been estimated under the assumption that 612 6 Doing so by combining all 56 observations yields SSE R 2887434 The F value is given by SSER SSEU J 2 2887434 28140324 SSEUN K 281403256 8 0313 The corresponding X2 value is X2 4x F 1252 These values are both much less than their respective 5 critical values 170954p482565 and X89574 9488 There is no evidence to suggest that the rms have different coef cients In the formula for F note that the number of observations N is the total number from both firms and K is the number of coef cients from both firms The above test is not valid in the presence of heteroskedasticity It could give misleading results To perform the test under the assumption that 612 at G We follow the same steps but We use values for SSE computed from transformed residuals For restricted estimation from part c the result is SSE 492412 For unrestricted estimation We have the interesting result SSE SSE N 1ltx sf N 1ltx s A2 A2 A2 A2 51 G2 G1 52 mg M mM Thus F 492412 484 03103 and 4848 8 1241 The same conclusion is reached There is no evidence to suggest that the firms have different coefficients The X2 and F test values can also be conveniently calculated by performing a Wald test on the coef cients after running Weighted least squares on a pooled model that includes dummy variables to accommodate the different coef cients Chapter 8 Exercise Solutions Principles of Econometrics 46 292 EXERCISE 815 8 b C To estimate the two variances using the variance model speci ed We rst estimate the equation WAGE 31 BZEDUCI B3EXPERl B4METROl el From this equation We use the squared residuals to estimate the equation ln l2 oc oc2METROl V The estimated parameters from this regression are 611 1508448 and 612 O33804l Using these estimates We have METRO 0 gt 6 exp1508448 O338041gtlt 0 4519711 METRO 1 gt 6 exp1508448 0338041 x 1 6337529 These error variance estimates are much smaller than those obtained from separate sub samples 6 3l824 and 6 15243 One reason is the bias factor from the exponential function see page 317 of the text Multiplying 6 63375 and 6 45197 by the bias factor exp12704 yields 6 22576 and 6 16100 These values are closer but still different from those obtained using separate subsamples The differences occur because the residuals from the combined model are different from those from the separate subsamples To use generalized least squares We use the estimated variances above to transform the model in the same Way as in 835 After doing so the regression results are with standard errors in parentheses W767 97052 12185EDUC 01328EDUC 15301METRO se 10485 00694 00150 03858 The magnitudes of these estimates and their standard errors are almost identical to those in equation 836 Thus although the variance estimates can be sensitive to the estimation technique the resulting generalized least squares estimates of the mean function are much less sensitive The regression output using White standard errors is W7GT 99140 12340EDUC 01332EDUC 15241METRO se 12124 00835 00158 03445 With the exception of that for METRO these standard errors are larger than those in part b re ecting the lower precision of least squares estimation Chapter 8 Exercise Solutions Principles of Econometrics 46 293 EXERCISE 816 8 b C d Separate least squares estimation gives the error variance estimates 6 2899215gtlt10 4 and 63 1536132gtlt10394 The critical values for testing the hypothesis H0 62 631 against the alternative H1 26 7 oil at a 5 level of signi cance are 7002515p15 0349 and 7097515p15 2862 The value of the F statistic is 4 A2 1536132gtlt104 25298 56 2899215gtlt10 Since 5298 gt 2862 We reject the null hypothesis and conclude that the error variances of the two countries Austria and Germany are not the same A2 GA The estimates of the coefficients using generalized least squares are estimated standard coef cient error B1 const 20268 04005 B2 lnINC 04466 01838 33 lnPRICE 02954 01262 34 lnCARS 01039 01138 Testing the null hypothesis that demand is price inelastic ie H0 2B3 21 against the alternative H1 33 lt 1 is a onetail ttest The value of our test statistic is t 02954 1 01262 558 The critical t value for a onetail test and 34 degrees of freedom is t00534 1691 Since 558 gt 1691 We do not reject the null hypothesis and conclude that there is not enough evidence to suggest that demand is elastic Chapter 8 Exercise Solutions Principles of Econometrics 46 294 EXERCISE 817 8 b C d The estimated regression is mmAWE 111195 003876SQFT100 001756AGE 00001734AGE2 se 0274 000087 000135 00000227 The residual plots are given in the gures below The absolute magnitude of the residuals increases as AGE increases suggesting heteroskedasticity with the variance dependent on the age of the house Conversely the absolute magnitude of the residuals appears to decrease as SQF T 1 00 increases although this pattern is less pronounced The variance might decrease as the house size increases but We cannot be certain Figure Xr817b 15 15 10 10 8 8 O 9 o g g 05 E g i E 05 O D E I 1 5 0 lt7 00 39 3 2 lt7 00 LIJ 0 Lu D 8 3 O E 8 0 o 8 05 0 E 9 05 o 0 8 0 I 2 is 10 9 10 6 1395 I I I I I I I I I I I I I I I 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 AGE SQFT100 Plot of residuals againstAGE Plot of residuals against SQF T I 00 We set up the model vare hOL1 012A GE OL3SQF T 100 and test the hypotheses II0oc2OOL3O H1oc20andoroL30 The test statistic value is X2 N XR2 1080gtlt 01082 116876 The critical chisquared value at a 1 level of signi cance is X89922 9210 Since 11688 is greater than 9210 We reject the null hypothesis and conclude that heteroskedasticity exists The estimated variance function is given as 612 eXp 47139 002177AGEl 0006377SQFT100l The robust standard errors for AGE and SQF T 1 00 are 000404 and 0006945 respectively Corresponding pvalues are 00000 and 03589 We can conclude that AGE has a signi cant effect on variance While SQF T 1 00 is not signi cant This conclusion agrees with our speculation from inspecting the gures in part b although in part b We did suggest the sign of SQF T 1 00 might be negative Chapter 8 Exercise Solutions Principles of Econometrics 46 Exercise 817 continued 6 f g The estimated generalized least squares model is 295 mzwRF12 11105 003881SQFT100 001540lGE 00001297IGEZ se 0024 000082 000136 00000272 b1 b2 b3 b4 i Least Squares 11120 003876 001756 00001734 0027 000087 000136 00000227 ii with HC standard errors 11120 003 876 001756 00001734 0033 000123 000175 00000372 iii GLS 11105 003881 001540 00001297 0024 000082 000136 00000272 iv with HC standard errors 11105 003881 001540 00001297 0028 000105 000144 00000314 The coef cient estimates from least squares and GLS are similar with the greatest differences being those for AGE and AGE2 The heteroskedasticityconsistent HC standard errors are higher than the conventional standard errors for both least squares and GLS and for all coefficients The conventional GLS standard errors are smaller than the least squares HC standard errors suggesting that GLS has improved the efficiency of estimation The GLS HC standard errors are slightly larger than the conventional GLS ones this could be indicative of some remaining heteroskedasticity The BreuschPagan test statistic obtained by regressing the squares of the transformed residuals on AGE and SQFT100 is X2 NXR2 1080gtlt0018169 1962 The 5 critical value is X89572 599 and the pvalue of the test is 00001 Thus We reject a null hypothesis of homoskedastic errors The variance function that We used does not appear to have been adequate to eliminate the heteroskedasticity Chapter 8 Exercise Solutions Principles of Econometrics 46 296 EXERCISE 818 a COKEU is a binary variable which assigns 1 if the shopper buys coke and zero otherwise Therefore the total number of shoppers who buy coke in store i is given by COKE U 3 and the proportion will be given by COK which is COKE I 42 b ECOKE E pi C01ltEjj J 2 I ECOKE 11 2 N N lpl pl I varCOKE 2varCOKElj 11 z varCOKEl zero covariance terms ll F1 l F1 NZ F1 F43 E 2 p391p39 R R 1121 I z ji 3 c pl is the population proportion of customers in store 139 who purchase Coke We can think of it as the proportion evaluated for a large number of customers in store i or the probability that a customer in store i will purchase Coke We can Write p 13 BZPRA T10 13015 COKE 134191510 PEPSI d The estimated regression is COKE 05196 006594PRA T10 008571DISPCOKE O1097DISPPEPSIl se 03207 031199 004671 00469 The results suggest that PRATIO and DISPPEPSI have negative impacts on the probability of purchasing coke although the coefficient of the price ratio is not significantly different from zero at a 5 signi cance level DISPCOKE has a positive impact on the probability of purchasing coke Both DISPPEPSI and DISPCOKE have significant coef cients if onetail tests and a 5 signi cance level are used Chapter 8 Exercise Solutions Principles of Econometrics 416 297 Exercise 818 continued 9 f 8 The null and alternative hypotheses are H 0 errors are homoskedastic H1 errors are heteroskedastic The test statistic is X2 NgtltR2 50gtlt0157747887 The critical chisquared value for the White test at a 5 level of signi cance is 2095 7 14067 Since 7887 lt 14067 We do not reject the null hypothesis There is insuf cient evidence to conclude that the errors are heteroskedastic The pvalue of the test is 0343 The variance of the error term is Va1il 1117 N I B1B2PRlTIOlB3DISP COKE B4DISPPEPSIl gtltB1B2PRATIOlB3DISP COKE B4DISPPEPSI NI The product in the above equation means that the variance will depend on each of the variables and their cross products Thus it makes sense to include the crossproduct terms when carrying out the White test It is surprising that the White test did not pick up any heteroskedasticity Perhaps the variation in pl is not suf cient or the sample size is too small for the test to be conclusive Or the omission of N could be masking the effect of the variables The estimated results are reported in the table below Maximum Minimum 05459 03385 Standard Deviation 0041 3 5 Mean 9 04485 The estimated GLS regression is COKE 05503 009673PRA T10 007831DISP COKE 01009DISPPEPSI se 03099 030205 004568 00449 The results are very similar to those obtained in part d both in terms of coef cient magnitudes and signi cance The coef cient of PRA T10 is a mild exception it is larger in absolute value than its least squares counterpart but remains insigni cant Given the relative importance of PRATIO this insigni cance is puzzling It could be attributable to the small variation in PRA T10 Chapter 8 Exercise Solutions Principles of Econometrics 416 298 EXERCISE 819 a The estimated least square regression with heteroskedasticityrobust standard errors is 1nVV Am 05297 O1272EDUC 006298EXPER OOOO7139EXPER2 se 02528 00170 001138 00000920 0001322EXPER gtlt EDUC 0000637 b Adding marriage to the equation yields 1nW 2F 05411 O1261EDUC 006137EXPER OOOO6933EXPER2 se 02542 00171 001159 00000956 0001309EXPER gtlt EDUC 00403MIRRIED 000063 8 003392 The null and alternative hypotheses for testing Whether married Workers get higher Wages are given by H0B6 O H1B6gtO The test value is 156 004029 1 r 188 seb6 000339 The corresponding pvalue is 01176 Also the critical value at the 5 level of signi cance is 1646 Since the test value is less than the critical value or because the p value is less than 005 We do not reject the null hypothesis at the 5 level We conclude that there is insuf cient evidence to show that Wages of married Workers are greater than those of unmarried workers c The residual plot EHKF I I I MARWED Figure xr819c Plot of least squares residuals against marriage The residual plot suggests the variance of Wages for married Workers is greater than that for unmarried workers Thus there is the evidence of heteroskedasticity Chapter 8 Exercise Solutions Principles of Econometrics 416 299 Exercise 819 continued d The estimated regression when MARRIED 1 is 1nVV39Am 09197 01008EDUC 005069EXPER 00007088EXPER2 se 03558 00222 001493 00001379 00004620EXPER gtlt EDUC 00007478 The estimated regression when MARRIED 0 is 1nVV Am 01975 01513EDUC 007284EXPER 00007014EXPER2 se 02945 00l94 00l271 0000l193 0002l45EXPER gtlt EDUC 0000654 T he G0ldfeld Qucmdt test The null and alternative hypotheses are H0 oi 6 against H1 6 7 6 The Value of the F statistic is i 021285 83 028658 0743 The critical Values are FLU 1110025 414576 0835 and FUC 170975414576 1194 Because 0743 F lt E6 0835 We reject H0 and conclude that the error Variances for married and unmarried women are different e The generalized least squares estimated regression is 1nVV Am 04780 01309EDUC 006452EXPER 00007128EXPER2 se 02212 00144 000932 00000862 0001443EXPER gtlt EDUC 0000484 There are no major changes in the Values of the coefficient estimates However the standard errors in the GLSestimated equation are all less than their counterparts in the least squaresestimated equation re ecting the increased ef ciency of least squares estimation Chapter 8 Exercise Solutions Principles of Econometrics 46 300 Exercise 819 continued f The marginal effect for a worker with 10 years of experience is given by 8ElnWAGE 8EDUC The estimate for the marginal effect calculated using the regression in part a is 13 B5EXPER 13 10B5 c 8ED U C Its standard error is seb 10b5 0011335 0127195 00013224gtlt10 011397 The estimate for the marginal effect calculated using the regression in part e is c 8ED UC 0478006 00014426 X 10 011643 Its standard error is se02 1005 0010193 The Ivalue for computing the interval estimates is re t0975995 1962 Thus the two interval estimates are as follows From the least squaresestimated equation in part a n1eJr tcseb1 10195 011397 i1962gtlt 0011335 00917 01362 From the GLSestimated equation in part e IT1 zcsef3 1005 011643 i1962gtlt 0010193 00964 01364 The interval estimate from the GLS equation is slightly narrower than its least squares counterpart but overall there is very little difference Chapter 8 Exercise Solutions Principles of Econometrics 46 301 EXERCISE 820 81 b C The residual plots against EDUC and EXPER are as follows 2 2 5 o o O O o o 6 0 0 0 1 E E 8 coo 8 0339 o 0 so 8 8gEo888 o E8 0 000 O 0 3 O 08g 0 0 0 0 8 3 l g Bz Ea g 8 o o 0 0 8 p 02 33 6 O 9 ego E g Q zoga9e 8 gooo D O g 1 0 ooggg 9009 o IEIKJ o 0 lt9 IlJJ s 83 8oog3 g o0 O 8 391 8 00 O 00 8 e 2 6 2 3 I I I I I 393 I I I I I I 0 4 8 12 16 20 24 0 10 20 30 40 50 60 70 EDUC EXPER Figure xr820 Residual polots against EDUC and EXPER Both residual plots exhibit a pattern in which the absolute magnitudes of the residuals tend to increase as the values of ED UC and EXPER increase although for EXPER the increase is not very pronounced Thus the plots suggest there is heteroskedasticity with the variance dependent on EDUC and possibly EXPER The null and alternative hypotheses are H 0 errors are homoskedastic H1 errors are heteroskedastic with H1 implying the error variance depends on one or more of EXPER EDUC or MARRIED The value of the test statistic is X2 NXR2 1000gtlt 001465 1465 The critical chisquared value at a 5 level of signi cance is x20953 7815 Since 1465 is greater than 7815 We reject the null hypothesis and conclude that heteroskedasticity exists The pvalue of the test is 00021 The estimated variance function is sf exp 30255 001391EDUC 000516EXPERl 004547MIRRIEDI The standard deviations for each observation are calculated by getting the square roots of the forecast values from the above equation The first ten estimates are presented in the following table Chapter 8 Exercise Solutions Principles of Econometrics 46 302 Exercise 820c continued Observation Standard deviation 027856 024957 026049 024982 027944 026470 027217 026745 027287 026123 399 F V 39gtquot d The generalized least squares estimated regression is 1nW71 05265 01274EDUC 006365EXPER 00007151EXPER2 se 02203 00144 000944 00000887 0001369EXPER gtlt EDUC 0000492 The least squares estimated equation with heteroskedasticityrobust standard errors is 1nVV Am 05297 01272EDUC 006298EXPER 00007139EXPER2 se 02528 00170 001138 00000920 0001322EXPER gtlt EDUC 0000637 The coef cient estimates in both equations are very similar However the standard errors in the GLSestimated equation are all less than their counterparts in the least squares estimated equation re ecting the increased ef ciency of least squares estimation Chapter 8 Exercise Solutions Principles of Econometrics 46 303 Exercise 820 continued 6 The marginal effect for a worker with 16 years of education and 20 years of experience is given by 8ElnWAGE WXPER 35 2B4EXPER B5EDUC 35 40B4 16B5 The least squares estimate for the marginal effect is c GED UC 0062981 40 X 00007l393 86 16gtlt 000l322388 0013265 Its standard error is se295 40294 16295 0002020 The generalized least squares estimate for the marginal effect is 8ED UC 0063646 40gtlt 00007151398 16 x 000136903 0013136 Its standard error is se295 40294 16295 0001898 The Ivalue for computing the interval estimates is re t409755995 1962 Thus the two interval estimates are as follows From the least squaresestimated equation me i tcse295 40294 16295 0013265 i1962 gtlt 0002020 000930 001723 From the GLSestimated equation in part d me i tCse295 40294 16295 0013136 i 1962 x 0001898 000941 001686 The interval estimate from the GLS equation is slightly narrower than its least squares counterpart but overall there is very little difference Chapter 8 Exercise Solutions Principles of Econometrics 46 304 EXERCISE 821 a Using the natural predictor the forecast Wage for a married Worker with 18 years of education and 16 years of experience is ME expO526482 Ol274l2gtlt 18 00636458gtlt 16 OOOO7l513983gtlt162 000136903402gtlt16gtlt18 26072 To compute the forecast using the corrected predictor We rst need to estimate the variance for a married worker with 18 years of education and 16 years of experience This estimate is given by 62 exp 3025504 00l39lgtlt 18 0005l605gtlt 16 00454734 00708577 Then the forecast from the corrected predictor is ME 675 exp s22 26072 x exp007085772 27012 b The 95 forecast interval is given by exp1nV tcsef exp3260868 1962 x x00708577 15464 43958 Chapter 8 Exercise Solutions Principles of Econometrics 46 305 EXERCISE 822 8 The estimated linear probability model is 1 06885 0001624L VR 005932REF O4816INSUR 003438RATE b se 021 12 0000785 0023 83 00236 000860 002377IMOUNT 000044l9CREDIT 00l262TERM 01283IRM 00l267 000020l8 000354 003 19 T he White test The null and alternative hypotheses are H 0 errors are homoskedastic H1 errors are heteroskedastic Under H1 We are assuming that the error variance depends on one or more of the explanatory variables their squares and their cross products The cross product terms are included because in the linear probability model varDELINQ UENT EDELINQ UENT x 1 EDELINQ UENT Where EDELINQUENT is a linear function of all the explanatory variables as expressed in the estimated equation The value of the test statistic is X2 Ngtlt R2 l000gtlt 021997 219974 The critical chisquared value for the White test at a 5 level of signi cance is x2095 40 55758 Since 219974 is greater than 55758 We reject the null hypothesis and conclude that heteroskedasticity exists The error variances are estimated using 1 x 1 The number of observations Where 21 is zero The number of observations Where S 0 is 135 The number of observations Where lt 001 is 15 8 Chapter 8 Exercise Solutions Principles of Econometrics 46 306 Exercise 822 continued 0 L VR REF INS UR RA TE AMOUNT CREDIT TERM ARM 1 000162 00593 04816 00344 00238 0000442 00126 01283 Ls 000078 00238 00236 00086 00127 0000202 00035 00319 ii 000162 00593 04816 00344 00238 0000442 00126 01283 LSHC 000068 00240 00304 00098 00145 0000207 00036 00277 iii 000159 00571 05016 00413 00258 0000382 00190 02089 lt001 000081 00211 00292 00082 00121 0000184 00041 00407 iv 000086 00327 04770 00204 00187 0000162 00065 00419 001 000038 00146 00297 00057 00099 0000118 00021 00140 V 000054 00267 05127 00002 00045 0000024 00018 00188 000001 000024 00105 04086 00048 00089 0000085 00018 00109 For most of the coef cients the least squares and generalized least squares estimates are similar providing the GLS estimates are obtained by discarding observations with variances less than 001 Moreover the standard errors from the rst three sets of estimates are suf ciently similar for the same conclusions to be reached about the signi cance of estimated coef cients an exception is AMOUNT Whose coef cient is not signi cantly different from zero in the least squares estimations The magnitudes of the coef cients change considerably when variances less than 001 or less than 000001 are set equal to one of these threshold values and the estimates are very sensitive to the threshold which is chosen In the extreme case Where variances less than 000001 are set equal to 000001 only two of the estimated coef cients are signi cantly different from zero In the other cases almost all of the 8 coef cients were signi cant Setting small and negative variances equal to a small number seems to be a practice fraught with danger It places very heavy Weights on a relatively few number of observations d LVR The estimated coef cient is 000086 This suggests that holding other variables constant a one unit increase in the ratio of the loan amount to the value of property increases the probability of delinquency by 000086 The positive sign is reasonable as a higher ratio of the amount of loan to the value of the property will lead to a higher probability of delinquency The coef cient of LVR is signi cantly different from zero at the 5 level REF The estimated coef cient is 00327 This suggests that holding other variables constant if the loan was for re nancing the probability of delinquency decreases by 00327 The negative sign is reasonable as re nancing the loan is usually done to make repayments easier to manage which has a negative impact upon the loan delinquency The coef cient of REF is signi cantly different from zero at the 5 level Chapter 8 Exercise Solutions Principles of Econometrics 46 307 Exercise 822d continued INSUR The estimated coef cient is 04770 This suggests that holding other variables constant if a mortgage carries mortgage insurance the probability of delinquency decreases by 04770 The negative sign is reasonable taking insurance is an indication that a borrower is more reliable reducing the probability of delinquency The coef cient of INS UR is signi cantly different from zero at the 5 level RATE The estimated coef cient is 00204 This suggests that holding other variables constant a one unit increase in the initial interest rate of the mortgage increases the probability of delinquency by 00204 The positive sign is reasonable as a higher interest rate will result in a higher probability of delinquency The coef cient of RATE is signi cantly different from zero at the 5 level AMOUNT The estimated coef cient is 00187 This suggests that holding other variables constant a one unit increase in the amount of the mortgage increases the probability of delinquency by 00187 The positive sign is reasonable because as the amount of the mortgage gets larger the borrower is more likely to face delinquency The coef cient of AMOUNT is not signi cantly different from zero at the 5 level CREDIT The estimated coef cient is 0000162 This suggests that holding other variables constant a one unit increase in the credit score decreases the probability of delinquency by 0000162 The negative sign is reasonable as a borrower with a higher credit rating will have a lower probability of delinquency The coef cient of CREDIT is not signi cantly different from zero at the 5 level TERM The estimated coef cient is 00065 This suggests that holding other variables constant a one yearincrease in the term between disbursement of the loan and the date it is expected to be fully repaid decreases the probability of delinquency by 00065 The negative sign is reasonable because given AMOUNT is constant the longer the term of the loan the less likely it is that the borrower will face delinquency The coef cient of TERM is signi cantly different from zero at the 5 level ARM The estimated coef cient is 00419 This suggests that holding other variables constant if the mortgage interest rate is adjustable the probability of delinquency increases by 00419 The positive sign is reasonable because with the adjustable rate the interest rate may rise above what the borrower is able to repay which leads to a higher probability of delinquency The coef cient of ARM is signi cantly different from zero at the 5 level CHAPTER 9 Exercise Solutions 308 Chapter 9 Exercise Solutions Principles of Ec0n0meIrics 46 309 EXERCISE 91 8 b If FFRATEZ 1 for I 1 234 then INVGWTH4 4 04FFRITE4 08FFRITE3 06FFRATE2 02FFRITE1 404x108x106x102x1 2 If FFR1TEt 15 for I 5 and FFR1TEt 1 forI6789 then For I 5 INVGWTH5 4 O4FFR1 TE5 O8FFRA TE4 06FFRA TE3 02FFRA TE 404gtltl508gtltl06gtltl02gtltl 18 For I 6 INVGWTH6 4 04FFRATE6 O8FFRATE5 06FFRATE4 O2FFRITE3 4 04gtlt1 08gtlt15 06gtlt1 02gtlt1 16 For I 7 INVGWTH7 4 04FFRATE7 O8FFRITE6 06FFRATE5 02FFRATE4 4 04gtlt1 08gtlt1 O6gtlt15 O2gtlt1 l7 For I 8 INVGWTH8 4 04FFRITE8 08FFRATE7 06FFRITE6 02FFR1TE5 404gtltl08gtltl06gtltl02gtltl5 19 For I 9 INVGWTH9 4 04FFR1 TE 08FFRITE8 06FFRA TE7 02FFRITE6 404gtltl08gtltl06gtltl02gtltl 2 Since FFRATE was increased from 1 to 15 in period 5 and then returned to its original level We use the impact and delay multipliers to examine the effect of the increase Using the notation B0 B1 B2 and 33 for the impact and delay multipliers and noting that the increase was 05 the effect of the increase in periods 5 6 7 and 8 is given by 05B005B1 05132 and 05133 respectively The estimates of these values are 02 04 03 and 01 Examining the forecasts given above We nd that relative to the initial value of IN VGWT H of 2 When I 4 IN VGWT H has declined by 02 04 03 and 01 in periods 5 6 7 and 8 respectively Thus our forecasts agree with the estimates We get from using the impact and delay multipliers Since the delay multiplier for period 4 is zero 34 0 IN VGWT H returns to its original level of 2 in period 9 Chapter 9 Exercise Solutions Principles of Econometrics 46 310 Exercise 91 continued 0 If FFRlTEt 15 fort 56789 then Fort5 INVGWTH5 4 04FFRATE5 08FFRITE4 06FFRITE3 02FFRATE2 404gtlt1508gtlt106gtlt102gtlt1 18 Fort6 INVGWTH6 404FFRITE6 08FFRATE5 06FFRITE4 02FFRITE3 404gtlt1508gtlt1506gtlt1 02gtlt1 14 Fort7 INVGWTH7 4 04FFRATE7 08FFRITE6 06FFRATE5 02FFRATE4 404gtlt1508gtlt1506gtlt1502gtlt1 11 Fort8 INVGWTH8 4 04FFRITE8 08FFRATE7 06FFRITE6 02FFRATE5 404gtlt1508gtlt1506gtlt15 02gtlt15 1 Fort9 INVGWTH9 4 04FFRITE9 08FFRITE8 06FFRATE7 02FFRITE6 404gtlt1508gtlt1506gtlt1502gtlt15 1 Since FFRA TE increased from 1 to 15 in period 5 and was then kept at its new level We use the impact and interim multipliers to examine the effect of the increase The impact and interim multipliers are B0 30 B1 B0 B1B2 and B0 B1B2 B3 for periods 5 6 7 and 8 respectively With an increase of 05 the estimated effects in periods 5 6 7 and 8 are given by 05190 02 05b0 b1 06 05b0 b1b2 09 and 05b0 b1b2 b3 1 Examining the forecasts given above We find that relative to the initial value of INVGWTH of 2 when t 4 IN VGWT H has declined by 02 06 09 and 1 in periods 5 6 7 and 8 respectively Thus our forecasts agree with the estimates We get from using the impact and interim multipliers The interim multipliers for i8 and t9 are the same as the total multiplier namely 1 and a value of IN VGWTH 1 becomes the new equilibrium value Chapter 9 Exercise Solutions Principles of Econometrics 46 311 EXERCISE 92 8 b 0 Overall advertising has a positive impact on sales revenue There is a positive effect in the current week and in the following two weeks but no effect after 3 weeks The greatest impact is generated after one week The total effect of a sustained 1 million increase in advertising expenditure is given by total multiplier 190 91 192 1842 3802 2265 7909 The null and alternative hypotheses are H0 3 0 against H1 3l 7 0 and the Ivalue is calculated from I 19 sebl for i 012 Relevant information for the signi cance tests is given in the following table The 5 and 10 critical values for a twotail test are t097599 1984 and t09599 1660 respectively The 5 and 10 critical values for a onetail test are t09599 1660 and t09099 1290 respectively We use to denote signi cance at a 10 level and to denote signi cance at the 5 level No implies a lack of signi cance We nd that 191 is signi cant for both types of test and for both signi cance levels 190 is only signi cant at the 10 level using a onetail test 192 is signi cant at the 10 level for a twotail test and signi cant at the 5 level using a one tail test Coefficient Standard Fvalue Twotail Onetail Error pvalue pVa1ue 190 11809 1560 0122 0061 191 14699 2587 0011 0006 192 11922 1900 0060 0030 Using tc t097599 1984 the 95 con dence interval for the impact multiplier is given by Z90 itc gtlt seb0 1842i1984gtlt 1181 05014185 The oneperiod interim multiplier is 90 191 18423802 5644 with standard error given by sebob1Jv Iv0gtv Iv1gt2ltFvTgt 13946216062gtlt 10406 1474 12141 The 95 con dence interval for the oneperiod interim multiplier is 19 bz gtltseb0 191 5644i1984gtlt1214 32358053 Chapter 9 Exercise Solutions Principles of Econometrics 46 312 Exercise 92c continued The total multiplier is 190 191 192 1842 3802 2265 7909 with standard error given by seb0 b1v a9gvEa9gv a922vU90Tm2c 1f52c9TET 139462160614214 2gtlt 104062gtlt 009842gtlt 10367 x10188 1009 The 95 con dence interval for the total multiplier is given by 190 191 b2JrtC gtlt seb0 191192 7909i1984gtlt1009 59079911 Chapter 9 Exercise Solutions Principles of Econometrics 46 313 EXERCISE 93 a For the rst allocation 87LE 6L 190rpm 19rpm bADI104 25341842gtlt 6 3802gtlt 1358 2265gtlt1313 4453 87LEm 6L 190rpm 19rpm b2ADI105 25343802gtlt 6 2265gtlt1358 5123 87LE08 6L 19rpm 1lone Z9IDKO6 2534 2265 x 6 3893 For the second allocation SALES106 6L 19rpm 1 rpm 1rpm 25343802gtlt13582265gtlt1313 3348 87LEm 6L 190rpm 19rpm b2ADI105 25341842gtlt62265gtlt1358 3947 87LE08 6L 19rpm 1lone Z9IDKO6 25343802gtlt6 4815 For the third allocation 87LE 6L 190rpm 19rpm 840 25341842gtlt23802gtlt13582265gtlt1313 3716 3907 6L 804014 b1ADI106 84010 25341842gtlt43802gtlt22265gtlt1358 4339 87LE08 6L 19rpm 1lone Z9IDKO6 25343802gtlt42265gtlt2 4508 Chapter 9 Exercise Solutions Principles of Econometrics 46 314 Exercise 93a continued b The total sales from each of the 3 allocations are 13469 12110 and 12563 respectively Thus the first allocation leads to the largest sales forecast over the 3 Weeks This outcome occurs because the first allocation allows time for the full effect of the 6 million expenditure to be realized The second allocation in which the marketing executive spends all 6 million in t 107 provides the highest sales revenue in t108 The coef cient for the rst lag is higher than the coef cients of the other lags suggesting that the effect of advertising on sales revenue is greatest one Week after the advertising expenditure is made The estimated variance of the forecast error f SILESIO8 SALES108 for the first allocation is varf 62 var6c 62 varb2 2 x 6 x COV6Lb2 238912559836gtltl42l4l2gtlt 0766l 429261 se f 42926 6850 The 95 con dence interval for the rst allocation is 87LE8 0 X se f 3893 i1984gtlt 6850 2534 5252 The estimated variance of the forecast error for the second allocation is 282 vE62 wQ2x6x 238912559836gtlt2160612gtlt 01317 811501 se f x81150 9008 The 95 con dence interval for the second allocation is 87LE8 6 X se f 4815 i1984gtlt 9008 30286602 The estimated variance of the forecast error for the third allocation is var f 62 var6c 42 varb1 22 varb2 2 x 4 x COV6Lb1 2x 2x COV6Lb2 2x 2x 4x covb2b1 23891 2559816gtlt 21606 4gtlt142148gtlt O1317 4gtlt O766116gtlt 10367 244989 se f x244989 4950 Chapter 9 Exercise Solutions Principles of Econometrics 46 315 Exercise 93b continued The 95 confidence interval for the third allocation is s7LE8 6 X se f 4508 1984 X 4950 3526 5490 The most favorable allocation is the second or the third If maximizing expected pro ts at t 108 is the objective then the second allocation is best However a risk averse marketing executive may prefer the third allocation because its expected profit is only slightly less than that for the second allocation and it has a much lower standard error of forecast error This is re ected in the forecast intervals where sales for the second allocation could be as low as 3028 whereas for the third allocation the lower limit of the forecast interval is 3526 Chapter 9 Exercise Solutions Principles of Econometrics 46 316 EXERCISE 94 8 b Using hand calculations i ii T Z a 016349 2 2 3T 01008 15436 A2 el 1 00653 tl Z The test statistic for testing H0p10 against the alternative H1p1 0 is Z TF1 M x 00634 0201 Comparing this value to the critical Z values for a two tail test with a 5 level of signi cance Z0025 196 and Z0975 196 We do not reject the null hypothesis and conclude that rl is not signi cantly different from zero The test statistic for testing H0 2p 0 against the alternative H1p2 70 is Z xTr E x 00653 0207 Comparing this value to the critical Z values for a 0025 196 and Z0975 196 We do not reject the null hypothesis and conclude that r2 is not signi cantly different two tail test with a 5 level of signi cance Z from zero 6 4 2 0 ljl ljl 2 4 5 1 I 2 The signi cance bounds are drawn at Jr196E i062 With this small sample the autocorrelations are a long Way from being signi cantly different from zero Chapter 9 Exercise Solutions Principles of Econometrics 46 317 EXERCISE 95 a The rst three autocorrelations are 250 1 Z D Gllc l G 1629753 2 04882 1Gt G 3338558 250 I Gr GGr 2 Gl 1124882 O 3369 2 39 2 Gt 52 3338558 39 39 t l 250 4Gr 39GGr 3 39G I 305802 0 0916 3 3 Gt g2 3338558 39 f To test Whether the autocorrelations are signi cantly different from zero the null and alternative hypotheses are H0 pk O and H1 pk 7 O and the test statistic is given by 2 xTr 15811413 At a 5 level of signi cance the critical values are i196 thus We reject the null hypothesis if lzk gt 196 The test results are provided in the table below Autocorrelations Zstatistic Critical value Decision 3 04882 7719 i 196 Reject H0 r2 03369 5327 i196 Reject H0 3 2 Q0916 1448 i 196 Do not reject H0 The signi cance bounds for the correlogram are Jr196x250 410124 It leads us to the same conclusion as the hypothesis tests Chapter 9 Exercise Solutions Principles of Econometrics 46 Exercise 95 continued b The least squares estimates for 91 and 5 are 250 ZGr 71Gr 1 51 162974 1t2 250 2 333 1119 04892 2 G14 G4 39 t2 8 G1 664 1662249 048925 gtlt 1664257 08480 318 The estimated Value is slightly larger than F1 because the summation in the denominator for F1 has one more squared term than the summation in the denominator for 9 The means are also slightly different Chapter 9 Exercise Solutions Principles of Econometrics 46 319 EXERCISE 96 81 b C A one percentage point increase in the mortgage rate in period I relative to what it was in period t 1 decreases the number of new houses sold between periods I and t 1 by 53510 units A 95 con dence interval for the coef cient of DIRA T EH is 19 6 X seb2 5351 i1971gtlt 1698 8698 2004 With 95 con dence we estimate that a one percentage point increase in the mortgage rate in period I relative to what it was in period t 1 decreases the number of new houses sold by a number between 20040 and 86980 The two tests that can be used are a ttest on the signi cance of the coef cient of 1 and the Lagrange multiplier test given by T gtltR2 The null and alternative hypotheses are H0 p 0 and H1p 7 0 The LMtest value is given by LMTgtltR2 2l8gtlt0l077 2348 The 5 critical value from a X31 distribution is 3841 Since the test statistic is greater than the critical value we reject the null hypothesis and conclude that there is evidence of autocorrelation Testing the signi cance of the coef cient of 1 we nd t 03306 0 5O9 00649 The 5 critical values are t097515 41197 since the Istatistic is less than 197 we reject the null hypothesis and conclude that there is evidence of autocorrelation The 95 con dence interval for the coef cient of DIRA T EH is given as fa itc gtlt sef3 5861i1971gtlt1410 8640 3082 Ignoring autocorrelation gave a lower value for the coef cient of interest and a slightly larger standard error resulting in a con dence interval with a similar lower bound but a larger upper bound When autocorrelation is ignored our inferences about the coef cient could be misleading because the wrong standard error is used Chapter 9 Exercise Solutions Principles of Econometrics 4e EXERCISE 97 3 19 Under the assumptions of the AR1 model corre ek pk Thus i ii iii i ii iii corree1 p 09 corree4 p4 094 06561 52 G3 1 5263 l p2 1 092 corree1 p 04 correle4 p4 044 00256 2 c V 1190 1 p 104 320 When the correlation between the current and previous period error is Weaker the correlations between the current error and the errors at more distant lags die out relatively quickly as is illustrated by a comparison of p4 06561 in part aii with p4 00256 in part bii Also the larger the correlation p the greater the Variance 6 as is illustrated by a comparison of 6 5263 in part aiii with G3 1190 in part biii Chapter 9 Exercise Solutions Principles of Econometrics 46 321 EXERCISE 98 a The forecasts for in ation are ITVTVWQ4 0100102354gtltlOOl2l3gtltO5Ol677gtltOl O28l9gtlt 03 O7902gtlt 02 04864 17vTv2Q 0lOOlO2354gtltO48640l2l3gtltlOOl677gtlt 05 O28l9gtlt Ol O7902gtlt 02 06060 ITVTVMQ2 01001 O2354gtlt 06060 01213 x 04864 01677 gtltlO O28l9gtlt O5 O7902gtlt 04 09265 b The standard errors of the forecast errors are For 2009Q4 f 63 0225103 6 047445 For 20lOQl 6 63 1 f 0225103 gtlt 1 023542 0237577 6 04874 For 20lOQ2 63 63 03 2 03 1 0225103 x 023542 012132 023542 1 0244606 6 04946 c The forecast intervals are 17vTFQ4 it x6 O4864i19897gtltO4745 04577 14305 097584 7 INF2010Q1it097581x 6 06060 i19897 x 04874 O3638 15758 7 INFMQ it097581 x 6 09265 i 19897 x 04946 00576 19106 These forecast intervals are relatively Wide containing both negative and positive values Thus the forecasts We calculated in part a do not provide a reliable guide to What in ation will be in those quarters Chapter 9 Exercise Solutions Principles of Econometrics 46 322 EXERCISE 99 a The ARDL model can be Written as 1 eL GL2 eL3 e4L y 55x 1 1 y 1 eL GL2 eL3 945 5 1 eL GL2 G3L3 945 5011 y on B11Igt1L B1L2 B1L3 B1L4x1 from which We obtain oc 1 eL eL2 eL3 945 5 1 91L eL2 eL3 e4L415 3 3L 3L2 3L3 1345 Thus 8 Q 1 e e e e4 and 5 1 eL eL2 eL3 G4L430 gt1L 3L2 33 1345 SOLO OL OL2 OL3 OL4 13 BIL 3L2 33L3 1345 e3L e3L2 933 91335 e3L2 e3L3 e3L4 91303 91315 9430L4 Equating coefficients of like powers in the lag operator yields B0 80 0 B0 80 31 911311 0 B1 911311 32 91131 911311 0 32 91131 921311 33 0182 G31 G330 0 gt 33 G13 G231G330 34 G183 GZBZ 0381 G430 O 34 GIB3 GZBZ 9331 G430 B5 G134 9233 G332 9431 0 B5 9134 9233 9332 9431 B5 GlBs l G2Bs 2 G3Bs 3 G4Bs 4 0 BS GlBs l G2Bs 2 G3Bs 3 G4Bs 4 Chapter 9 Exercise Solutions Principles of Econometrics 46 Exercise 99 continued b C 323 The estimated weights up to 12 lags and their graph are given below MULTIPLIERS Weight Estimate 0 0790 1 0186 2 0140 3 0188 4 0315 5 0173 6 0150 7 0162 8 0174 9 0135 10 0122 11 0120 12 0115 O l 2 5 4 5 8 9 10 11 1392 13 The multipliers are negative at all lags In absolute Value terms an unemployment change has its greatest effect immediately and then drops away quickly at lag 1 It increases again at lags 3 and 4 and then drops away again After that the effect is small although there is a slight increase at lag 8 The increases at lags 4 and 8 suggest a quarterly effect If the unemployment rate is constant in all periods then DU 0 in all periods and the estimated in ation rate is St 8 16 6 293 4 01001 210235401213 01677 02819 0517 Chapter 9 Exercise Solutions Principles of Econometrics 46 324 EXERCISE 910 8 b C The forecasts for DURGWTH are 2010Q1 00103 O1631gtlt 01 07422gtlt 06 03479gtlt 09 07524 2010Q2 00103 O1631gtlt 07524 07422gtlt 08 03479gtlt 06 06901 Since this model has the same lags as the example in Section 98 of POE4 the formulas given in that section for the lag weights are relevant They are B0 Z 50 B1 2 51 9130 Bs Z elBs l The lag weights for up to 12 quarters are as follows 522 Estimate 07422 02268 00370 00060 98gtlt10 4 l6gtlt10 4 26gtltl0 5 43gtltl0quot6 69 x 1077 llgtltl0quot7 l9gtltl0quot8 30gtltl0quot9 49gtltl010 T 2 00 S3 Oo1OU1lgtL Jlgt ltD The one and twoquarter delay multipliers are A an URGWTHl 01 g 02268 6INGRWTHH 32 6DURGWTHl 2 00370 7 a1NGR WTHl2 These values suggest that if income growth increases by 1 and then returns to its original level in the next quarter then growth in the consumption of durables will increase by 0227 in the next quarter and decrease by 0037 two quarters later Chapter 9 Exercise Solutions Principles of Econometrics 46 325 Exercise 910c continued The one and twoquarter interim multipliers are 0 3 07422 02268 0969 30 0 0 0969 00370 0932 These Values suggest that if income growth increases by 1 and is maintained at its new level then growth in the consumption of durables will increase by 0969 in the next quarter and increase by 0932 two quarters later Since the coef cients in the table in part b become negligible by the time lag 12 is reached the total multiplier can be obtained by summing all the coefficients in that table Doing so yields 8 0 09373 0 J This Value suggests that if income growth increases by 1 and is maintained at its new level then at the new equilibrium growth in the consumption of durables will increase by 0937 Chapter 9 Exercise Solutions Principles of Econometrics 46 326 EXERCISE 911 a To Write the AR1 in lag operator notation We have ex Pez 1 V et pet 1 Vt 1 pLe V 0 Since 1 pL1 PL 1 we can show that 1 pL 1 1 pL p2L2 p3L3 by showing 1 pL1pLp2L2 p3L3 1pLp2L2 p3L3 pLp2L2 p3L3 1 Thus we have 1 pLe V e 1 pL391v 1 pL p2L2 p3L3 v v ptH p2v p3v3 Chapter 9 Exercise Solutions Principles of Econometrics 46 327 EXERCISE 912 3 Coef cient Estimates and AIC and SC Values for Finite Distributed Lag Model q0 q1 q2 q3 q4 q5 q6 31 04229 05472 05843 05828 06002 05990 05239 30 03119 02135 01974 01972 01940 01940 01830 31 01954 01693 01699 01726 01728 01768 32 00707 00713 00664 00662 00828 03 00021 00065 00062 00192 134 00222 00225 00475 85 00015 00169 86 00944 AIC 31132 34314 34587 34370 34188 33971 34416 AIC 02753 05935 06208 05991 05809 05592 06037 SC 30584 33492 33490 32999 32543 32052 32223 SC 02205 05113 05111 04620 04165 03673 03844 Note AIC AIC 1 ln27c and SC SC 1 ln27c The AIC is minimized at q 2 While the SC is minimized at q 1 b i A 95 confidence interval for B0 is given by 00 0975D88se1 0 01974 1987 X 00328 02626 0 1322 ii The null and alternative hypotheses are H0303132 05 H1303132gt 05 The test statistic is 90 191 92 05 0062656 t 1815 seb0 191 192 0034526 The critical value ist 1662 Since t1815gt1662 We reject the null 09588 hypothesis and conclude that the total multiplier is greater than 05 The pvalue is 00365 iii The estimated normal growth rate is GN 058427 0437344 1336 The 95 con dence interval for the normal growth rate is GN r097588sec N 1336 1987 X 00417 12531419 Chapter 9 Exercise Solutions Principles of Econometrics 46 328 EXERCISE 913 a The graphs for SALES and ADV follow Both appear not to be trending and both uetuate around a constant mean SALES 34 32 ll W 11 7111 1 IV Il HN V 22 26 24 ADV 14 13 in 1111 W1 11 1111 2quot Will 39 V 07 O6 39 39 39 39 I 39 39 39 39 I 39 39 39 39 I 39 39 39 39 I 39 39 39 39 I 39 39 39 39 I 25 50 75 100 125 150 b Lag SC SC 1 ln27r Total Multiplier 0 05949 3433 6020 1 04269 3265 7275 2 03756 3214 8067 3 03736 3211 8634 4 04015 3239 8863 5 04288 3267 8595 The total multiplier is sensitive to lag length up to lag 3 for lag 3 and longer lags there is little Variation Chapter 9 Exercise Solutions Principles of Econometrics 4e 329 Exercise 913 continued c Of the six possible lag lengths the SC reaches a minimum when the lag length equals three The estimates for this lag length appear below The lag structure is such that the greatest impact from advertising on sales is felt immediately and the lag Weights decline thereafter with the exception of the Weight at lag 3 which is greater than that at lag 2 The declining lag Weights are sensible We expect the effect of advertising to diminish over time however the increase at lag 3 is not expected The lag Weight at lag 2 is not signi cantly different from zero at a 5 level the remaining lags Weights are signi cant Dependent Variable SALES Method Least Squares Date 05727711 Time 0816 Sample 6 157 Included observations I52 Variable Coef cient Std Error t Statistic Prob C 1921618 0688875 2789502 00000 ADV 2756383 0805025 3423970 00008 AD u1 2473420 0997601 2479388 00143 AD uquot 2 1526656 1019415 1497581 01364 AD w393 1877676 0819567 2291050 00234 d i The oneweek delay multiplier is 6SILES 1 24734 8ID VH The 95 con dence interval for the oneweek delay multiplier is b iz04seb 24734 il976gtlt 09976 o5o24445 ii OneWeek interim multiplier 90 bl 27564 24734 52298 The 95 con dence interval for oneWeek delay multiplier is 90 b1it0975D147seb0 bl 52298 i 1976 X 08249 3600 6860 iii TwoWeek delay multiplier as7LFs 15267 6lDK The 95 con dence interval for the twoWeek delay multiplier is 6 it seb 15267 i1976gtlt 10194 O4883541 097 Chapter 9 Exercise Solutions Principles of Econometrics 46 330 Exercise 913d continued 6 0 iv TwoWeek interim multiplier 911 191 92 27564 24734 15267 67565 The 95 con dence interval for the twoWeek interim multiplier is b0 b1 Z92 i t119751471seb11 b1 b2 67565 i 1976 x 08387 50998414 A 1 million increase in advertising expenditure in each Week will increase sales by 311 in the first Week by B11 131 in the second Week and by 311 131 132 in the third Week Thus the total increase over 3 Weeks is 31311 2131 132 Its estimate is 1911 1911 191 1911 191 191 27564 52298 67565 14743 with se3b11 2191 192 17035 We wish to test H113B11 2131 132 S 6 versus H1 231311 2131 132 gt 6 The value of the Istatistic is t 147426 6 17035 1655 We reject H11 and conclude that the CEO s strategy will 513 Since 513 gtt10951471 increase sales by more than 6 million over the 3 Weeks The estimated equation is S 12 19216227564ADV1 24734ADV1115267ADV1 18777ADV13 For forecasting 1 2 3 and 4 Weeks into the future We set t158159160 and then 161 The required sample values ofADV are ADK55 0889 ADK56 0681 ADK57 0998 The forecast values for each part are presented in the table below Forecast Values millions t158 t159 t160 t161 i 24394 22018 21090 19216 ii 35419 31912 27197 26727 iii 27150 27248 27847 27850 In the rst set of forecasts SALES gradually declines as the effect of the advertising expenditure during the sample period Wears off with the forecast in the last period equal to the intercept In the second set of forecasts the large initial expenditure on advertising leads to a large initial increase in SALES which then declines over the forecast horizon Having a uniform expenditure of 1 million in each year leads to SALES that are more uniform and which achieve a value equal to the intercept plus the total multiplier in the final period 27850 19216 8634 Chapter 9 Exercise Solutions Principles of Econometrics 46 331 EXERCISE 914 8 The estimated model is lnIRE1 38241 07746 lnPRICE 02175lnPRICE1 00026 lnPRICE2 se 01006 03129 03185 03221 058681nPR1CE 00143lnPRICE4 03153 O2985 The interim and delay elasticities are reported in the table below Lag Delay Elasticities Interim Elasticities 0 07746 07746 1 02175 05572 2 00026 05546 3 05868 11414 4 00143 11271 Only 190 the coef cient of lnB is signi cantly different from zero at a 5 level of signi cance All coef cients for lagged values of lnB namely bl 192193194 are not signi cant at a 5 level This result is symptomatic of collinearity in the data When collinearity exists least squares cannot distinguish between the individual effects of each independent variable resulting in large standard errors and coef cients which are not signi cantly different from zero Interpreting the delay multipliers if the price is increased and then decreased by 1 in period t there is an immediate increase of 077 in area planted In period t1 that is one period after the price shock there is a decrease in area planted of 022 In period t2 there is practically no change in the area planted In period t3 there is an increase in area planted by 059 and in period t4 there is a decrease of 001 The interim multipliers represent the full effect in period tS of a sustained 1 increase in price in period 1 Thus if the price increases by 1 in period 1 there is an immediate increase in the area planted of 077 The total increase when period t1 is reached is 056 at period t2 it is 055 at period t3 it is 114 and after t4 periods there is a 113 increase The different signs attached to the delay multipliers the relatively large Weight at t 3 and the interim multipliers that decrease and then increase are not realistic for this example The pattern is likely attributable to imprecise estimation Chapter 9 Exercise Solutions Principles of Econometrics 46 332 Exercise 914 continued b Using the straight line formula the lag Weights are 30 oco i 0 B1 oco oz 139 1 B2 oco 20L 139 2 B3 oco 30L 139 3 B4 oco 40L 139 4 Substituting these Weights into the original model gives lnAREA oc OL0lI1PRICEtOL0 OL1lI1PRCEl1OL0 2oc1lnPRICE2 oco 3oc1lnPRICE3 OL0 4oc1lnPRICE4 e or oco lnPRICE lnPRICE1 lnPRICE2 lnPPRICE3 lnPRICE4 ocl lnPRICE1 2lnPRICE 3 lnPRICE3 41nPR1CE e 0L OL0Zt0 oclz e Where 2 lnPRICEl lnPRICEl1 lnPRICE2 lnPPRICE3 lnPRICE4 2 lnPRICEl1 2 lnPRICE2 3 lnPRICE3 4 lnPRICE4 c The least square estimates for OL0 and ocl are 610 O4247 and a1 OO996 d The estimated lag Weights are 0 6 042467 3 a0 6 042467 009963 03250 fs 6 26 042467 2 X 009963 02254 3 6 36 042467 3 X 009963 01258 4 6 46 042467 4 X 009963 00261 These lag Weights satisfy expectations as they are positive and diminish in magnitude as the lag length increases They imply that the adjustment to a sustained price change takes place gradually with the biggest impact being felt immediately and with a declining impact being felt in subsequent periods The linear constraint has xed the original problem Where the signs and magnitudes of the lag Weights Varied unexpectedly Chapter 9 Exercise Solutions Principles of Econometrics 46 333 Exercise 914 continued e The table below reports the delay and interim elasticities under the new equation Lag Delay Elasticities Interim Elasticities 0 04247 04247 1 03250 07497 2 02254 09751 3 01258 11009 4 00261 11270 These delay multipliers are all positive and steadily decrease as the lag becomes more distant This result compared to the positive and negative multipliers obtained earlier is a more reasonable one It is interesting that the total effect given by the 4year interim multiplier is almost identical in both cases and the 3year interim multipliers are very similar The earlier interim multipliers are quite different however with the restricted weights leading to a smaller initial impact Chapter 9 Exercise Solutions Principles of Econometrics 46 334 EXERCISE 915 The leastsquares estimated equation is 8 b C d 1111If1 38933 07761lnPRICE 00613 02771 00624 03 782 least squares se39s HAC se39s The correlogram for the residuals is 87 W H H U H UUHH I I I I I I I I I I I I I I I I I I I 2 4 6 8 10 12 14 16 18 20 22 24 The signi cant bounds used are Jr196x3 4 i0336 Autocorrelations 1 and 5 are signi cantly different from zero The null and alternative hypotheses are H0 p 0 and H0 p 7 0 and the test statistic is LM 54743 yielding a pvalue of 00193 Since the pvalue is less than 005 We reject the null hypothesis and conclude that there is evidence of autocorrelation at the 5 percent signi cance level The 95 con dence intervals are i Using least square standard errors gtlt seI92 07761 i 20369gtlt 02775 02109l34l3 72 i t097532 ii Using HAC standard errors 92 it gtlt seI92 07761 i 20369gtlt 03782 0005715465 097532 The Wider interval under HAC standard errors shows that ignoring serially correlated errors gives an exaggerated impression about the precision of the leastsquares estimated elasticity of supply The estimated equation under the assumption of AR1 errors is l 38988 08884lnPRICE se 00922 02593 e 04221e1 V 01660 Chapter 9 Exercise Solutions Principles of Econometrics 46 335 Exercise 915d continued 6 The Ivalue for testing whether the estimate for p is signi cantly different from zero is t 04221 01660 2542 with a pvalue of 00164 We conclude that 8 is signi cantly different from zero at a 5 level A 95 con dence interval for the elasticity of supply is 92 it 1gtlt seb2 08884 i 20423 x 02593 03588l4l79 097530 This con dence interval is narrower than the one from HAC standard errors in part c re ecting the increased precision from recognizing the AR1 error It is also slightly narrower than the one from least squares although we cannot infer much from this difference because the least squares standard errors are incorrect We write the ARDL11 model as lnAREA1 8 01 lnAREA11 811 lnPRICE1 81 lnPRICE11 e1 The estimated model is m1ze g 23662 04043 lnAREA11 07766lnPRICE1 06109lnPRICE11 06557 01666 02798 02966 For this ARDL11 model to be equal to the AR1 model in part d we need to impose the restriction 81 0181 Thus we test H11 81 0180 against H1 81 7 0181 The test value is 1 81 0180 O6109 04043 x 07766 A AA l0559 se810180 02812 with pvalue of 0300 Thus we fail to reject the null hypothesis and conclude that the two models are equivalent The correlogram presented below su ests the errors are not serially correlated The signi cance bounds used areJr19635 03412 The LM test with a pvalue of 0423 con rms this decision 1 11 11011101 4 H 4llllll 1012141618 20 22 24 Chapter 9 Exercise Solutions Principles of Econometrics 46 336 EXERCISE 916 8 b C The forecast values for lnARE11 in years T 1 and T 2 are 404899 and 382981 respectively The corresponding forecasts for AREA using the natural predictor are Z71tE11 eXp404899 5734 2TtE1 eXp382981 4605 Using the corrected predictor they are 27371 2FaE711exp622 573395 gtlt eXp028489922 5971 2TEl1 1 27371 exp622 460539gtlt eXp028489922 4796 The standard errors of the forecast errors for lnAREA are seu1 6 028490 seu2 em 028490 1 0404282 03073 The 95 interval forecasts for lnAREA are T1 4111019751291 gtlt seu1 404899 20452 x 028490 3 4663 463 167 T2 4171019751291 gtlt seu 382981 20452 x 03073 320132 445830 The corresponding intervals for AREA obtained by taking the exponential of these results are For 71 e3 46630e43963167 320210269 For 72 e3392 me4394583 24568634 The lag and interim elasticities are reported in the table below Lag BS Lag Elasticities Interim Elasticities 0 B11 50 07766 07766 1 B1 51 011311 02969 04797 2 32 01131 01200 03597 3 33 01B1 00485 03112 4 B1 91133 196 02916 The lag elasticities show the percentage change in area sown in the current and future periods when price increases by 1 and then returns to its original level The interim elasticities show the percentage change in area sown in the current and future periods when price increases by 1 and is maintained at the new level Chapter 9 Exercise Solutions Principles of Econometrics 46 337 Exercise 916 continued d The total elasticity is given by EB 8 8 O77663 O61086 O 2783 0 139 1 1 1 O40428 39 If price is increased by 1 and then maintained at its new level then area sown will be 028 higher when the new equilibrium is reached Chapter 9 Exercise Solutions Principles of Econometrics 46 338 EXERCISE 917 8 b The estimated model is E O7316O4249G 01332G se 00633 00636 The correlogram of the residuals is shown below The signi cance bounds are drawn at i196x248 i01245 There are a few signi cant correlations at long lags speci cally at lag orders 5 9 10 11 and 19 but apart from lag 5 they are relatively small 15 10 HHU H H x20 8 I10 3915 I1 3916 I16 iO 5239 4 2 4 6 The test value for the LM test with two lags is LM 7405 and the corresponding pvalue is 00247 Since the pvalue is less than 005 we reject the null hypothesis that autocorrelation does not exist and conclude that there is evidence of autocorrelation at the 5 signi cance level The estimated model is at 08386 04432G1 01995G2 01533G3 se 00627 00676 00635 The correlogram of the residuals is shown below The signi cance bounds are drawn at i196x247 i01247 There are two signi cant correlations at the long lags of 10 and 16 but they are relatively small Chapter 9 Exercise Solutions Principles of Econometrics 46 339 Exercise 917b continued 0 15 10 O5 U H U N IJHU 10 3915llIllIllIllllll 2 4 6 8 10 12 14 16 18 20 22 24 The test value for the LM test with two lags is LM 0916 and the corresponding p value is 0632 Since the pvalue is greater than 005 We do not reject the null hypothesis of no autocorrelation We conclude there is no evidence of autocorrelation at the 5 signi cance level The results are presented in the table below The tvalue used to compute the forecast intervals was t0975j247 19696 Period Forecasts Standard Errors Forecast Intervals Actual Figures 2009Q4 13371 09899 0613 3287 115 2010Q1 16214 10827 0511 3754 118 2010Q2 17014 11515 0567 3969 0914 The actual gures fall Within the intervals Chapter 9 Exercise Solutions Principles of Econometrics 46 340 EXERCISE 918 a The estimated AR2 model is s7Ls 1 1614 03946SALES1 01926SALES The correlogram below shows no evidence of serially correlated errors LM tests at various lags similarly show no evidence of serial correlation Auto correlation AC 0006 0025 0080 0058 0004 0044 0125 0034 0055 0131 0040 0045 sx OOOquotJO39U1 3LOINJ x R3 b to e The following table contains the oneperiod ahead forecasts and forecast errors for both the AR2 and exponential smoothing models after reestimating both models for each period Both methods tend to over or under forecast at the same time In two periods the absolute value of the forecast error is lower for exponential smoothing and in the other two periods the forecast errors for the AR2 model are smaller AR2 Exp Sm Forecast Observed AR2 Exp Sm Forecast Forecast Period Value Forecast Forecast Error Error 154 28963 282011 283925 07619 05705 155 26430 285364 286896 21064 22596 156 25900 276452 275187 17452 16187 157 28020 269021 266542 11 179 13658 f The meansquare prediction errors for each set of forecasts is MSPEAR2 2328 MSPEExp sm 2479 Using this criterion the AR2 model has led to the more accurate forecasts Chapter 9 Exercise Solutions Principles of Econometrics 46 341 EXERCISE 919 8 120 160 200 b 0 1400 1200 1000 800 600 400 200 120 The four graphs are as follows HOMES IRATE 11 11 1992 1994 1996 1998 2000 2002 2004 2006 2008 1992 1994 1996 1998 2000 2002 2004 2006 2008 DHOMES DIRATE 80 40 40 30 1111 139939939 39quotH 39 i39quotH 39 39quotH2355quot39Eb39ddquot39Eb39d 39quotEb39dA39quot b39d quot39 b39d quotquot 13993993923939 39 39 391399399394quot 39 39 39139939939equot 39 39 39139939939s39 39 39 39 39o39o39o39 39 39 39 39o39o39239 39 39 39 39o39o39i 39 39 39 39o39o39e39 39 39 39 39o39o39squot The series for HOMES and RA TE exhibit trends HOMES trends upwards until 2005 and then trends downwards IRATE wanders up and down but overall trends downwards On the other hand the series for DHOMES and DIR1 TE do not appear to be trending but uctuate around constant means The estimated model is 24912 O3350DH0MES1 5O7878DIRATE1 288550D1RATE2 se 33327 00649 169283 171278 All estimates except for the intercept and DIRlTE2 are signi cantly different from zero at the 5 level The test statistic for testing H0 0151 52 against the alternative H0 0151 7 52 is 118408 2 0615 192621 The 5 critical value is t0975p212i1971 and the corresponding pvalue is 05394 Since the pvalue is greater than 005 we do not reject the null hypothesis and conclude that the data are compatible with the hypothesis H 0 0151 52 Chapter 9 Exercise Solutions Principles of Econometrics 46 342 Exercise 919c continued d 6 f If H0 is true the model can be written as DHOMES 8 01DHOMES1 81DIRATE1 0181DIRA TE2 V which is equivalent to the AR1 error model DHOMES 5 5101124 TEH e e 0164 v The correlogram of residuals is displayed below The signi cance bounds are i196 216 410133 It suggests that there are two signi cant correlations at lags at 5 and 21 20 I I I lll 8 10 12 14 16 18 20 22 24 The LM X2 test value with two lagged errors is 48536 with a corresponding pvalue of 00883 At a 5 signi cance level we fail to reject the null hypothesis that the errors are serially uncorrelated If we used a 10 signi cance level we would conclude there is evidence of serial correlation The estimated ARDL model is zfozfs 29215 O3O73DHOMESH O2O69DHOMES5 se 32841 00635 00633 64324DIRA TEH 46631DIRATE3 15974 16094 Using the signi cance bounds Jr196x213 i01343 the correlogram of residuals for this model does not suggest any autocorrelation except at lag 21 which is suf ciently distant to ignore Also the AIC and SC values for this model are slightly lower than those for the model in 992 And there are no coef cients except the constant that are not signi cantly different from zero In 992 the coef cient of DIR1TE2 was not signi cant These four things the lack of serial correlation the improved AIC and SC the exclusion of a lag with an insigni cant coef cient and the inclusion of signi cant lags lead us to conclude the new model is an improvement Chapter 9 Exercise Solutions Principles of Econometrics 46 343 EXERCISE 920 a Recognizing that DHOMES HOMES HOMESH We can Write the equation as HOMES HOMESH 5 91HOMES1 HOMESt2 95DHOMESt5 SODIRA TEH 53DIRA TEt3 V Rearranging yields HOMES 5 GIHOMESH 91HOMESt2 HOMESH 95DHOMESt5 ESODIRA T EH 83DIRA T Et3 V 5 91 1HOMES1 91HOMES2 95DHOMES5 ESODIRA T EH 83DIRA T Et3 V b The estimated equation is zfozfs 29215 O3O73DHOMESH O2O69DHOMES5 se 32841 00635 00633 64324DIRATE1 46631D1RATE 15974 16094 The equation to be used for forecasting is 29215 O6927HOMES1 O3O73HOMES2 O2O69DHOMES5 64324DIRA T EH 46631DIRA T E3 The forecasts for April May and June 2010 are 1 1EAR 29215 O6927gtlt 411 03073 x 324 O2069gtlt 38 64324gtlt 002 46631gtlt 01 370 MAY 29215 06927 x 370 03073 x 411 O2069gtlt 9 64324gtlt 00 46631gtlt 004 380 JUNE 29215 06927 x 380 03073 X370 O2069gtlt 15 64324gtlt 00 46631gtlt 002 372 Chapter 9 Exercise Solutions Principles of Econometrics 46 344 Exercise 920 continued c The standard errors of the forecast errors are seu1 6 47502 A quot 2 12 2 12 seu61e1 475o2106927 57785 seu3 6U9112 912 112 1 12 47502069272 03o732 069272 1 68827 12 The three forecast intervals are 1 r3EA z0508 gtlt seu3 370 i1971gtlt 47502 276 464 HOMESMAY i t0975208 gtlt seu2 380 i 1971 X 57785 266 494 JUNE z0508 gtlt seu3 372 i1971gtlt 68827 236508 Chapter 9 Exercise Solutions Principles of Econometrics 46 345 EXERCISE 921 a The estimated equation is 13 03870 03501DU 01841G 00992G se 0058700846 00307 00368 b The residual correlogram for lags up to 24 is presented below No serious problems of error autocorrelation are apparent The only slightly signi cant autocorrelation is at lag 13 The signi cance bounds used are i1967 4102 393 I I I I I I I I I I I I I I I ll l 2 4 6 81012141618 20 22 24 c The following table gives the LM test results for lags up to 4 In all cases the pvalues are greater than 01 Using any significance level up to 10 We conclude there is no evidence of serial correlation in the errors Lags X2 value pvalue 1 0170 0680 2 0271 0873 3 3896 0273 4 6141 0189 d i The estimated model with DU2 added is 13 03742 03230DU 00458DU 018236 00971G se 00586 01060 00990 00314 00374 ii The estimated model with G2 added is 15 03876 03391DU 0183261 00991G 00082G se 0072000979 00311 00370 00360 Chapter 9 Exercise Solutions Principles of Econometrics 46 346 Exercise 921d continued 6 iii The estimated model with both DU2 and G2 added is 13 03778 03208DU 00429DU 018216 00970G 00030G se 0075801103 01065 00316 00376 00389 For all three estimated equations the coef cient estimates found to be signi cant at the 5 percent level were those for DU G and GH Whenever DU2 or G2 or both t 19 were added to the original equation their estimated coef cients were insigni cant In parts b and c We concluded that error autocorrelation is not signi cant Both the correlogram and the LM tests supported such a conclusion Also in part d adding DU2 andor G2 did not improve the model Their coef cients were not signi cantly different from zero For these reasons We conclude that the Olltun s law speci cation given in 959 is satisfactory Chapter 9 Exercise Solutions Principles of Econometrics 46 347 EXERCISE 922 8 b os wv I nIi 39Ai II 39 0 W vl 3quotquotquotquotquotquot 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 quot 2IIIIIIllllllll The times series graphs for CONGWT H and INCGWT H follow While both exhibit considerable serial correlation they do appear to uctuate around their respective constant means CONGWTH INCGWTH 0 Mi il4i miVlilHu l q 1 I1 I I I VrI ny Iii 39H 3 A AA 2A 1 1 llllllllllllllllllllll 25 50 75 100 125 150 175 200 25 50 75 100 125 150 175 200 The estimated model is 09738 04496 INCGWTH se 009 oo497 The estimate 80 04496 suggests that a 1 increase in the income growth rate increases the consumption growth rate by 046 The correlogram below shows signi cant serial correlation in the errors at lag 2 There is also some slight evidence of serially correlated errors at some longer lags 6 10 and 11 For the LM test we nd Xfz 2193 with a pValue less than 000005 a strong indication of serially correlated errors Autocorrelation AC III 1 0025 I 1 2 0327 I III 3 0084 I III 4 0091 I III 5 0102 I II 5 0143 I ll 7 0046 I ll 8 0071 I 3 9 0128 I II 10 0153 I21 11 0145 I III 12 0084 Chapter 9 Exercise Solutions Principles of Econometrics 46 348 Exercise 922 continued C d The estimated model after adding CONGWT H H is 06716 02714C0NGWTH 035011NCGWTH se 01188 00635 00530 The estimate 01 02714 is signi cantly different from zero at the 5 signi cance level t 427 The AIC and SC values for this model are 01250 and 00750 respectively compared to 00452 and 00119 for the model discussed in part b the lower values suggest this model is an improvement The corresponding EViews AIC and SC values are 20197 and 20697 for the above model and 20995 and 21328 for the model in part b See footnote 12 on page 366 of POE4 However the correlogram of the residuals displayed below suggests there is still signi cant serial correlation in the errors at lags 1 and 2 The LM test also rejects the null hypothesis that the errors are not serially correlated Ex 3445 pvalue 00000 Autocorrelation AC I3 I 1 0255 I II 2 0255 I I 34 0014 I I 4 0012 I HI 5 0048 I DI 0 0108 I I 7 0005 I I 8 0010 I DI 9 0089 I BI 10 0077 IDI 11 0087 I 1 12 0038 We conclude that the model is an improvement over that in part b but it is still not satisfactory The estimated model after adding C ONGWT H H is 04249 01594C0NGWTH 02806C0NGWTH 032161NCGWTH se 01254 00653 00615 00509 The estimate 02 02806 is signi cantly different from zero at the 5 signi cance level t 457 The AIC and SC values for this model are 02174 and 01508 respectively compared to 01250 and 00750 for the model discussed in part c the lower values suggest this model is an improvement The corresponding EViews AIC and SC values are 19273 and 19940 for the above model and 20197 and 20697 for the model in part c Chapter 9 Exercise Solutions Principles of Econometrics 4e 349 Exercise 922d continued 6 In the correlogram of the residuals given below the first autocorrelation is signi cantly different from zero although its magnitude r1 0143 is not large The LM test gives a 9amp2 value of 1546 with corresponding pvalue 00004 suggesting that serially correlated errors are still a problem Auto correlation AC 0143 0014 0021 0092 0018 0104 I 0045 U 0071 ll 0057 D 10 0079 ll 11 0062 139 12 0035 DI LOOD JO7l J J LJfJ i We conclude that adding CONGWTHH has improved the model but the existence of serially correlated errors means that it is still not satisfactory The estimated model after adding INCGWT H H is 03320 00233C0NGWTH 02101CONGWTH se 01219 00699 00610 03493INCGWTH 023341NCGWTH 00491 00539 The estimate 31 02334 is significantly different from zero at the 5 significance level t 433 The AIC and SC values for this model are 03004 02170 respectively lower than that for the model discussed in part d The EVieWs values are 18444 and 19277 Auto correlation AC 0014 0014 0009 0160 0043 0045 0085 0070 Eli 0075 39 D39 0078 ll 11 0044 I I 12 0022 I I I I I I jI I II I In 3 C39ltOm39JOU1vF LJf3 5 Chapter 9 Exercise Solutions Principles of Econometrics 46 350 Exercise 922e continued 0 g The correlogram above shows a signi cant but not large autocorrelation at lag 4 However performing the LM test with 2 and 4 lags gives X32 0220 pvalue 08957 and X34 7204 pvalue 01255 suggesting serial correlation is no longer a problem We conclude that this model is an improvement over that in part d Adding CONGWTHX3 or INCGWTHH cases the extra coef cient was not signi cantly different from zero and the AIC and SC values increased Furthermore the correlgrams and LM statistics led to the same conclusion about serially correlated errors as was reached in part e did not improve the model in part e In both Dropping CONGWT H H from the model in part e and reestimating gives 03407 02143 C0NGWTH 035551NCGWTH se 01188 00596 00454 02414 INCGWTHH 00480 The AIC and SC values are 03099 and 02433 respectively values that are lower than those for the model estimated in part e EViews values are 18348 and 19015 The correlogram below shows some evidence of serially correlated errors at lag 4 but the LM test values X32 0145 pvalue 09301 and X34 6593 pvalue 01591 do not suggest serial correlation is a problem Auto correlation AC 0005 0017 0011 0152 0044 0043 0087 3912 0059 39 039 0075 39039 I0 0081 39ll 11 0045 I I 12 0022 II II II 1 I l I LOCO JO39U1J LOfJ Chapter 9 Exercise Solutions Principles of Econometrics 46 351 EXERCISE 923 The estimated equation is 03407 02143 CONGWTHH 03555 INCGWTH se 01188 00596 00454 024141NCGWTH 00480 The forecasts the standard errors of the forecasts and the forecast intervals are given in the following table The intervals are relatively wide showing that there is a great deal of uncertainty about future consumption growth Period Forecasts Standard Errors Forecast Intervals 2010Q1 10499 05995 0059 2041 2010Q2 09842 05995 0007 1975 2010Q3 10077 06132 0006 2021 Using C as an abbreviation for CONGWT H and I as an abbreviation for INCGWT H the forecasts are obtained as follows fQ 034074 021428 CQ 0355451Q 0241441Q4 034074 021428gtlt 13 035545gtlt 06 024144gtlt 09 104987 Q 034074 021428 CQ 03 5545 1Q 0241441Q 034074 021428gtlt 10 035545gtlt 08 024144gtlt 06 098424 Q 034074 021428 Q 035545 1Q 0241441Q 034074 021428gtlt 104987 035545gtlt 07 024144gtlt 08 100767 The standard errors of the forecast errors are 61 8 059954 G2 8 059954 8 810 059954x 102142772061315 The forecast intervals are given by C 1 i t0951936 where I 16528 095193 Chapter 9 Exercise Solutions Principles of Econometrics 46 352 EXERCISE 924 8 b The model in 994 without the error term is given by CONGWTH 8 GZCONGWTHH ESOINCGWTH 81INCGWTH1 It can be written in lag operator notation as 1 eL2C0NGWTH 5 5 5L1NCGWTH or CONGWTH 1 02L2 18 1 02L2 180 81LINCGWTH Equating this equation with the in nite lag representation CONGWTH on 30 31L BZLZ 33L3 34L 3SL INCGWTH implies 1 0L2180 81L 30 31L BZLZ 33L3 34L Thus 5 81L 1 02L230 3L 3L2 3L3 1345 p r 30 31L3L2 33L3 34L4 0230L2 e3L3 e3L4 giving B0 50 B1 51 B2 9280 B3 9231 BS 92Bs 2 S 2 2 The estimated multipliers are presented in the table below Lag Delay Interim Multiplier Multiplier 1 03 555 03 555 2 02414 05969 3 00762 06731 The total multiplier estimate is 33 J0 07597 8 8 035545 024144 1 1 O21428 The delay multipliers show that if the growth rate of income is increased by 1 and then retumed to its original level then the growth rate of consumption will increase by 036 in the current quarter by 024 in the next quarter and by 008 in the quarter after that The interim multipliers show that if the growth rate of income is increased by 1 and then maintained at this new level then the growth rate of consumption will increase by 036 in the current quarter by 060 in the next quarter and by 067 in the quarter after that When a new equilibrium is reached consumption growth will have increased by the total multiplier namely 086 Chapter 9 Exercise Solutions Principles of Econometrics 46 353 EXERCISE 925 81 b INF I I25I I I I 50I I I I 75 I I 100 I I 125 I I 150 WGWTH 35 30 25 20 15 N 00 0 25 50 75 100 125 150 Neither of the series appears to be trending over the given time period However an assumption of a constant mean over the whole period could be questioned for both series Both appear to have a higher mean for the earlier period up to about observation 50 l982Q3 and a lower mean after that The estimated equation is ITVTV 00215 10254WGWTH se 00942 The coef cient of WGWT H suggests that an increase in wage growth of 1 results in a 1025 percent increase in the in ation rate The residual correlogram that follows shows signi cant autocorrelations at lags 1 2 3 and 4 The signi cant bounds are i2x160 410158 The LM test for AR2 errors yields a test value of LM 3356 with corresponding p value of 00000 Thus we conclude that the errors are autocorrelated Chapter 9 Exercise Solutions Principles of Econometrics 416 354 Exercise 925b continued 0 Auto correlation AC 0448 0254 0347 0185 0059 0101 0003 0077 0055 0048 0174 12 0152 nnaquotquotquot uUUH iE moooonnwms The estimated equation is EVE 00352 05405INF1 04914WGWTH se 00652 01021 To nd the impact and total multipliers We need to rewrite the model in terms of the in nite distributed lag representation NE on EBSWGWTHH e s0 Working in this direction We have NE 1 e5 1 eL5 WGWTH e on 30 B1L BL2 333 WGWTH e and 1 9119150 B0 B1L B2112 B3113 39 39 g or 50 1 e1LB0 B114 B2142 B3143 39 39 ie B0 B1L B2112 B3143 39 39 e B09114 B1911 B2911 39 39 b B0 B1 B0e1LB2 B1e1L2 B3 B2e1L3 quot Equating coef cients of equal powers in the lag operator gives 8000 B 01B10 for 1 21 Thus the impact multiplier is given by 00 30 04914 And the total multiplier is given by 0 04914 g j 00 B001B00fB00f1quote1105405 1069 In part b the total multiplier and the impact multiplier were both equal to 10254 Introducing a lagged value of INF has led to an impact multiplier that is much less but a total multiplier that is approximately the same Chapter 9 Exercise Solutions Principles of Econometrics 46 355 Exercise 925 continued dampe The residual correlograms for models with INE1 added and then INF2 and then 0 N113 and the results of the various LM tests are given below I IU39I0C0ff 3 fi0f1 AC Autocorrelation AC Autocorrelation AC I I 1 0053 1 0029 I I 1 0023 I I 2 0107 2 0152 I I 2 0001 I 3 0251 3 0228 I I 3 0072 I I 4 0000 4 0002 I I 4 0055 I I 5 0037 5 0050 I I 5 0017 I I 6 0097 5 0078 I I 0 0025 I I 7 0043 7 0040 I I 7 0081 I I 8 0071 3 0030 I I 8 0050 I I 9 0041 9 0044 I I 9 0091 I I 10 0025 10 0020 I I 10 0002 I I 11 0135 11 0127 I 11 0152 I n 12 0045 12 0052 I I 12 0043 UNFL1 INFi 19INFi 2 INFi 19INFi 29INFi 3 LM Test and p Values Lags included in test Lags included in 2 3 equation LM value p value LM value p value INFH 6439 0040 12246 0007 INFINF 8137 0017 12064 0007 INE1INE2INF3 1143 0565 2342 0505 After adding INF1 a signi cant autocorrelation remains at lag 3 but those at lags 1 2 and 4 are no longer signi cant The LM tests con rm that serial correlation remains with X2 values that are signi cant at the 5 level for error processes involving 2 and 3 lags Adding INF does nothing to improve the situation The signi cant autocorrelation at lag 3 remains and the LM test values do not improve Adding N113 eliminates the serial correlation at all lags There are no signi cant autocorrelations at the 5 level and the pvalues for the LM test for processes involving 2 and 3 lags are 0565 and 0505 respectively The estimated equation is EVE 00504 045371NF 02174INE3 03728 WGWTH se 00691 00676 01068 In the model NE 8 01INFH 0INE2 03INE3 SOWGWT H I v the coef cient 0 was not signi cantly different from zero pvalue 04497 and so it was Worth considering dropping it Omitting it led to a fall in the SC of 0028 and a fall in the AIC of 0009 and did not introduce any serial correlation in the errors Adding WGWTHH did not improve the equation Its coef cient was not signi cantly different from zero and the AIC and SC both increased Chapter 9 Exercise Solutions Principles of Econometrics 46 356 EXERCISE 926 The estimated equation used for forecasting is given by ITVTV 00504 04537INFH 02174INE3 037281VGWTH The forecast values are 17v12o1oQ2 00504 045371NF00Q O2174IN17009Q3 03728WGWTH00Q 00504 04537 x 038 02l74gtlt 09l 03728gtlt 06 05435 ITVTVMQ3 00504 045371NF00Q O2174IN17009Q4 03728WGWTH00Q 00504 04537 x 05435 02l74gtlt 065 03728 x 05 05239 17v12o1oQ4 00504 O4537IN172010Q3 021741NF00Q 03728WGWTH00Q4 00504 04537 x 05239 02174gtlt 038 03728gtlt 07 05309 GHQ 00504 O4537INF010Q4 021741NFQ 03728WGWTHQ 00504 04537 x 05309 02174 x 05435 03728 x 04 04578 INF The standard errors of the forecast errors are seu1 8 05111 seu 6 f12 O511151O45369212 05613 56043 02 1 as 1 f f12 O5111510453692 O45369412 05711 A 2 4 3 2 cs10 0 0 0 O5928 36044 The 95 forecast intervals are ITVTVMQ2 it x seu1 05435i1976gtlt05111 04661553 0975153 I77772o1oQ3it0975153gtltseu2 05239i1976gtlt05613 O5851633 ITVTVMQ4 it gtltseu3 05309i1976gtlt05711 O5981659 0975153 17vTF2o11Q1iz gtlt seu4 04578 i1976gtlt 05928 O7141629 0975153 These forecast intervals are very Wide containing both positive and negative values and hence do not contain much information about likely values of future in ation Knowing wage growth might help predict in ation but it still leaves a great deal of uncertainty Chapter 9 Exercise Solutions Principles of Econometrics 46 357 EXERCISE 927 a The equation is NE 8 GIINFH 93INE3 501VGWTH v Applying the lag operator to this equation we have 1 91L 93L3 1INF 5 SOWGWTH and NE 1 91L 93L3 5 1 91L 93L3 50WGWTH 0L 3 3L 3L2 3L3 1VGWTH e Thus 5 or 1 eL eL3 15 j 1 9 9 and 1 91L 93L3 150 30 3L 3L2 3L3 or 50 1 91L 93143 X60 B1L B2112 B3143 B4144 39 39 8 B0 B1L B2112 B3143 B4144 39 39 B09114 B1911 B2911 B3e1L4 39 39 B09313 B1e3L4 39 39 B0 B1B0e1LB2 B1e1L2 B3 B2e1 B0e3L3 B4 B3e1 B1e3L3 quot Equating coef cients of equal powers in the lag operator gives 50 B0 B1 91B0 0 B2 91B1O B 91BJ1 93Bj3 0 for j 23 Thus expressions that can be used to calculate 0c and the BS are 50 B0 B1 9130 B2 9131 B 91B193B3 for j 23 b When WGWT H remains constant at zero estimated in ation is 5 00504 O1532 Oi 1 1 O45369 O21743 39 To test H0 oc O we can use t dcse6c or alternatively since OL 0 when 5 O we can use I seS The test values from these two alternatives are t dcse6L O153247028758 0533 r Sse 3 005o4009345 0539 At oc005 the critical values are it i1976 Thus we do not reject H0 0975153 There is no evidence to suggest that in ation will be nonzero when wage growth is zero Chapter 9 Exercise Solutions Principles of Econometrics 46 358 Exercise 927 continued C The rate of in ation when Wage growth is constant at 025 is iv 6L ozsifs i0 Computing the total multiplier 2601 numerically We nd Z613111335 Thus an estimate of the in ation rate is iv 01532025gtlt11335 01301 An EVieWs program that can be used to compute the total multiplier is vector200 b b1c4 b2c2b1 b3c2b2 for i4 to 200 bic2bi1c3bi3 next scalar totmusumb d The delay and interim multipliers for up to 12 quarters are Delay multiplier Estimate Interim multiplier 139 59 03728 03728 131 01139 01691 05419 131 01131 00767 06187 133 01139 03136 01159 07345 134 01133 03131 00893 08239 133 01134 03131 00572 08811 136 01133 03133 00511 09322 131 01136 03134 00426 09749 133 01131 03136 00318 10067 139 01133 03136 00255 10322 1319 01139 03131 00209 10531 1311 011319 03133 00164 10694 1311 011311 03139 00130 10824 Chapter 9 Exercise Solutions Principles of Econometrics 46 359 Exercise 927d continued 6 f The graph for the delay multipliers for up to 12 quarters follows 40 35 30 25 20 BETA LAG An increase in wage growth increases the in ation rate However the effect decreases as the lag increases with the exception of a spike at lag 3 After 12 quarters the effect is nearly zero The graph for interim multipliers for up to 12 quarters is 11 10 09 08 07 INTERIM 06 05 04 03 0 1 2 3 4 5 6 7 8 9 10 11 12 LAG If wage growth increases to a new level and then is held constant at that new level in ation increases at a diminishing rate approaching the total multiplier which is approximately 11 The estimated changes in in ation are given in the following table Quarter T1 T2 T3 T4 T5 Change in In ation 02130 02131 03130 0213 03131 02133 0313 02134 03133 Estimate 00746 01457 00661 00462 00526 CHAPTER 1 0 Exercise Solutions 360 Chapter 10 Exercise Solutions Principles of Econometrics 46 361 EXERCISE 101 3 b C The price of housing and rent paid are determined by supply and demand forces in the market place The omitted factors from this regression include macroeconomic forces such as unemployment rates interest rates population growth etc all of which might well affect not only rent paid but also the median house value If there is correlation between median house value and the regression error term then median house value is endogenous The model in column 1 contains one potentially endogenous variable MDHOUSE In order to carry out instrumental variables estimation we require at least one strong instrument There are 4 potential instruments We test for strong instruments by computing the joint F test of signi cance of these variables in the first stage regression Column 2 contains the first stage regression results including all instruments Column 3 contains the first stage regression omitting FAMINC REG REG2 and REG3 Using the sum of squared residuals SSE in columns 2 and 3 we can compute the F statistic as F SSER SSEU 83222 37676 45546 133 J f 43767650 6 4856 By the StaigerStock rule of thumb we are satisfied because the calculated F is greater than 10 A more informative answer is obtained by examining the critical values for the weak instrument tests of Stock and Yogo in Table 10E1 and 10E2 If we adopt the Maximum Test Size criterion for a test of the coefficient on the endogenous variable and are willing to accept a test size of 010 for a 5 test then the critical value for the F statistic is 2458 B1 L4 The null hypothesis is that the instruments are weak so that under this criterion we cannot conclude that we have strong instruments In order to make such a conclusion we would have to be willing to accept a test size of 020 for a nominal 5 test for in that case the relevant F critical value is 1026 If we adopt the Maximum Relative Bias criterion comparing the bias of the IV estimator to the bias of the least squares estimator and a relative bias of 010 then the relevant F critical value is 1027 Under this metric we can conclude that the instruments are strong The regression based Hausman test for endogeneity augments the regression of interest with the least squares residuals from the first stage regression The null hypothesis is that the variable MDHOUSE is exogenous and the alternative hypothesis is that MDHOUSE is endogenous The Hausman test is a t test for the significance of the coef cient of VHA T The 2tail critical value of the t distribution with 46 degrees of freedom is 201 The calculated value of the Istatistic is 399 Since 399 lt 201 we reject the null hypothesis that the coef cient of VHAT is zero using the 005 level of signi cance We conclude that MDHO USE is endogenous Chapter 10 Exercise Solutions Principles of Econometrics 46 362 Exercise 101 continued d 6 We note two important changes when we compare the least squares estimates in column 1 and the instrumental Variables estimates in column 5 First the IV estimate of the coef cient of PCT URBAN is much smaller than the corresponding least squares estimate and its standard error is larger The coef cient of PCT URBAN is now insigni cant whereas the least squares estimate s t Value of 211 is signi cant at the 005 level Secondly the I V estimate of the effect of MDHOUSE on RENT is larger in magnitude indicating a larger effect than we first estimated The standard error of the I V coefficient is larger 0339 than the corresponding least squares estimate but the I 661 is Very signi cant That the estimates for the structural parameters are the same in columns 4 and 5 is not an accident The rst stage least squares residuals VHAT are uncorrelated with PCT URBAN because it is an explanatory Variable in the rst stage regression and it is a property of the least squares residuals that they are uncorrelated with model explanatory Variables Also VHAT is uncorrelated with the tted Value of MDHOUSE that is used to compute the ZSLSI V estimates as explained below equation 1OD8 The test for the Validity of the 3 surplus instruments the oyeridentifying restrictions is computed as NR2 from the arti cial regression of the ZSLSI V residuals on all available instruments The resulting statistic under the null hypothesis that the surplus instruments are Valid uncorrelated with the regression error is distributed as xfLB413 The Value of the test statistic is NR2 50gtlt 0226 113 From Table 3 at the end of the book the 095 percentile of the X33 distribution is 7815 We conclude that at least one of the extra instruments is not Valid and therefore that the I V estimates in column 5 are questionable The test does not identify which instrumental Variable might be the problem Chapter 10 Exercise Solutions Principles of Econometrics 46 363 Insert Correction of IV standard errors Bonus material In the simple linear regression model yl B1 52x e the ZSLS estimator is the least squares estimator applied to yl B1525cle Where fr is the predicted Value from a reduced form equation So the ZSLS estimators are A CJ139 7 B2 A 2 2 xi X r 7 329 In large samples the ZSLS estimators have approximate normal distributions In the simple regression model A32 N 52 Zx139 35 The error Variance 62 should be estimated using the estimator A2 Zltyj 1 2xi2 GZSLS N2 with the quantity in the numerator being the sum of squared ZSLS residuals or SSEZSLS The problem with doing ZSLS with two least squares regressions is that in the second estimation the estimated Variance is A2 Ex 31 lA32Ez392 Gwrong X The numerator is the SSE from the regression of y on Sci which is SSEwmg Thus the correct 2SLS standard error is 63513 V CASESLS 82SLS A 2 Zx x 332 c2 2 c2 and the Wrong standard error calculated in the 2nd least squares estimation is A 2 62 A z Gwrong 2 V W 0 g Gwrong Z izquot xZ 972 Z3 W2 Given that We have the Wrong standard error in the 2nd regression We can adjust it using a correction factor A 2 A 52sLs 52sLs SeltB2 82 Xsewrong B2 6 Xsewrong p b wrong wrong Chapter 10 Exercise Solutions Principles of Econometrics 46 364 EXERCISE 102 3 b C d 6 In this labor supply function of married Women We expect the coefficient of WAGE to be positive as increased Wage offers induce a greater quantity of labor supplied The coef cient of EDUC in this supply equation re ects the competing forces of i more persistent and intelligent Workers may have an inclination to Work more or ii more educated Workers may be more efficient and choose to Work less The coef cient of AGE might be positive or negative as We anticipate a lifecycle Work pattern of increasing labor effort up to some point in middleage and then decreasing Work effort thereafter The presence of children should have a negative effect on the labor supply of married Women The coef cient of N WIFEINC should be negative as increased household income reduces the need for the vvife s income This supply equation cannot be consistently estimated by least squares Recall that supply and demand jointly determine the hours and Wages In this case WAGE is endogenous just like HOURS is endogenous An endogenous variable on the right hand side of an equation makes the least squares estimator inconsistent An argument could also be made on the basis that a measure of ability is not included in the equation Ability bias is a form of omitted variable bias Where the effect of an individual s ability is not measured but captured in the error term Since one s ability is usually correlated with their education and Wage these variables may be correlated with the error term and this endogeneity will result in the failure of the least squares regression To satisfy the logic of instrumental variables they must be correlated with the endogenous variable and uncorrelated with the error term We expect there to be a correlation between WAGE and EXPER and WAGE and EXPER2 since Workers with more experience can demand higher Wages Because they are demand factors rather than supply factors they are probably exogenous relative to the supply equation and uncorrelated with the supply equation error term The supply equation is identi ed because We have only specified one endogenous variable and there is at least one instrumental variable With EXPER and EXPER2 as instrumental variables We satisfy the requirement L 2 B Estimate the reduced form equation by least squares WAGE y1 y2EDUC y3AGE y4KIDSL6 y5KIDS618 y6NWIFEINC GIEXPER BZEXPERZ u Obtain the fitted values of the reduced form equation WEE Replace its endogenous counterpart in the original supply model and apply least squares The estimated parameters for this last regression will be the ZSLSI V estimators The standard errors based on this two step process are incorrect See the solution to Exercise 101 for more and a correction factor Chapter 10 Exercise Solutions Principles of Econometrics 4e 365 EXERCISE 103 3 b The estimated least squares regression with standard errors in parentheses is v O234210331MONEY 166200UTPUT se 09799 00090 O2506 i Testing H 0 B1 0 52 1 B3 1 against the alternative that at least one of these equalities is not true gives an F statistic of 1052 and a p value of 00000 The F95373 critical value is 273 Since F 1052 gt 273 We reject the strong null hypothesis Or since the p value is less than the level of signi cance 005 We reject the null hypothesis and conclude that this data does not follow the quantity theory of money ii Testing the Weaker hypothesis H 0 02 1 B3 1 We obtain an F statistic of 1264 and a p value of 00000 The 05 critical value is F95373 312 We reject the Weak joint null hypothesis A scatter diagram of the least squares residuals against the variable MONEY GROWTH x is shown in Figure Xr103b It shows a tendency for the residuals to get larger in magnitude as x increases which suggests that heteroskedasticity exists I I 5 0 f 0 5 E 10 quot I 9 15 20 25 0 30 0 4390 8390 120 100 200 240 280 320 360 X Figure xr103b Scatter plot for least squares residuals To use the LM test for heteroskedasticity described in Chapter 8 of POE page 214 obtain the least squares residuals and regress their squared values on MONEY GROWTH The LM statistic is NR2 from this regression Under the null hypothesis of homoskedasticity the test statistic has the X51 distribution In this case the value of the test statistic is 17838 and Z a pvalue of 0000024 Thus We can reject the null hypothesis of homoskedasticity Based on the gure it appears that the problem arises because of one severely unusual observation Chapter 10 Exercise Solutions Principles of Econometrics 46 366 Exercise 103 continued 0 d 6 The robust standard errors are compared to the least squares standard errors in the following table Coef cient Least Squares Robust Standard Estimate Standard Errors Errors White s b1 02342 0979925 0619615 2 10331 0009042 0023694 b3 16620 0250566 0175914 For b1 and b3 the robust standard errors are much smaller than the least squares standard errors This suggests that the least squares method will understate the precision of the estimate under heteroskedasticity For 2 the robust standard errors are much smaller than the least squares standard errors so least squares overstates the precision of the estimate under heteroskedasticity The I V2SLS estimated model of the in ation equation is v 1094010351MONEY 139420UTPUT se 1858200098 05515 i Testing the strong hypothesis H0 B1 0 3 1 33 1 using 2SLS estimates which have not been corrected for heteroskedasticity gives an F statistic of 82331 and a p value of 00001 Since the p value is less than the level of signi cance 005 We reject the null hypothesis and conclude that this data does not follow the quantity theory of money Testing the same hypothesis using robust standard errors gives an F statistic of 97457 and a p value of 00000 Since the p value is less than the level of signi cance 005 We reject the null hypothesis again ii Testing the Weaker hypothesis H0 32 1 53 1 using 2SLS estimates which have not been corrected for heteroskedasticity We obtain a F statistic of 926 and a p value of 00003 Since the p value is less than the level of signi cance 005 We reject the Weak joint null hypothesis Testing the Weaker hypothesis using robust standard errors returns an F statistic of 23028 and a p value of 01072 In this case We do not reject the null hypothesis and conclude that the Weaker joint hypothesis is not rejected Chapter 10 Exercise Solutions Principles of Econometrics 46 367 Exercise 103 continued 0 g h To perform the Hausman test the first step is to obtain the residuals from the reduced form equation of the endogenous variable in question In this case We estimate the reduced form equation OUTPUT y1 yMONEY 01INITIAL 02SCHO0L 03INV 04POPRATE v Obtain the least squares residuals 9 0UTPUT WETT estimate an auxiliary regression which is the original model specification augmented with 13 and test Whether the coefficient of the residuals 19 is signi cantly different from zero The estimated auxiliary regression is reported with t value for the key coef cient using the usual least squares standard errors and the robustt using White s robust standard errors v 10940 10351MONEY 13942OUTPUT 0338813 0 055 t robust 095 The nal step of the Hausman test requires us to test the null hypothesis H0 8 0 against the alternative hypothesis H1 5 0 Where 5 is the coefficient of the residuals 13 This is equivalent to testing the null hypothesis H0 covOUT PUT e 0 The robustt statistic and the p value for this sample are 09512 and 03447 respectively Since the p value is larger than the level of significance 005 We do not reject the null hypothesis and cannot conclude that OUT P UT is endogenous To test the null hypothesis H 0 all the surplus moment conditions are valid We follow the steps outlined in Section 1043 For this part We will keep to the assumption that MONEY is exogenous and OUT P UT is endogenous The test statistic obtained is NR2 76 X 0032305 24552 The critical value x2LB x241 78147 is much larger than the test statistic therefore We do not reject the null hypothesis that all surplus moment conditions are valid The test p value is 04834 Applying the joint F test described in Section 1042 requires us to test the null hypothesis H 0 01 02 03 0 4 0 in the reduced form equation from part f The F test values are 464 p 00022 and 321 p 00178 for the least squares and robust tests respectively We reject the null hypothesis that all the coef cients are zero at the 5 level However simply rejecting the null hypothesis is not adequate evidence of strong instruments The rule of thumb states that the F test value must be greater than 10 to be strong the results of both of our joint F tests suggest that We should be concerned that We are using Weak instruments Chapter 10 Exercise Solutions Principles of Econometrics 46 368 Exercise 103 continued Bonus material Using the analysis of Weak instruments in Appendix 10E We can be more precise The critical values for the Weak instrument test using the maximum IV test size criterion are in Table 10E1 For this case they are L 010 015 020 025 4 2458 1396 1026 831 We cannot reject the null hypothesis that the instruments are Weak even if We can tolerate a 5 test on the coef cient of the endogenous variable having actual size up to 025 The critical values for the Weak instrument test using the maximum IV relative bias criterion are in Table 10E2 For this case they are L 005 010 020 030 4 1685 1027 671 534 Once again We see that We cannot reject the null hypothesis that the instruments are Weak even if We can tolerate up to 030 of the least squares estimator s bias Note however that these tests are not valid under heteroskedasticity Chapter 10 Exercise Solutions Principles of Econometrics 46 369 EXERCISE 104 a As a check of your work the summary statistics are Vari able 035 Mean St d Dev M n Ilax x 25 1770892 1 110162 2 42647 2 4822 e 25 1671932 1 158174 2 57634 2 50074 y 25 6557176 2 216547 3 65135 5 98294 ey 25 8229108 1 110162 1 42647 34822 b As shown in the gure below the data tend to fall below the regression line for x lt 0 and above the regression line for x gt 0 I I I I I I 2 1 0 1 2 3 x valuesgenerated data Figure xr104b Data values and regression function E02 c The leastsquares estimated equation is given by y 10009 19490x se 00996 00903 The estimate for 51 which is 10009 is very close to the true value of 1 However the estimate for 52 which is 1949 is quite different from the true value of 1 The tstatistics for testing whether 31 and 52 are 1 are 000872 and 1050 respectively We do not reject the null hypothesis of B1 1 but do reject the null hypothesis of 5 1 Chapter 10 Exercise Solutions Principles of Econometrics 46 370 Exercise 104 continued d In contrast to the plot in part b Figure Xr104d shows a tted regression line that runs through the center of the observations Thus it is not a good estimate of the true regression function 0 1 x vaues generated data C y O Fitted values Figure xr104d Fitted regression and observations e The sample correlation matrix of the variables x e and is as follows x e 100000 090968 100000 000000 041531 100000 gt gtlt There is a high correlation between x and e The zero correlation between x and is a characteristic of the least squares estimation procedure In real problems the variable e is not observable and therefore we cannot calculate the correlations between x and e and e and Chapter 10 Exercise Solutions Principles of Econometrics 46 371 EXERCISE 105 8 b C d The leastsquares estimated equation is sf s 43428 00052INCOME se 08561 00112 1 507 046 The estimated equation using the instrumental Variables estimator with instrument 2 A VERA GEINCOME is 9 09883 00392INCOME se 15240 00200 1 O6484 19550 To perform the Hausman test We estimate the arti cial regression as 1 09883 O3918INCOME 0075519 se 1 172000154 00201 7 0843 525430 3757 To perform the Hausman test We test the null hypothesis that the coef cient for 13 is zero The Istatistic is 3757 At the 001 level of signi cance We reject the null hypothesis and conclude that x and e are correlated The reduced form estimation yields 0 350220 164171 VERlGEINCOME 1 183 580 The second stage regression replaces INCOME with the tted Value from the reduced form The result of the estimation is 9 09883 00392INCOME se 12530 00165 1 079 238 The standard errors are lower than those in part b which causes the tstatistics to be higher In particular using the incorrect standard errors from part d makes the estimated slope appear statistically signi cant at the 005 level of signi cance Chapter 10 Exercise Solutions Principles of Econometrics 46 372 EXERCISE 106 a The correlation between x and e is coVx e 09 rxe VarxVare 21 b The sample correlation between x and e is 065136 only slightly higher than the true Value in a 06364 C Figure Xr106c shows us that the data tends to fall below the regression line for x lt 0 and above the regression line for x gt 0 o o 0 393 000 0 0 39 0 0 S 9 0 o 4 o O Q on LI 394 39 6 39 A x vaues generated data 0 yvaues generated data 0 ey Figure xr106c Fitted regression and observations d The results from least squares are presented below Sample Range Estimate Standard error B1 1 10 27775 03608 1 20 30169 02036 1 100 30078 00787 1 500 30183 00341 B2 1 10 13722 01727 1 20 13876 01211 1 100 14016 00533 1 500 14535 00237 Chapter 10 Exercise Solutions Principles of Econometrics 46 373 Exercise 106d continued 6 f The estimate for 31 moves closer to the true value as the sample size increases from 10 to 20 and to 100 However it does not get closer when the sample increases from 100 to 500 For 32 the estimates move away from the true value as the sample size increases As expected for both cases the standard errors decrease as the sample size increases The estimates do not get closer to the true values as sample size increases because of the inconsistency caused by the correlation between x and e The inconsistency does not disappear as the sample size increases The sample correlations between 21 22 x and e are Z1 Z2 9 e Z1 10000 Z2 00153 10000 x 06208 02894 10000 e 00034 00277 06514 10000 The nonzero correlations between x and 21 06208 and between x and 22 02894 coupled with the essentially zero correlations between e and 21 00034 and e and 22 00277 mean that both 21 and 22 will be satisfactory instrumental variables However because the correlation between x and 21 is greater than the correlation between x and 22 21 is the better instrumental variable Using 21 as an instrumental variable the estimates are Sample Range Estimate Standard error B1 1 10 27144 04277 1 20 30810 02500 1 100 29771 01051 1 500 30315 00451 52 1 10 10640 02526 1 20 10263 01966 1 100 09363 01132 1 500 09961 00504 The IV estimates for both 31 and 32 are getting closer to the true values as the sample size increases re ecting the consistency of the IV estimator Chapter 10 Exercise Solutions Principles of Econometrics 46 374 Exercise 106 continued 23 h Using 22 as an instrumental variable the estimates are Sample Range Estimate Standard error B1 1 10 18923 1106 1 20 32433 06975 1 100 29902 00887 1 500 30295 00424 52 1 10 29503 5170 120 01110 2471 1 100 11349 01470 1 500 10666 01014 Using 22 as an instrumental variable gives estimates that are very far away from the true values when the sample sizes are small less than 20 for 32 and less than 10 for B1 When the sample size is larger the estimates move closer to the true values particularly those for B2 Comparing the results using 21 alone to those using 22 alone those using 21 alone lead to more precise estimation even when the sample size is small This result occurs because the correlation between 21 andx is much higher than the correlation between 22 andx Using both 21 and 22 as instrumental variables the estimates are Sample Range Estimate Standard error 51 1 10 27114 04337 1 20 30852 02555 1 100 29808 00997 1 500 30311 00446 52 1 10 10491 02549 1 20 10026 01987 1 100 09921 00932 1 500 10090 00449 Using both 21 and 22 as instrumental variables the estimates are getting closer to the true values as the sample size increases The results are very similar to those obtained using only 21 as an instrumental variable although there has been a slight improvement in precision for sample sizes T 100 and 500 Chapter 10 Exercise Solutions Principles of Econometrics 46 375 EXERCISE 107 3 b The least squares estimated equation is Q 17623 O1468XPER 043 8OCAP 023 9222113 se10550 00634 O1176 00998 The signs of the estimates are positive as expected All the standard errors are relatively low except that for the constant term thus all estimates of the slope coefficients are signi cant The sample averages for labor and capital are 100467 and 78347 respectively The error variance is 62 75965 The variancecovariance matrix for the estimates is b1 b2 b3 b4 bl 11138 22 00468 00040 23 00049 00012 00138 24 00322 00000 00087 00100 Using XPER 10 and LAB and CAP equal to their sample averages the predicted Wine output is Q0 17623 01468 x 10 04380 x 78347 02392 gtlt100467 90647 The variance of the prediction error for this case is vEfvEevEa9XPER vEaoCA1fvEa9LAB vEao4 2XPER0 N6 2C113 Filip 2L113 c 2XPER0CAP0 ZXPEROLABO c 2C11gtL113 Substituting the values of the variances and covariances We obtain Q 7756 and therefore sef VET 2785 Alternatively the predicted value and the standard error of the prediction error can be obtained using automatic software commands The 95 interval prediction uses tc 997571 19939 Q tcse f 90647 19939 x 2785 351 1462 Chapter 10 Exercise Solutions Principles of Econometrics 46 376 Exercise 107b continued 19 C d ii Using your computer software we can calculate the predicted wine output and the standard errors given 20 years experience as Q0 10533 and se f 2802 A 95 interval prediction is 10533 i 19939 x 2802 495 1612 iii For 30 years experience Q0 12001 and se f 2957 The interval prediction is 12001i19939 x 2957 6111790 The estimated arti cial regression is Q 24867 05121XPER 03321CAP 02400LAB 0415819 I 21978 The Hausman test to test whether the variable XPER and the error term are correlated is the same as testing whether the coef cient for 19 is zero The results suggest that the coef cient for 13 is signi cant The pvalue of the test is 0031 so at a 5 level of signi cance we can conclude that there is correlation between XPER and the error term The IV estimated equation is Q 24867 05121XPER 03321CAP 02400LAB se 27230 02205 01545 01209 1 091 232 215 199 As in part a the estimates have the expected positive signs Relative to the least squares results the values of the estimated coef cients for XPER and CAP have changed considerably but that for LAB is approximately the same All coef cients are signi cant at a 10 level of signi cance or at a 5 signi cance level when using onetailed tests Bonus material The rst stage F value is 981361 which is close to the rule of thumb value of 10 However using the StockYogo maximum test size criterion from Table 10E1 the test critical values are L 010 015 020 025 1 1638 896 666 553 We can reject the null hypothesis that the instrument is weak if we are willing to accept a test size on the coef cient of the endogenous variable of up to 015 For the lower maximum test size of 010 we are unable to reject the null hypothesis that the instrument is weak Chapter 10 Exercise Solutions Principles of Econometrics 46 377 Exercise 107 continued e The following results are obtained using automatic software commands for forecasting i For 10 years experience Q0 76475 and se f 3468 The interval prediction is 76475 i 19939 x 3468 O731456 ii For 20 years experience Q 12768 and se f 3621 The interval prediction is 12768 i19939 x 3621 5551999 iii For 30 years experience Q0 17890 and se f 4891 The interval prediction is 1789 i19939 x 4891 814 2764 A comparison of these prediction intervals with those from least squares estimation suggests that ignoring the correlation between XPER and the error term will 0 yield intervals that are too narrow which in turn leads to false reliability about wine output 0 overpredict wine output for a manager with 10 years experience 0 underpredict wine output for managers with 20 and 30 years experience Chapter 10 Exercise Solutions Principles of Econometrics 4e EXERCISE 108 a The Hausman test is carried out by rst estimating the reduced form for ln WAGE These estimation results are Dependent Variable LOGWAGE Method Least Squares Sample 1 753 IF LFP1 Included observations 428 Coefficient Std Error tStatistic Prob C 0357997 0318296 1124729 02613 EDUC 0099884 0015097 6615970 00000 AGE 0003520 0005415 0650176 05159 KIDSL6 0055873 0088603 0630591 05287 KIDS618 0017648 0027891 0632765 05272 NWIFEINC 569E06 332E06 1715373 00870 EXPER 0040710 0013372 3044344 00025 EXPERquot2 0000747 0000402 1860055 00636 Rsquared 0164098 Mean dependent var 1190173 Denote the residuals from the reduced form as 9logWAGE The estimated supply equation augmented with the residuals from the reduced form with t statistics in the parentheses is HEIJTS 243220 154482lnWAGE 177449EDUC 107841IGE r 73388 57611 54716 20187 210834KIDSL6 475571KIDS618 00092NWIFEINC 1623601 21363 14980 25585 59394 The Hausman test is used to test for endogeneity by considering the null hypothesis that the coef cient of 13 is signi cantly different from zero The estimates suggest that the coef cient for 13 is signi cant since the pValue of the test is 00000 so we can conclude that there is correlation between ln WA GE and the error term b The estimated reduced form equation is shown in part a The Fstatistic of the joint hypothesis H0 75 7c8 0 is 825 yielding a pValue of 00003 At a 5 level of signi cance we reject the null hypothesis and conclude that these instruments have a signi cant correlation with lnWAGE However using the rule of thumb the F statistic is less than 10 which implies that these instruments are not strong instruments Chapter 10 Exercise Solutions Principles of Econometrics 46 379 Exercise 108b continued 0 C1 Bonus material Using the StockYogo critical Values for testing if instruments are weak we can be more precise Using the maximum test size criterion from Table 10E1 the test critical Values are L 010 015 020 025 2 1993 1159 875 725 We can reject the null hypothesis that the instrument is weak if we are willing to accept a test size on the coef cient of the endogenous Variable of up to 025 For the lower maximum test sizes we are unable to reject the null hypothesis that the instruments are weak Following the steps outlined for testing surplus instrument Validity in Section 1042 we test the null hypothesis that all surplus moment conditions are Valid The test statistic calculated is NR2 428 x 00020 08581 Under the null hypothesis the test statistic has a chisquare distribution with 1 degree of freedom The 05 critical Value for the X3 is 384 Since 08581 lt 384 we do not reject the null hypothesis indicating that our surplus moment conditions are Valid The pValue of the test is 03543 The potential endogeneity problem with ED UC is that a measure of ability is omitted from the equation and is thus in the error term of the supply equation It is likely that more able people also attend school longer thus inducing a correlation between the regression error and EDUC A Valid instrument must be uncorrelated with the regression error and ability in particular and should be strongly correlated with EDUC It is likely that MOTHEREDUC FATHEREDUC and HEDUC are correlated with a woman s years of education The argument for SIBLINGS is less obvious though perhaps in larger families each child attends school for fewer years The only problem with MOT HEREDUC and FAT HEREDUC as instruments is that more intelligent parents and more educated have more intelligent and educated children To test the suitability of the instruments MOT HEREDUC FAT HEREDUC HEDUC and SIBLINGS we rstly test for significant correlation between the endogenous and instrumental Variables The reduced form equation which we use to conduct the joint test is EDUC 7 71GE 7c3KIDSL6 7c4KIDS618 7c5NWIFEINC 7c6M0THERED UC 7c7FA THERED UC 7c8HED UC 7c9SIBLINGS v For a joint hypothesis test of all possible instruments H0 7c6 7 78 7 0 the F statistic is 6067 with a pValue of 00000 This Fstatistic is much greater than 10 implying that at least some of our instruments are strong instruments and we should not be too concerned that we are using weak instruments Chapter 10 Exercise Solutions Principles of Econometrics 46 380 Exercise 108d continued 6 Testing the strength of each individual instrumental Variable We use the regression estimation results These are presented in the following table Instrumental Variable Istatistic pValue 76 MOTHEREDUC 38647 00001 77 FATHEREDUC 32427 00013 78 HEDUC 11004 00000 79 SIBLINGS 09057 03656 All instruments are signi cantly different from zero except for SIBLINGS Furthennore the Istatistics of HEDUC and MOTHEREDUC are greater than 33 This suggests that HEDUC and MOT HEREDUC are strong instrumental Variables and FAT HEREDUC is a Weaker instrumental Variable The other requirement of an instrumental Variable is instrument Validity This can only be tested on surplus instruments and when all other endogenous Variables have been fully specified This test is conducted in part h of this exercise The estimates of the reduced form equations ED UC or lnWA GE 71 72IGE 73KIDSL6 74KIDS618 75NWIFEINC 76EXPER 77EXPER2 78M0THERED UC 79FA THERED UC 710HED UC 711SIBLINGS v are presented in the following table Dependent Variable ED UC Dependent Variable ln WA GE PSr r Istatistic pValue Pjsrggzizr Istatistic pValue TE1 55378 67882 00000 05551 16757 00945 72 00003 00235 09812 00058 10025 03167 73 04794 21031 00361 00035 00373 09702 74 01096 15206 01291 00342 11675 02437 75 000002 25560 00109 00000 26713 00079 TE6 00403 11697 02428 00450 32161 00014 TE7 00007 06371 05244 00008 19449 00525 78 01179 37857 00002 00012 00939 09253 759 00988 33547 00009 00069 05795 05625 710 03416 10953 00000 00256 20200 00440 7511 00320 09093 03637 00067 04715 06375 Chapter 10 Exercise Solutions Principles of Econometrics 46 381 Exercise 108e continued The Ftest of joint signi cance of EXPER EXPER2 MOT HEREDUC FAT HEREDUC HEDUC SIBLINGS H0 7c6 P 7 0 results in Fstatistics of4102 and 413 for the EDUC and ln WAGE reduced form equations respectively Both F tests are signi cant at a 1 level of signi cance However We should be concerned about using Weak instruments for ln WAGE since 413 lt 10 Bonus Material This example illustrates the problems of evaluating instrument strength when there is more than one endogenous variable The two Fvalues in part e are not adequate We should use the CraggDonald test statistic in equation 1OE3 The Stata 111 calculation of this value and the critical values reported by Stata are Minimum eigenvalue statistic 313616 Critical Values of endogenous regressors 2 Ho Instruments are weak of excluded instruments 6 5 10 20 30 2SLS relative bias 15 72 9 48 6 08 4 78 10 15 20 25 I 2168 1233 910 742 2SLS Size of nominal 5 Wald test Stata calls the CraggDonald statistic the Minimum eigenvalue statistic Its value is 314 which We compare to the critical values using the IV relative bias or IV maximum test size criteria We see that We cannot reject the null hypothesis that the instruments are Weak Chapter 10 Exercise Solutions Principles of Econometrics 46 382 Exercise 108 continued f The Hausman test uses the residuals from the reduced form equation for EDUC called VHlTEDUC and the residuals from the reduced form for lnWAGE called VHATLNWAGE These variables are added to the HOURS equation The estimated arti cial regression is Dependent Variable HOURS Method Least Squares Included observations 428 Coefficient Std Error tStatistic Prob C 1836672 4324070 4247554 00000 LOGWAGE 1452066 2491999 5826912 00000 EDUC 1233164 3211899 3839361 00001 AGE 9242835 5359770 1724484 00854 KIDSL6 2488949 9832368 2531383 00117 KIDS618 3965773 3280008 1209074 02273 NWIFEINC 0011856 0004012 2955198 00033 2 2 2 Z 2 2 2 X To test the null hypothesis that both EDUC and lnWAGE are endogenous We conduct a joint test on the coefficients of the two residual terms We arrive at a Fstatistic of 1825 with a pValue of 00000 and therefore We reject the null hypothesis and conclude endogeneity exists in at least one of ED UC and ln WA GE g The ZSLS estimated model using all instrumental variables is Dependent Variable HOURS Method TwoStage Least Squares Included observations 428 Coefficient Std Error tStatistic Prob C 1836672 7473748 2457498 00144 LOGWAGE 1452066 4307186 3371264 00008 EDUC 1233164 5551466 2221331 00269 AGE 9242835 9263858 0997731 03190 KIDSL6 2488949 1699432 1464577 01438 KIDS618 3965773 5669185 0699531 04846 NWIFEINC 0011856 0006934 1709783 00880 Chapter 10 Exercise Solutions Principles of Econometrics 46 383 Exercise 108g continued h i Most estimates have the expected sign with the only exception being EDUC A negative sign of the coef cient estimate for ED UC suggests that women with more education work fewer hours than those who do not Another surprising result is that AGE KIDSL6 KIDS6I8 and N WIFEINC are not signi cant at a 005 level of signi cance leaving only lnWAGE and EDUC as statistically signi cant To test the validity of the overidentifying instruments we regress the twostage least squares residuals upon all exogenous and instrumental variables and calculate the test statistic NR2 428 x 0006232 26673 With a 05 critical value of x2LB x262 949 we do not reject the null hypothesis that the surplus instruments are valid since 267 lt 949 The pvalue of this test is 06149 We have used a sample of 428 working women from 1975 to determine the in uence of wage education age kids and other sources of income on the labor supply of married women Because this is a supply equation we know that hours and wages are jointly determined by supply and demand and thus wages are an endogenous variable and correlated with the regression error Also the equation omits a measure of ability and ability is likely positively correlated with both wages and years of education A Hausman test veri es our prior reasoning we reject the null hypothesis that these two variables are not correlated with the regression error term The presence of a regression error that is correlated with one or more righthand side explanatory variables means that the usual least squares estimator is both biased and inconsistent To carry out twostage least squares we required instruments that are correlated with the endogenous variables yet uncorrelated with the regression errors Because there are two endogenous variables we need at least two instrumental variables We employed experience experience squared the years of education of mother father and husband as well as the number of siblings The instruments are not strong jointly for logwage but experience is a strong single IV with a I value of 322 The instruments are jointly strong for education We cannot reject the validity of the 4 surplus instruments using the Sargan NR2 test for the validity The two stage least squares estimation found that several of the explanatory variables were not statistically signi cant implying that the drivers behind the labor supply of married women are wages and the education levels The household income from other sources than the woman s employment NWIFEINC is statistically signi cant at a 10 level Education and log wage statistically signi cant at a 5 level and all other explanatory variables are insigni cant From the model estimates we nd that each additional year of education decreases labor supply by 123 hours and a 1 increase in wages increases labor supply by about 15 hours We might expect that a 100 dollar increase in N WIFEINC is associated with a decrease in the labor supply of 118 hours And lastly the number of children in a household and the age of the woman have no in uence over labor supply Chapter 10 Exercise Solutions Principles of Econometrics 46 384 EXERCISE 109 8 b The least squares estimates of the supply equation are Dependent Variable LOGQPROD Method Least Squares Sample 1960 1999 Included observations 40 Coefficient Std Error tStatistic Prob C 2109688 0799153 2639905 00123 LOGP 0009110 0067941 0134083 08941 LOGPF 0090195 0042646 2114962 00416 TIME 0011171 0005149 2169632 00369 LOGQPROD1 0732689 0106635 6871012 00000 All coefficients are signi cant at a 5 level of signi cance except for the coef cient of lnB which is disappointing in this supply relationship All signs are as expected as the price of broilers increases We expect production to increase as the price of feed inputs increases We expect production to decrease over time We expect the quantity produced to increase to feed an increasing demand due to population growth and We expect that an increase in production in the previous period will be associated with an increase in production in the current period Using ZSLS and instrumental variables lnY lnPB POPGRO lnPH and lnEXPT S Dependent Variable LOGQPROD Method TwoStage Least Squares Sample 1960 1999 Included observations 40 Coefficient Std Error tStatistic Prob C 2974702 1025654 2900298 00064 LOGP 0289120 0133000 2173826 00366 LOGPF 0163530 0058689 2786393 00086 TIME 0020679 0007202 2871371 00069 LOGQPROD1 0598974 0139139 4304855 00001 The ZSLS estimate of the coefficient of lnP is larger and is signi cant at the 05 level Other coef cients maintain their signs and significance Chapter 10 Exercise Solutions Principles of Econometrics 46 Exercise 109 continued c The rst step in the Hausman test is to estimate the reduced form equation Dependent Variable LOGP Method Least Squares Sample 1960 1999 Included observations 40 Coefficient Std Error tStatistic Prob C 1154092 5954765 1938098 00618 LOGY 1235581 0624824 1977487 00569 LOGPB 0020084 0210586 0095370 09246 POPGRO 0061159 0085777 0712994 04812 LOGP1 0342212 0153288 2232477 00329 LEXPTS 1679849 0740066 2269863 00303 LOGPF 0148438 0100821 1472287 01510 TIME 0062288 0022306 2792487 00089 LOGQPROD1 0160882 0284871 0564755 05763 Save the residuals VHAT from the reduced form and add to the original regression equation The Hausman test checks the significance of the variable VHA T Dependent Variable LOGQPROD Method Least Squares Sample 1960 1999 Included observations 40 Coefficient Std Error tStatistic Prob VHAT 0457227 0117771 3882331 00005 C 2974702 0710735 4185386 00002 LOGP 0289120 0092164 3137022 00035 LOGPF 0163530 0040669 4021011 00003 TIME 0020679 0004990 4143641 00002 LOGQPROD1 0598974 0096418 6212285 00000 The estimated coef cient of VHAT has a Ivalue of 388 and is signi cant at the 001 level of signi cance Thus We conclude that as suspected lnPRICE is an endogenous Variable in this supply equation Chapter 10 Exercise Solutions Principles of Econometrics 46 386 Exercise 109 continued d 6 To test that the instruments are adequate We must identify at least one strong instrumental variable In the reduced form We nd that lnPH and lnEXPT S are signi cant at the 05 level and lnY is signi cant at the 10 level These are not extremely strong The F test on all of the instrumental variables in the reduced form equation shown in part c yields an F statistic of 392 with and the pvalue is 00072 Thus the instruments are jointly signi cant at the 01 level but do not attain the rule of thumb value of 10 We conclude that the instruments We have are signi cantly correlated with the endogenous variable lnP but may not be strong enough so that twostage least squares will reliable Bonus material The StockYogo critical values for this example are not included in Tables 10E1 and 10E2 Consult the StockYogo paper The critical values provided by Stata 111 are Critical Values of endogenous regressors 1 Ho Instruments are weak of excluded instruments 5 5 10 20 30 2SLS relative bias I 1837 1083 677 525 10 15 20 25 2SLS Size of nominal 5 Wald test 2687 1509 1098 884 We cannot reject the null hypothesis that the instruments are Weak using either the relative bias or maximum test size criteria One might expect the log of exports of chicken could also be endogenous As domestic price rises the exports of chicken should fall as domestic price falls exports should rise If exports and domestic price are jointly determined then lnEXPT S is endogenous and not a valid instrument To check instrument validity We test the null hypothesis that the excess moment conditions are valid Obtain the twostage least squares residuals Regress these on all exogenous variables and instruments The test statistic NR2 3671 has a X34 distribution if all surplus instruments are valid The pvalue is 04523 and the critical chi squared value is 949 Thus based on this test We fail to reject the validity of the overidentifying restrictions CHAPTER 1 1 Exercise Solutions 387 Chapter 11 Exercise Solutions Principles of Econometrics 4e 388 EXERCISE 111 The ratio of the expressions for 71 and 72 is B1O 2B10 1 751 0 2B1 0 1 ZBI Thus one Way to estimate 31 is to first obtain estimates 771 and 772 by applying least squares to the reduced form equations and to then estimate 31 from A31 ft 7 If 13cX18xX and Qfc2X5gtltX then fs 7 7 518 02778 Chapter 11 Exercise Solutions Principles of Econometrics 4e 389 EXERCISE 112 a Let the estimated demand curve be Q2611 6c2P6c3PS6c4DI Solving for P and inserting values PS and DI We have A 1 A A A P f 1 A Q f 3 PS i 4D139 052 052 052 052 42795 1 Q 12960 PS 50140 DI 03745 03745 03745 03745 114284 26705gtlt Q34611x 195 133899gtlt 01 114284 26705gtlt QA34611gtlt 22133899gtlt 35 1115801 26706gtlt Q Similarly solving the supply curve Q A31 A3219 A33PF for P yields 01 1 A33 P T T TPF B2 B2Q B2 200328 1 A 10009 PF 03380 03380 03380 59271929587 x Q 29614gtlt PF 59271929587 x Q 29614 x 23 8841129587gtltQ Figure Xr112a is a sketch of the demand and supply equations for the given set of exogenous variable values 120 100 80 Demand Suppy 40 I 20 Figure xr112a Demand and supply graph Chapter 11 Exercise Solutions Principles of Econometrics 4e 390 Exercise 112 continued b The equilibrium values can be found by equating the demand and supply equations at the given exogenous variable values One can equate either the equations derived in a or those with quantity as the lefthand side variable The latter are Demand Q 42795 03745P 12960 x 22 50140 x 35 417822 03745P Supply Q 200328 03380P 10009 x 23 29881 033 80P Solving these two equations we have QEQM 182509 and PEQM 628407 c Using the reduced form estimates in Tables 112a and 112b the predicted equilibrium values are QEQMR 78951 06564gtlt 22 21672 x 35 05070gtlt 23 182604 PEQMR 32512417081gtlt2276025gtlt3513539gtlt 23 628154 These values are very close to those calculated in part b d Figure Xr112d is a plot of the two demand curves and the supply curve 140 120 100 80 Demand pxO Suppy 60 Demand 40 20 Figure xr112d Demand and supply graph following a change in income Chapter 11 Exercise Solutions Principles of Econometrics 46 391 Exercise 112 continued 6 f The new equilibrium price and quantity are given by equating the new demand equation with the old supply equation The new demand equation is Q 42795 03745P 12960 x 22 50140 x 45 467962 03745P Therefore the new equilibrium is QZEM 206295 and PEZM 698784 and the changes in equilibrium price and quantity are APEQM 698784 628407 70377 AQEQM 206295 182509 23786 The income elasticity of demand is the percentage change in quantity demanded due to a percentage change in income and can be derived from the equation OAQ 4Q Q ADI ADIDI D The income elasticity of demand implied by the shift in part d is the percentage change in equilibrium quantity demanded given a percentage change in income at the given exogenous variable values We calculate this as AQEQM AQEQM QEQM 23786182509 ADIDI 45 3535 3D 04561 ADI Using the reduced form estimates we first calculate the quantity demanded after income is increased from D 35 to DI 4 This new equilibrium quantity demanded is QQM 204276 Combining this value with the equilibrium quantity demanded from part c we calculate the income elasticity of demand at the given exogenous variable values as AQEQM AQEQM QEQM 204276 182604182604 SD ADI ADIDI 45 3535 04l54 The elasticity calculated using the graphical solution is similar to the elasticity calculated using reduced form estimates They are not exactly the same because the result from the reduced form estimates does not take into account whether each of the variables PS D or PF appears in the demand equation or the supply equation or both The graphical solution uses information on the location of these exogenous variables in the demand and supply equations Chapter 11 Exercise Solutions Principles of Econometrics 46 392 EXERCISE 113 8 b C The wage equation cannot be estimated satisfactorily using the least squares estimator because it is part of a simultaneous equation system Having identified an auxiliary relationship which has lnWAGE as an explanatory Variable and HOURS as the dependent Variable tells us that lnWAGE and HOURS are endogenous Variables The wage equation is subject to endogeneity and the least squares estimator is biased and inconsistent The wage equation is identi ed because 1 Variable KIDS is omitted In this context there are two simultaneous equations Therefore to be identified the equation must have M 1 2 1 1 Variable absent M being the number of equations in the simultaneous model system The alternative to least squares estimation is twostage least squares estimation The steps for conducting a twostage least squares regression are outlined in Section 1151 For this simultaneous equation system the steps are 0 Least squares estimation of the reduced form equation for HOURS where the exogenous Variables are ED UC EXPER and KIDS 0 Calculate the predicted Values for the Variable 0 Replace HOURS with HOTIR in the wage equation and then estimate this new wage equation by least squares 0 Note that the standard errors calculated using this method will not be correct but the estimator is consistent See the insert on the following page for how to correct the standard errors Chapter 11 Exercise Solutions Principles of Econometrics 4e 393 Insert Correction of IV standard errors Bonus material In the simple linear regression model yl B1B2xl el the ZSLS estimator is the least squares estimator applied to yl B1B2fcl el where 2 is the predicted Value from a reduced form equation So the ZSLS estimators are 3 ZACz 7J1quot 2 2 2i if IA31 7 3237 In large samples the ZSLS estimators have approximate normal distributions In the simple regression model 2 A cs 52 N ham Zx139 X The error Variance 62 should be estimated using the estimator A2 Zlty139i31 i32x1392 GZSLS N2 with the quantity in the numerator being the sum of squared ZSLS residuals or SSEWS The problem with doing ZSLS with two least squares regressions is that in the second estimation the estimated Variance is A2 Zlty l 1 l 2 i2 WIquot0I lg p S The numerator is the SSE from the regression of yl on 2 which is SSEwr0ng Thus the correct 2SLS standard error is S6632 cAi SLS 2 2 62SLS 2 Z quot xZ997 xZ997 and the Wrong standard error calculated in the 2nd least squares estimation is W lt33mng cm 202 39c2 zpe c2 Zoe c2 Given that we have the Wron standard error in the 2nd re ression We can ad39ust it usin 9 a correction factor A 2 A GZSLS GZSLS SeltB2 82 Xsewrong B2 6 Xsewrong U wrong wrong Chapter 11 Exercise Solutions Principles of Econometrics 46 394 EXERCISE 114 3 b Least squares should be used to estimate the parameter 3 because there are no endogenous explanatory Variables in the rst equation The parameter 3 is identi ed because it can be consistently estimated Twostage least squares should be used to estimate the parameter OL because there is an endogenous Variable yl on the righthand side of the second equation There are M 2 equations in this model which implies that M 1 1 Variables should be absent for the model to be identi ed The parameter Ot is identi ed because x is absent from the second equation and it is present in the rst equation Chapter 11 Exercise Solutions Principles of Econometrics 46 395 EXERCISE 115 a The demand and supply curve estimates are XR 115 2SLS estimations 1 2 DEAND2SLS SUPPLY2SLS C 4 2795 20 O328 5 54 1 22 P 0 3745 0 3380 0 15 0 02 P5 1 2960 0 35 0 5 0140 2 28 PF 1 0009 0 08 v 30 30 Standard errors in parentheses plt005 plt001 plt0001 Reporting these equations in the usual format We have Demand Q 4280 O3745P 1296PS 501401 se O1648 03552 2284 1 2273 03552 2196 Supply Q 2003 O338OP 1001PF se 002492 008252 1 1356 1213 Chapter 11 Exercise Solutions Principles of Econometrics 46 396 Exercise 115 continued b The price elasticities of supply and of demand at the mean are calculated as AQAQQ23 8 AQAQQZOL 5 AP APP 2Q D AP APP 2Q Using our estimates 5 31 O3380gtlt 62724 11485 Q 18 8 661 03745 gtlt 6272 12725 The signs of the elasticities are as expected we expect 83 to be positive because quantity supplied increases as price increases and We expect 8D to be negative because quantity demanded decreases as price increases Both elasticities have a magnitude greater than 1 which indicates that both supply and demand considered elastic and therefore responsive to prices a percentage increase in price leads to a larger than 1 change in supply and demand Chapter 11 Exercise Solutions Principles of Econometrics 46 397 EXERCISE 116 The least squares estimates of the demand and supply equations are XR 116 LS estimations 1 2 Denends Suppys C 10910 200328 371 122 F 00233 O3380 008 002 ps 07100 021 7 00764 119 RF 10009 008 A 30 30 Standard errors in parentheses plt005 plt001 plt0001 Reporting these equations in the usual format We have Demand Q 1091 002330P 071OOPS 007644DI se 007684 O2143 1191 I O3032 3313 006419 Supply Q 2003O338OP 1001PF se 002175 007639 1 1554 1310 Considering the supply equation rst the coef cients are almost equal to the estimates in 113b The standard errors of the least squares estimates are all smaller than those in Table 113b On the other hand the least squares demand coef cient estimates are very different to the estimates in Table 113a The intercept and coef cient of P have the opposite sign to their twostage least squares counterparts and the coef cient estimates of PS and D1 are much smaller than those in Table 113a Once again the least squares standard errors are smaller than the twostage least squares standard errors but even though they are smaller the coefficients of P and D1 are not signi cantly different from zero All coef cients have signs which agree with economic reasoning except for the positive coef cient of P in the least squares demand equation Economic reasoning suggests that it should be negative since the quantity demanded decreases when price increases Chapter 11 Exercise Solutions Principles of Econometrics 46 398 EXERCISE 117 la b Rearranging the demand equation Q 011 ocP oc3PS oc4DI ed yields 1 3 Pa Q a1a3PSa4DIe 2 5152Q53PS54DIud According to economic theory it is expected that there is an inverse relationship between price and quantity demanded so we expect 52 lt O If the price of a substitute increases the demand for truf es increases increasing the price of truf es so We expect 53 gt0 If disposable income increases and if truf es are a normal good then demand increases and equilibrium price increases We expect 54 gt O Rearranging the supply equation Q 31 BP B3PF e yields PBiQ 1313PFeS l1l2Ql3PFuS According to economic theory there is a positive relationship between quantity supplied and price Thus We expect 1 gt0 An increase in the price of a factor of production reduces supply and increases equilibrium price so We expect 13 gt O The estimated demand equation is Dependent Variable P Method TwoStage Least Squares Included observations 30 Instrument list C PS DI PF 0139 Coefficient Std Error tStatistic Prob C 1142841 1359161 0840843 04081 Q 2670519 1174955 2272869 00315 PS 3461081 1115572 3102517 00046 DI 1338992 2746707 4874899 00000 13 114234 26705Q 34611PS 133 399191 se 135916 11750 11156 27467 Chapter 11 Exercise Solutions Principles of Econometrics 4e 399 Exercise 117b continued 0 d The estimated supply equation is Dependent Variable P Method TwoStage Least Squares Included observations 30 Instrument list C PF DI PS Coefficient Std Error tStatistic Prob C 5879822 5859161 1003526 00000 Q 2936711 0215772 1361027 00000 PF 2958486 0155964 1896905 00000 or 587982 29367Q 29585PF 13 se 58592O2158 O1560 The signs are as We expected in part a and all coefficients are signi cantly different from zero since all pvalues are less than the level of signi cance of 005 The price elasticity of demand at the mean is calculated as 8 AQAQ 2X D AP APF 52 Q Using our estimates DLgtlt 1 gtlt6239724 l2725 52 Q 26705 18458 Figure xr117d is a sketch of the supply and demand equations using the estimates from part b and the given exogenous variable values The lines are given by linear equations Demand 13 114284 26705Q 34611gtlt 22 133899gtlt 35 1115801 26705Q Supply P 587982 29367Q 29585 x 23 9247029367Q Chapter 11 Exercise Solutions Principles of Econometrics 46 400 Exercise 117d continued 120 100 80 Demand SupIoy 60 40 20 Figure xr117d Demand and supply graph e The estimated equilibrium Values from part d are given by equating the supply and demand equations after substituting in the given exogenous Variable Values Therefore equating these equations yields 1115801 267O5QEQM 92470 29367QEQM QEQM 182503 When QEQM is substituted into the demand equation substituting into the supply equation will yield the same result We nd the equilibrium Value of P thus PEQM11158O1 26705 x 182503 628427 Using the reduced form estimates in Tables 112a and 112b the predicted equilibrium Values are QEQM RF 78951 06564 x 22 21672 x 35 05070 x 23 182604 PEQMRF 325124 17081gtlt 22 76025gtlt 35 l3539gtlt 23 628154 Comparing the equilibrium Values calculated using the results from part d to those calculated using the reduced form estimates We nd them to be almost equal Chapter ll Exercise Solutions Principles of Econometrics 46 401 Exercise 117 continued f The estimated leastsquares estimated demand equation is I3136195O1512Q136O7PS123582DI se 90872 O4988 05940 18254 All estimated coefficients are signi cantly different from zero except for the intercept term and the coef cient of Q The sign for the coefficient of Q is incorrect because it suggests that there is a positive relationship between price and quantity demanded Compared to the results from part b the coefficient of Q has the opposite sign and the estimated intercept and the coefficient of PS are much smaller The estimated supply equation is 13 52876326613Q29217PF se 50238 01712 O1482 All estimates in this supply equation are signi cantly different from zero All coefficient signs are correct and the coefficient Values do not differ much from the estimates in part b Chapter 11 Exercise Solutions Principles of Econometrics 46 402 EXERCISE 118 8 b C The summary statistics are presented in the following table Mean Standard Deviation Variable LFP 1 LFP 0 LFP 1 LFP 0 AGE 419720 432831 77211 84678 KIDSL6 01402 03662 03919 06369 FAMINC 24130 21698 11671 12728 On average women who work are younger have fewer children under the age of 6 and have a higher family income Also the standard deviation across all variables is smaller for working women 32 gt 0 A higher wage leads to an increased quantity of labor supplied 33 The effect of an increase in education is unclear B4 This sample has been taken for working women between the ages of 30 and 60 It is not certain whether hours worked increases or decreases over this age group 35 lt 0 36 lt 0 The presence of children in the household reduces the number of hours worked because they demand time from their mother 37 lt 0 As income from other sources increases it becomes less necessary for the woman to work N WIFEINC measures the sum of all family income excluding the wife s income The least squares estimated equation is Dependent Variable HOURS Method Least Squares Sample 1 753 IF LFP1 Included observations 428 Coefficient Std Error tStatistic Prob C 2114697 3401307 6217309 00000 LNWAGE 1740781 5421544 0321086 07483 EDUC 1444486 1796793 0803925 04219 AGE 7729976 5529450 1397965 01629 KIDSL6 3425048 1000059 3424845 00007 KIDS618 1150205 3082925 3730889 00002 NWIFEINC 0004246 0003656 1161385 02461 Chapter 11 Exercise Solutions Principles of Econometrics 46 403 Exercise 118c continued or Written out in full 177R 2115 17411nWAGE 1444EDUC 7730IGE se 34015422 1797 5530 3425KIDSL6 1150KIDS618 000425NWIFEINC 1000 3083 000366 The negative coefficient for lnWAGE is unexpected We expected this coefficient to be positive d The estimated reduced form equation is Dependent Variable LNWAGE Method Least Squares Sample 1 753 IF LFP1 Included observations 428 Coefficient Std Error tStatistic Prob C 0357997 0318296 1124729 02613 EDUC 0099884 0015097 6615970 00000 AGE 0003520 0005415 0650176 05159 KIDSL6 0055873 0088603 0630591 05287 KIDS618 0017648 0027891 0632765 05272 NWIFEINC 569E06 332E06 1715373 00870 EXPER 0040710 0013372 3044344 00025 EXPERquot2 0000747 0000402 1860055 00636 An additional year of education increases Wage by 00999gtlt 100 999 e The presence of EXPER and EXPER2 in the reduced form equation and their absence in the supply equation serves to identify the supply equation Assuming that this supply equation is part of a demand and supply simultaneous equation system M 1 2 1 1 Therefore only one exogenous variable needs to be absent from the supply equation for it to be identified and having 2 exogenous variables absent is sufficient for this requirement if these variables are strongly signi cant We see that EXPER is significant at the 01 level and EXPER2 is significant at the 5 level using a one tail test The Ftest of their joint significance yields an F value of 825 which gives a pvalue of 00003 While the joint test leads us to reject the null hypothesis that the coef cients of both EXPER and EXPER2 are zero the F value is less than the rule of thumb value for strong instrumental variables of 10 Chapter 11 Exercise Solutions Principles of Econometrics 46 404 Exercise 118 continued f The twostage least squares estimated equation is Dependent Variable HOURS Method TwoStage Least Squares Sample 1 753 IF LFP1 Included observations 428 Instrument list C EDUC AGE KIDSL6 KIDS618 NWIFEINC EXPER EXPERquot2 Coefficient Std Error tStatistic Prob C 2432198 5941718 4093425 00001 LNWAGE 1544818 4807387 3213426 00014 EDUC 1774490 5814259 3051961 00024 AGE 1078409 9577347 1125999 02608 KIDSL6 2108339 1769340 1191596 02341 KIDS618 4755708 5691786 0835539 04039 NWIFEINC 0009249 0006481 1427088 01543 0139 177K 2432 15451nWAGE 177EDUC 1078IGE Se 59424807 581 9577 211KIDSL6 4756KIDS618 000925NWIFEINC 177 5692 000648 The statistically signi cant coefficients are the coef cients of ln WAGE and EDUC The sign of ln WAGE has changed to positive and so is now in line with our expectations The other coef cients have signs that are not contrary to our expectations Chapter 11 Exercise Solutions Principles of Econometrics 46 405 Exercise 118 continued Bonus material Additional analysis of identification 6 In the solution above We noted that the F test of their joint signi cance yields an F Value of 825 which gives a pValue of 00003 While the joint test leads us to reject the null hypothesis that the coef cients of both EXPER and EXPER2 are zero the F Value is less than the rule of thumb Value for strong instrumental Variables of 10 If We use the StockYogo critical Value We can be more precise Testing the null hypothesis that the instruments are Weak against the alternative that they are not the critical Value for the F statistic is 1159 choosing the criteria based on the size of nominal 5 test having maximum size of 15 We cannot reject the null hypothesis that the instruments are Weak based on this criterion Indeed We cannot reject the hypothesis that the instruments are Weak unless We are Willing to accept a 25 rejection rate for a nominal 5 test Cri t i cal Val ues of endogenous regressors 1 Ho Inst rurrent s are weak of excluded inst rurrent s 2 I 1 0 1 5 20 25 2SLS Size of norri nal 5 WEI d test I 19 93 1159 8 75 7 25 Chapter 11 Exercise Solutions Principles of Econometrics 46 406 EXERCISE 119 3 b C d The endogenous Variables in this demand equation are lnQ and lnP as price and quantity are jointly determined by supply and demand The exogenous Variables are lnY and lnPB as income and the price of beef are determined outside the model or exogenously i ii iii W V Vi i ii iii W The intercept falls out of the model and the Variables are in differenced form The parameters of interest are not affected just attached to transformed Variables The generalized least squares transformation is discussed in Appendix 9A If p 1 then the transformed error vf is not serially correlated The serial correlation problem is solved The approximation 1OOAlny E Ay is accurate if the changes in the Variable are not too large Because the Variables in the equation are time series the Variables are growth rates The parameter 0L2 is the income elasticity of demand since its interpretation is the same as in the loglog demand model Since poultry is a normal good we anticipate OL2 gt0 The law of demand implies that OL3 lt 0 An increase in the price of the substitute good beef will increase the equilibrium price and quantity of poultry thus we expect OL4 gt O The endogenous Variables in this supply equation are lnQPROD and lnPRICE because these Variables are jointly determined by supply and demand The exogenous Variables are the price of broiler feed PF TIME and lagged production QPRODH B2 is the price elasticity of supply The law of supply suggests 32 gt0 An increase in the price of an input reduces equilibrium quantity thus we anticipate 33 lt 0 If there is technical progress there should be more output from unchanged inputs so we expect 34 gt 0 If a year of high production follows a previous year of high production then 35 gt O In this system of M 2 equations there must be at least M 2 1 1 Variable omitted from an equation for identi cation In the demand equation there are 3 Variables omitted price of broiler feed PF TIME and lagged production QPRODH The supply equation omits two Variables the changes in income Y and the price of beef Thus both equations are identified according to the order condition which is a necessary but not sufficient condition Chapter 11 Exercise Solutions Principles of Econometrics 46 Exercise 119 continued 6 The estimated reduced form for the change in lnP DLP is given below Dependent Variable DLP Method Least Squares Sample 1950 2001 IF YEARgt1959 AND YEARlt2000 Included observations 40 Coefficient Std Error tStatistic Prob C 2167566 1536048 1411132 01673 DLY 1963925 0632990 3102618 00038 DLPB 0453689 0195732 2317904 00266 LOGPF 0142191 0077109 1844021 00739 TIME 0007787 0009202 0846152 04034 LOGQPROD1 0259794 0202478 1283072 02081 i The reduced form shows that increases in the growth of income DLY and in the growth of beef price DLPB have positive and signi cant at the 05 level effects on the equilibrium growth rate of price The effect of growth in the price of feed logPF has a positive and signi cant at the 10 level effect on equilibrium growth rate of price The other variable are not significant ii The actual growth in price in 2000 was 26 The predicted value based on the reduced form in i is 3134 The 95 interval estimate is 1107 1734 using the Icritical value 20322 34 degrees of freedom The actual value is inside this rather wide interval 407 YEAR DLP DLPFSEF DLPLB DLPF DLPUB 2000000 0026290 0069893 0110700 0031340 0173381 Chapter 11 Exercise Solutions Principles of Econometrics 46 Exercise 119 continued 0 The estimated reduced form for lnP is i ii Dependent Variable LOGP Method Least Squares Sample 1950 2001 IF YEARgt1959 AND YEARlt2000 Included observations 40 Coefficient Std Error tStatistic Prob C 2811041 1852558 1517384 01384 LOGPF 0272105 0092998 2925939 00061 TIME 0031646 0011098 2851342 00074 LOGQPROD1 0437906 0244199 1793231 00818 DLY 0246556 0763420 0322963 07487 DLPB 0400223 0236064 1695400 00991 408 The estimates show that increasing the price of feed logPF has a positive and signi cant at the 01 level effect on equilibrium lnPRICE The effect of TIME is negative and signi cant at the 01 level implying signi cant technological progress Lagged production and growth in the price of beef have positive and signi cant at the 010 level effects on lnPRICE The real price of chicken is 0946 The 95 interval estimate is 0701 0987 The point prediction using the natural predictor is 0831 The observed value is Within the interval YEAR P PHATLB PHAT PHATU B 2000000 0945990 0700588 0831498 0986869 To obtain this prediction interval We follow the procedure outlined in Chapter 455 ofPOE page 155 Chapter 11 Exercise Solutions Principles of Econometrics 46 409 Exercise 119 continued g The two stage least squares estimates are Demand Dependent Variable DLQ Method TwoStage Least Squares Sample 1950 2001 IF YEARgt1959 AND YEARlt2000 Included observations 40 Instrument list C LOGPF TIME LOGQPROD1 DLY DLPB Coefficient Std Error tStatistic Prob DLY 0856237 0150318 5696173 00000 DLP 0453350 0110838 4090210 00002 DLPB 0311649 0106347 2930493 00058 Supply Dependent Variable LOGQPROD Method TwoStage Least Squares Sample 1950 2001 IF YEARgt1959 AND YEARlt2000 Included observations 40 Instrument list C LOGPF TIME LOGQPROD1 DLY DLPB Coefficient Std Error tStatistic Prob C 2784102 1158856 2402458 00217 LOGP 0227421 0245024 0928162 03597 LOGPF 0147371 0077867 1892606 00667 TIME 0018584 0009831 1890218 00670 LOGQPROD1 0628437 0164478 3820798 00005 The demand equation estimates are the correct signs and signi cant at the 01 level The income elasticity of demand is estimated to be 0856 The price elasticity of demand is estimated to be 0453 and the crossprice elasticity of demand is 0312 The supply estimates reveal that the price elasticity of supply is not estimated very precisely and it is statistically insigni cant The estimated coef cient of the price of feed implies that a 1 increase in the price of feed decreases supply by 0147 percent The estimate is signi cant at the 10 level The estimated coef cient of TIME is positive and signi cant at the 10 level showing that technology has increased the quantity produced by about 18 per year Finally lagged production is very signi cant and positive Chapter 11 Exercise Solutions Principles of Econometrics 46 410 Exercise 119 continued h g Adding the log of exports as an instrument yields the following estimates of the supply equation Dependent Variable LOGQPROD Method TwoStage Least Squares Sample 1950 2001 IF YEARgt1959 AND YEARlt2000 Included observations 40 Instrument list C LOGPF TIME LOGQPROD1 DLY DLPB LEXPTS Coefficient Std Error tStatistic Prob C 3342003 1240020 2695120 00107 LOGP 0408017 0193525 2108346 00422 LOGPF 0194669 0074501 2612968 00131 TIME 0024716 0009232 2677104 00112 LOGQPROD1 0542197 0170358 3182687 00031 The effect of using this instrument is to increase the magnitudes of the coefficients and reduce their pvalues except for lagged production Exports are a good instrument in the sense that they should be strongly correlated with the endogenous variable PRICE However if exports are jointly determined with price and domestic consumption then exports are endogenous and correlated with the supply equation making it an invalid instrument Using exports as an instrument means that we have two surplus instruments Testing their validity using Sargan s NR2 test yields a pvalue of 0657 indicating that we cannot reject the validity of the overidentifying surplus instruments Bonus Material on instrument strength In part g we noted that the demand equation estimates are of correct sign and significant However for the demand equation the first stage F statistic is only 373 which is far less than the desired rule of thumb The StockYogo critical values are 534 for maximum relative bias of 30 Table 1OE2 and 831 for 25 test size for a nominal 5 test Table 1OE1 Similarly for the supply equation the first stage F is 188 The StockYogo critical value is 725 for 25 test size for a nominal 5 test Table 1OE1 Chapter 11 Exercise Solutions Principles of Econometrics 4e 411 Exercise 119 continued Bonus Material on instrument strength h The log of exports is statistically signi cant in the following rst stage regression Stata output Fi rststage regression of Ip 08 estimation Esti mat es efficient for hormskedasti ci ty only Statistics consi st ent for hormskedast i ci ty only Nunber of obs 40 F 6 33 5590 Prob gt F 00000 Total centered S8 1763650454 Centered F2 09104 Total uncentered S8 3267532368 Uhcentered R2 09516 Residual S8 1579906212 Root NBE 06919 Ip I Coef Std Err t Pgtt 9596Cbnf Interval Ipf I 2777852 0864871 321 0003 1018258 4537447 tine I 0248097 0106693 2 33 0026 0465165 0031028 qprod1 I 0330933 278201 012 0906 5329108 5990974 dly I 1278641 7112976 018 0858 1319282 157501 dlpb I 4939115 2225954 222 0033 0410377 9467853 Iexpts I 1042761 4141909 2 52 0017 2000836 1885439 cons I 5263764 2173379 0 24 0810 3895396 4948148 Included instruments Ipf time qprod1 dly dlpb Iexpts The rst stage F for the significance of the instruments is 356 which is far less than the desired rule of thumb The StockYogo critical Values are 539 for maximum relative bias of 30 Table 1OE2 and 780 for 25 test size for a nominal 5 test Table 1OE1 We cannot reject the null hypothesis that the instruments are Weak Chapter 11 Exercise Solutions Principles of Econometrics 46 412 EXERCISE 1110 8 b The twostage least squares estimation of the supply equation 1114 is Dependent Variable LQUAN Method TwoStage Least Squares Included observations 111 Instrument list MON TUE WED THU STORMY Coefficient Std Error tStatistic Prob C 8628354 0388970 2218256 00000 LPRICE 0001059 1309547 0000809 09994 STORMY 0363246 0464912 0781321 04363 01 1 86280001061nPRICE O363STORMY se O3890 131 0465 The signs of these estimated coef cients are as expected The coef cient 02 is positive suggesting that there is a positive relationship between price and quantity supplied However this coef cient is virtually zero and is not signi cant at any level The coef cient 03 is negative agreeing that less sh are supplied in stormy weather but it is also not signi cant The elasticity of supply is estimated as the coef cient of lnPRICE since this is a loglog equation Thus 85 00011 implying that supply is inelastic The new demand equation is lnQUAN OL1 OL2lI1PRICE OL3M0N OL3TUE OL4WED oL5THU oc6RAINY oc7C0LD ed The algebraic reduced form for lnPRICE is 1nPRICE 71 7cM0N 7c3TUE 7c4WED 7c5TH U 7c6STORMY oc7RAINY oc8COLD v Chapter 11 Exercise Solutions Principles of Econometrics 46 413 Exercise 1110 continued c The estimated reduced form equation is Dependent Variable LPRICE Method Least Squares Sample 1 111 Included observations 111 Coefficient Std Error tStatistic Prob C 0290228 0082069 3536396 00006 MON 0121576 0108589 1119604 02655 TUE 0056677 0106981 0529786 05974 WED 0028360 0108520 0261330 07944 THU 0040420 0105824 0381961 07033 STORMY 0312658 0081793 3822553 00002 RAINY 0016733 0093620 0178737 08585 COLD 0080989 0074359 1089155 02786 0139 1n77uT 02902 01216MON 00567TUE 00284WED 00404THU se 00821 O1086 01070 O1085 O1058 03127STORMY 00167RAINY 00810COLD 00818 00936 00744 The degrees of freedom for the F test of the joint signi cance of all Variables except for ST ORMY are 6 103 The test returns a pValue of 07229 which is much larger than the level of significance 005 This implies that We cannot reject the null hypothesis that all coefficients are equal to zero Thus the instrumental Variables are not adequate for estimation of the supply equation The Value of the F statistic is only 061 far below the rule of thumb Value of 10 Chapter 11 Exercise Solutions Principles of Econometrics 4e 414 Exercise 1110 continued d The least squares and twostage least squares estimates of the demand and supply equations are XR1110d 2SLS and LS estimations 1 2 3 4 DEM4NDLS DEM4ND2SLS SUPPL 1 LS SUPPL 1 2515 C 8 8189 8 4417 8 5009 8 5848 018 022 010 032 LPHOE 0 5448 1 2228 0 4381 0 1489 018 053 019 108 MW 0 0318 0 0333 0 21 023 TUE 0 4935 0 5328 020 022 MED 0 5392 0 5758 021 022 THU 0 0948 0 1179 020 022 H4NY 0 0888 0 0720 0 18 0 19 can 0 0818 0 0881 0 13 0 17 S739O39391W 0 2180 0 3130 0 18 039 Standard errors in parentheses plt005 plt001 plt0001 Discussion of demand equation The estimated sign for lnPRICE is as expected and the coefficient is statistically signi cant in both estimations We see that the twostage least squares estimate is more negative suggesting that using least squares has an upwards bias on the coefficient of lnPRICE The coefficient estimates of the dummy variables TUE WED and THU have the same signs in both estimations and the dummy variable MON has a different sign MON and THU are not signi cantly different from zero in both model estimations The signs of the ZSLS estimates of the coef cients of RAINY and COLD are not as expected since rainy and cold days are meant to deter people from eating out However We note that these coefficient estimates are not significantly different from zero at a 5 level of significance in both model estimations Discussion of supply equation The coefficient for lnPRICE does not have the expected coefficient in either estimation The negative coefficient estimate is not consistent with economic theory which says that quantity supplied and price are positively related In addition to having the Wrong sign the twostage least squares estimate has a large standard error which could be a result of inadequate instrumental variables The coefficient for ST ORMY has the expected negative coefficient but is not significantly different from zero in both estimations Chapter 11 Exercise Solutions Principles of Econometrics 46 415 Exercise 1110 continued e The augmented supply equation is lnQUAN B B lnPRICE B3STORMY B4MIXED e The demand equation is as speci ed in part b The least squares estimated reduced form equation for lnPRICE is Dependent Variable LPRICE Method Least Squares Included observations 111 Coefficient Std Error tStatistic Prob C 0373905 0084282 4436355 00000 MON 0114093 0104875 1087897 02792 TUE 0076200 0103509 0736169 04633 WED 0060763 0105366 0576681 05654 THU 0033436 0102202 0327152 07442 STORMY 0416731 0086674 4808031 00000 RAINY 0004910 0090483 0054265 09568 COLD 0063500 0072045 0881394 03802 MIXED 0231099 0079313 2913736 00044 0139 1n7R1T o3739 01141MON 00762TUE 00608 WED 00334THU se 00843 o1o49 01035 01054 01022 O4167STORMY 00049RAINY 0063 SCOLD 02311MIXED 00867 00905 00720 00793 The F test for the joint signi cance of the coef cients of MON TUE WED THU RAINY and COLD has an F statistic Value of 05432 and a pValue F 6 102 of 07742 Since this pValue is larger than the level of signi cance 005 and 0542 lt170956102 2189 We cannot reject the null hypothesis that these coef cients are equal to zero Therefore since the instrumental Variables that are required to identify the supply equation are not statistically signi cant the addition of MIXED does not increase the chances of estimating the supply equation by twostage least squares Chapter 11 Exercise Solutions Principles of Econometrics 4e 416 Exercise 1110 continued 0 The least squares and twostage least squares estimates of the demand and supply equations are XR1110f 2SLS and LS estimations 1 2 3 4 E ALL539 E AL2SZS 51 ZXLS39 EMTTZXLZSZS 86169 85130 85570 91348 016 019 013 057 LFTWCZF 05446 09470 04021 10723 018 041 020 141 AITV 00316 0 0069 021 021 DJ 04935 05168 020 021 1497 05392 05608 021 021 771 00948 01085 020 021 HMA0 00666 00698 0 18 0 18 CCZL7 00616 00153 0 13 0 15 STOWW 0 2738 0 9178 019 065 A4XEZ7 0 1062 0 4541 0 17 039 Standard errors in parentheses plt005 plt001 plt0001 Discussion of demand equation The estimated sign for lnPRICE is as expected and the coef cient is statistically signi cant in both estimations We see that the twostage least squares estimate is more negative suggesting that using least squares has an upwards bias on the coefficient of lnPRICE The coefficient estimates of the dummy variables TUE WED and THU have the same signs in both estimations and the dummy variable MON has a different sign MON and THU are not signi cantly different from zero in both estimations The signs of the coefficients of RAINY and COLD are not as expected with the exception of the least squares coefficient estimate of COLD We can note that these coef cient estimates are not significantly different from zero at a 5 level of significance in both estimations Discussion of supply equation The coefficient for lnPRICE has the expected sign in the two stage least squares estimation and an unexpected sign in the least squares estimation This is a slight improvement on the results obtained in part d However this coefficient remains statistically insignificant in the twostage least squares estimation The coefficients for ST ORMY and MIXED have the expected negative coefficient but are not significantly different from zero in both estimations Chapter 11 Exercise Solutions Principles of Econometrics 4e EXERCISE 1111 a The estimated reduced form equation is 417 Dependent Variable LPRICE Method Least Squares Sample 1 111 IF CHANGE1 Included observations 77 Coefficient Std Error tStatistic Prob C 0481515 0097537 4936764 00000 MON 0037860 0122927 0307990 07590 TUE 0067762 0123268 0549712 05842 WED 0142633 0129110 1104736 02730 THU 0257394 0126655 2032245 00459 STORMY 0443856 0082429 5384721 00000 0139 mHieF15 04815 00379MON 00678TUE 01426 WED 02574THU se 00975 01229 01233 01291 O1267 04439STORMY 00824 The pValue for testing the null hypothesis H0 2762 0 is 00000 Since this Value is less than the level of signi cance 005 We reject the null hypothesis and conclude that this coef cient is signi cantly different from zero The F test Value is 2900 Well above the rule of thumb threshold of 10 It is important to test for the statistical significance of ST ORMY because it is the supply equation s shift variable It is required to be statistically significant for the demand equation to be identified If ST ORMY is not statistically signi cant then the twostage least squares regression and the estimation procedure will be unreliable Bonus material The StockYogo test for Weak instrument critical Value using the criteria of test size is 1638 Table 10E1 if We can tolerate a test with Type I error of 10 for a 5 nominal test Chapter 11 Exercise Solutions Principles of Econometrics 46 418 Exercise 1111 continued b The null hypothesis of this Hausman test is H0 coVlnPRICEe O which is tested by testing for the significance of the coefficient of 192 in 1nQUAN ocl oc2 1nPRICE oc3MON oc4TUE oc5WED oc6TH U 5192 error In the results below 132 is denoted VHAT2 Dependent Variable LQUAN Method Least Squares Included observations 77 after adjustments Coefficient Std Error tStatistic Prob C 8362892 0180431 4634949 00000 LPRICE 1018517 0316737 3215656 00020 MON 0295299 0209392 1410268 01629 TUE 0345265 0209403 1648803 01037 WED 0361873 0220716 1639538 01056 THU 0394476 0226110 1744621 00854 VHAT2 0821220 0375889 2184743 00323 The t statistic and p Value for the null hypothesis H0 5O are 21847 and 00323 respectively Since this pValue is less than the level of signi cance 005 We reject the null hypothesis and conclude that lnPRICE is endogenous The robust Version of this test yields Istatistic of 227 and thus our conclusion is unchanged Chapter 11 Exercise Solutions Principles of Econometrics 46 419 Exercise 1111 continued c The least squares and twostage least squares estimates of the demand equation are XR 11110 2SLS and LS estimations 1 2 DEM4NDLS DEM4ND2SLS C 3 5327 3 3629 0 17 0 19 LPHOE 0 4354 1 O185 0 13 0 34 MW 0 2923 0 2953 0 21 0 22 TUE 0 3500 0 3453 021 0 22 MED 0 4031 0 3619 0 23 0 23 THU 0 2690 0 3945 0 22 0 24 N 77 77 Standard errors in parentheses plt005 plt001 plt0001 These estimates have the expected signs The twostage least squares and least squares estimates are Very similar in Values with the exception of the coefficient of lnPRICE Both estimation procedures conclude that the day indicator Variables are not signi cant at a 5 level of significance Compared to Table 115 all estimated coef cients have the same sign except for the coefficient of MON Also the intercept estimate and the coef cient estimate of lnPRICE are similar but the coefficient estimates for TUE WED and THU are quite different Furthermore all of the part c twostage least squares estimates of the Weekday indicator Variables are insignificant Whereas Table 115 shows that TUE and WED are statistically signi cant Exercise 1111 continued Chapter 11 Exercise Solutions Principles of Econometrics 46 d The estimated reduced form equation is 0139 These results are Very different to those obtained in part a All the coef cients of the Dependent Variable LPRICE Method Least Squares Sample 1 111 IF CHANGE 0 Included observations 34 Coefficient Std Error tStatistic Prob C 0010302 0112051 0091936 09274 MON 0171103 0218527 0782984 04402 TUE 0032193 0185680 0173381 08636 WED 0190059 0171072 1110988 02760 THU 0244165 0164665 1482796 01493 STORMY 0149337 0166718 0895749 03780 420 mmi2Fm 00103 O1711MON 00322TUE O1901WED O2442THU 56 01121 O2185 O1857 01493 STORMY 01667 01711 O1647 Weekday indicator Variables have opposite signs and the coef cient for ST ORMY is smaller In addition in part a the only Variables which were not statistically signi cant were MON TUE and WED In part d all exogenous Variables are statistically insigni cant Comparing these results to Table 114b all of the estimated coef cients have Very different Values although the only estimated coef cient with the opposite sign is the coef cient of THU All Weekday indicator Variables are statistically insigni cant in both estimated regressions However ST ORMY is statistically signi cant in Table 114b and not statistically signi cant in the above regression Chapter 11 Exercise Solutions Principles of Econometrics 46 421 Exercise 1111 continued e As described in part b the Hausman test is a test for the endogeneity of lnPRICE which is tested by testing for the signi cance of the coefficient of 192 in lnQUAN OL1 OL2lI1PRICE OL3M0N OL4TUE OLSWED oc6THU 5192 ed The variable V2HAT 132 Dependent Variable LQUAN Method Least Squares Sample 1 111 IF CHANGE 0 Included observations 34 Coefficient Std Error tStatistic Prob C 8776691 0270090 3249548 00000 LPRICE 0868443 2676883 0324423 07481 MON 0901467 0548862 1642430 01121 TUE 0842788 0427418 1971814 00590 WED 0872323 0607449 1436043 01625 THU 0354528 0719563 0492699 06262 V2HAT 0109989 2714966 0040512 09680 The t statistic and p value for the null hypothesis H0 50 are 00405 and 09680 respectively Since this pvalue is greater than the level of significance 005 We do not reject the null hypothesis and conclude lnPRICE does not show signs of endogeneity This is consistent with Graddy and Kennedy s expectation that when inventory changes are small simultaneity between demand and supply does not exist Chapter 11 Exercise Solutions Principles of Econometrics 46 422 Exercise 1111 continued f The least squares and twostage least squares estimates of the demand equation are XR1111f 2SLS and LS estimations 1 2 DEM4NDLS DEM4ND2SLS 0 8 7756 8 7767 026 027 LPHI 05 0 9754 0 8684 044 2 63 MN 09118 0 9015 048 054 TUE 0 8409 0 8428 042 042 MED 0 8904 0 8723 041 060 THU 0 3786 0 3545 0 40 0 71 N 34 34 Standard errors in parentheses plt005 plt001 plt0001 All the estimates have the expected signs and are almost identical The major difference between the two sets of estimates is that as a consequence of the smaller least squares standard errors all of the least squares coefficient estimates are signi cantly different from zero except those for MON TUE and THU whereas none of the twostage least squares coef cient estimates are signi cantly different from zero Comparing these Values to those in part c we find that the coefficient estimates for lnPRICE appear to be quite similar with the exception of the least squares coef cient estimate of lnPRICE in part c which is likely to exhibit simultaneous equation bias Also the coefficient of lnPRICE is always significantly different from zero in part c and only significant in the least squares part f estimation The estimated Values of the coefficients of the weekday indicator Variables are Very different Chapter 11 Exercise Solutions Principles of Econometrics 46 423 Exercise 111 1 f continued Part c models the demand for sh when there are large changes in inventory and part f models the demand for fish for small changes in inventory It has been postulated that when more fish are sold and bought causing large changes in inventory sellers are more responsive to prices and therefore endogeneity is present and on the days Where there is little change in inventory endogeneity should not be present This is supported by our estimates which show that the two stage least squares and least squares coefficient estimates of lnPRICE are similar when CHANGE 0 but very different when CHANGE 1 This discrepancy suggests that a coef cient bias exists when CHANGE 1 due to endogeneity Also note that the least squares estimate of the price elasticity of demand when CHANGE O is similar in magnitude to the twostage least squares estimate of the price elasticity of demand when CHANGE 1 CHAPTER 1 2 Exercise Solutions 424 Chapter 12 Exercise Solutions Principles of Econometrics 46 425 EXERCISE 121 a The AR1 model yr pyr1 vr can be rewritten as a function of lagged errors Y1 P0 V1 Y2 P1V2 PP0 V1 V2 P20 PV1V2 2 yr Vt pVt 1p Vt 2 39 pty0 The mean of yr is Eyr Evr pvrr p2vr2 O since the error vr has zero mean and the Value of p y0 is negligible for a large I The variance of y is Varyr Evr pvrr p2vr2 2 Evr pvr1 p2vr2 vr pvr1 p2vr2 Evr2 p2vr21 p4vr22 since Evrrvrr Oj 7 k o2lp2p2 SlI1C EVr2EVf1 Gi 621192 Where 1 p2 p2 11 p2 is the sum of a geometric progression The covariance between yr and yr2 is coVyr yr2 Evr pvrr p2vr2 p3vr3 vr2 pvr3 p2vr4 Ep2vi2 p v53 p6v54 62p21p2 94 62p2 192 Chapter 12 Exercise Solutions Principles of Econometrics 46 426 Exercise 121 continued b The random Walk model yr yrr vr can be Written as a function of lagged errors Y1 Y0 V1 2 32 31V2 00 V1V2 30 ZVs s1 t y yH 14 yo 214 s1 Where yo is the initial Value The mean of yr is Eyry0Evrv2vry0 since EvrO The variance of yr is Va1 yt Ely Ey2 Ey0 vr 122 vr y0 l vrv2wgf 2 2 2 2 2 EhfH5mg toi since Evr2 Evr2r 62 V The covariance of yr and yr2 is coVyr yr2 Evr vrr vr2 vr3 vr2 vr3 vr4 2 2 2 y z 39 oir 2 Chapter 12 Exercise Solutions Principles of Econometrics 46 427 EXERCISE 122 For W since the tau 3178 is less than the 5 critical Value of 286 the null of nonstationarity is rejected and We infer that W is stationary For Y since the tau 1975 is greater than the 5 critical Value of 286 the null of nonstationarity is not rejected and We infer that Y is not stationary For X since the tau 3099 is greater than the 5 critical Value of 341 the null of nonstationarity is not rejected and We infer that X is not stationary For Z since the tau 1913 is greater than the 5 critical Value of 341 the null of nonstationarity is not rejected and We infer that Z is not stationary Chapter 12 Exercise Solutions Principles of Econometrics 46 428 EXERCISE 123 Consider the time series of form yr or yrr vr vr N Oo2 Subtract yr1 from both sides of the equation yr yrr Ayr oc vr Hence yr is integrated of order 1 since it had to be differenced once yr yrr to achieve stationarity Now consider the time series of form yr 2 yr1 yr2 x vr Subtract yr1 from both sides of the equation Ayr Ayr1 or vr Where Ayr1 yr1 yr2 Thus Ayr is integrated of order 1 since its rst difference Ayr Ayrr is stationary In other Words yr is integrated of order 2 because it had to be differenced twice to achieve stationarity Chapter 12 Exercise Solutions Principles of Econometrics 46 429 EXERCISE 124 a A plot of the data is shown below The data appears to be uctuating and it may be stationary OH 55 SU 45 40 35 30 25 10 15 I I I I II I I I I I I I Ill I I I I I I I Ill I I I I I I I Ill I I I I I l I III I I I I I I I Ill l I I I I I I Ill I I I I I I I II I I I I I I I II 90 DD 10 20 30 40 50 80 YD Figure xr124a Plot of time series for oil b Since the data appears to uctuating around a constant term We use the Dickey Fuller test which includes a constant term A0IL 026901L 0942 tau 3625 Since the tau 3625 is less than the 5 critical Value of 286 the null of nonstationarity is rejected and We infer that OIL is stationary c Since OIL is stationary it is integrated of order 0 Chapter 12 Exercise Solutions Principles of Econometrics 46 430 EXERCISE 125 a A plot of the data is shown below The data appears to be trending and hence may be nonstationary BOND 900 850 800 I50 700 650 e lllllllllllllllllllIlllllll 58 69 TU 71 T2 3 74 75 Figure xr125a Plot of time series for bond yields b Since the data appears to uctuating around a trend we use the Dickey Fuller test which includes a constant term and a trend OO35BOND1 27866 0015t O459ABOND1 tau 1 83 5 An augmented DickeyFuller test with one lagged term ABONDH was needed to ensure that the residuals were not autocorrelated Since the tau 1835 is greater than the 5 critical Value of 341 the null of nonstationarity is not rejected and we infer that BOND is not stationary c The rst difference of the series DBOND BOND BOND 1 is shown below DBOND 40 30 20 20 30IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 68 69 U 1 72 3 4 5 Figure xr125a Plot of time series for bond yields Chapter 12 Exercise Solutions Principles of Econometrics 46 431 Exercise 125c continued Since the data appears to uctuating around a constant We use the Dickey Fuller test which includes a constant term O532DBONDH 0871 tau 5955 Since the tau 5955 is less than the 5 critical Value of 286 the null of nonstationarity is rejected and We infer that DBOND is stationary d Since BOND has to be differenced once to achieve stationarity We conclude that BOND is integrated of order 1 Chapter 12 Exercise Solutions Principles of Econometrics 46 432 EXERCISE 126 8 b Plots of the data CONSUMPTION and INCOME are shown below CONSUMPTION INCOME uocm 3950 mm nonm 1mm 12001 encm mm mm sumo wcm 5 13 39039 mm an IIIIIIIIIIIIIII 39as39ue39 oquotgequot9 quotss39 e39139o3939n39iu39 563958quot909e9t96960D0r20t Figure xr126a Plots of time series for CONSUMPTION and INCOME Since CONSUMPTION appears to be trending we use the Dickey Fuller test which includes a constant term and a trend 0024CONSUMPTIONH 67176 16188t tau 1550 Since the tau 1550 is greater than the 5 critical Value of 341 the null of nonstationarity is not rejected and we infer that CONSUMPTION is not stationary Since INCOME appears to trending we the Dickey Fuller test which includes a constant term and a trend 0040INCOME1 2378300 54248t tau 0894 Since the tau 0894 is greater than the 5 critical Value of 341 the null of nonstationarity is not rejected and we infer that INCOME is not stationary To determine the order of integration we need to test the first differences A plot of the differences in CONSUMPTION DC and in INCOME DD is shown below DC DI 31113 21113 0 mm mm Illlllllllllllllllll39 9lm020C E llllllllllllllllllll m9Cmm0l Figure xr126b Plots of differenced series Chapter 12 Exercise Solutions Principles of Econometrics 46 433 Exercise 126b continued Since DC appears to uctuating around a constant we use the DickeyFuller test which includes a constant term In this case the test results are sensitive to the number of augmentation terms that are included and in turn the number of augmentation terms included will depend on the selection criterion used by your software We present results for the case with no augmentation terms and the case with 2 augmentation terms With no augmentation terms the estimated equation is AEC 0714DC 86074 tau 6579 In this case since the tau Value 6579 is less than the 5 critical Value of 286 the null of nonstationarity is rejected and we infer that DC is stationary The order of integration is determined from the number of times a series has to be differenced to render it stationary Since we concluded that the rst difference of CONSUMPTION is stationary it follows that CONSUMPTION is integrated of order 1 Using an augmented DickeyFuller test with two lagged terms ADCH ADCH to ensure that the residuals are not autocorrelated leads to a different result AEC 0295DC 359944 O617ADC1 O428ADC tau 2228 In this case since the tau 2228 is greater than the 5 critical Value of 286 the null of nonstationarity is not rejected and we infer that DC is not stationary To nd the order of integration we have to difference the series again to check for stationarity The difference of a difference DDC DC DC 1 is also known as the second difference of CONSUMPTION Its graph appears below DDC 3000 2000 1000 0 1000 2000 3000 4000 IIIIIIIIIIIllllIllIIIIIIIIIIllllllllllIIIIIIIIIIIlllllllllIIIIIIIIIIIIIIIIIIII 85 88 90 92 94 96 98 00 02 04 Figure xr126b Plot of second difference of CONSUMPTION Chapter 12 Exercise Solutions Principles of Econometrics 46 434 Exercise 126b continued 0 Since the second difference of CONSUMPTION DDC appears to uctuating around zero we the Dickey Fuller test without a constant term ADDC 2339DDC1 O524ADDCt1 tau 13661 Since the tau 13661 is less than the 5 critical Value of 194 the null of nonstationarity is rejected and we infer that DDC is stationary In this case it follows that since CONSUMPTION had to be differenced twice to be stationary that CONSUMPTION is integrated of order 2 Turning now to INCOME since DI appears to uctuating around a constant we use the Dickey Fuller test which includes a constant term A131 1187DI1 1464252 tau 10676 Since the tau 10676 is less than the 5 critical Value of 286 the null of nonstationarity is rejected and we infer that DI is stationary Since INCOME had to be differenced once to render it stationary it follows that INCOME is integrated of order 1 If we conclude that CONSUMPTION is 12 and INCOME is 11 then any estimated relationship between them will be spurious because they are not of the same order of integration However if we have concluded that CONSUMPTION and INCOME are both 11 then we need to test the stationarity of the residuals from a regression of CONSUMPTION on INCOME to determine whether the Variables are spuriously related or cointegrated The estimated equation is 9084 O9884INCOME The estimated DickeyFuller test equation for the residuals from this regression is Xe 0316 tau 3909 The tau Value of 3909 is less than the 5 critical Value of 337 and so we reject the null hypothesis that the residuals are not stationary Given the residuals are stationary in this case we conclude that the Variables CONSUMPTION and INCOME are cointegrated Chapter 12 Exercise Solutions Principles of Econometrics 46 435 EXERCISE 127 8 b The series TXNAG and USNAG are graphed below um uosnzo 65m mourn 61m 95550 55m 90013 51ID 35533 5m soma mn 75cm I I I I I I I I I I I I I 7 I I76 I I78 I lm I Iml I8 I l l P 4 T5 O5 m a P 4 m Figure xr127a Plots of time series for TXNAG and USNAG Both series TXNAG and USNAG are trending upwards The DickeyFuller tests with a constant and a trend are shown below OO24TXNIGH 123960 O804t 0763ATXNAG1 tau 1213 OO69USNAGH 5313248 33471 O798AUSNAGt1 tau 2792 For TXNAG since the tau 1213 is less than the 5 critical Value of 341 the null of nonstationarity is not rejected and we infer that TXNAG is not stationary For USNAG since the tau 2792 is less than the 5 critical Value of 341 the null of nonstationarity is not rejected and we infer that USNAG is not stationary Changes in the Variables DTX TXNAG TXN1G1 and DUS USNAG USN1G1 are shown below DTX DUS 1 7 I I76 I I78 I la I la I I8 I l I 7 I I76 I I78 I I I E I I8 I I I Figure xr127b Plots of rst differences DT S and DUS Both series DTX and DUS are uctuating around a constant The DickeyFuller tests with a constant are Chapter 12 Exercise Solutions Principles of Econometrics 46 436 Exercise 127b continued 0 d 6 TE 0226DTX1 8117 tau 2549 pvalue 01097 A7559 0230DUs 120547 tau 2587 pvalue 01017 Because the pvalues are greater than 010 at the 5 and 10 levels of signi cance we do not reject the null hypothesis of nonstationarity However using a level of significance of 11 we conclude that the change variables DTX and DUS are stationary This is an example where it would be prudent to gather more information so that a more decisive inference about the property of the data can be made Assuming for illustrative purposes that TXNAG and USNAG are 11 variables we can check whether they are cointegrated or spuriously related by testing the property of the regression residuals TXNAG 0096USNAG 2859739 1 19191 A 0015 10780A 1 t tau 0811 Since the tau 0811 is greater than the 5 critical value of 337 the null of no cointegration is not rejected The variables TXNAG and USNAG are spuriously related The regression of DTX on DUS is as follows ET 0036DUS 23258 1 3412 This result shows that the change in TXNAG is significantly related to the change in USNAG In c we are testing the relationship between nonstationary variables with a view to establishing their long run relationship In d we are testing the relationship between stationary variables with a view to establishing their short run relationship Chapter 12 Exercise Solutions Principles of Econometrics 46 437 EXERCISE 128 a The data series real gross domestic product GDP and the in ation rate INF are shown below GDP INF 16000 14000 12000 10000 8000 8000 4000 2000 I39lquotlquotlquotlquotlquotquotll39l39 IIIIIIIIIIIIIIIIIIIIIIIII 34 3 9 93 95 99 0 05 03 34 3 so 93 96 99 02 05 as Figure xr128a Plots of time series for GDP and INF Since GDP is trending We apply the DickeyFuller test with a constant and a trend E5 OO24GDPH 105797 2866t O552AGDPH tau 1961 Since the tau 1961 is greater than the 5 critical Value of 341 the null of nonstationarity is not rejected The Variable GDP is not stationary Since INF is Wandering around a constant We apply the DickeyFuller test with a constant ETVTV OO26INF1 0104 O6O8AINF1 O194AINF 0553A1NE tau 1350 O722AINE4 O4O6AINE5 O1OOAINE6 0277A1NE 0400A1NE An augmented Dickey Fuller test with 8 lagged terms AINP1 to AINE8 was needed to ensure that the residuals were not autocorrelated Since the tau 1350 is greater than the 5 critical Value of 286 the null of nonstationarity is not rejected The Variable INF is not stationary Chapter 12 Exercise Solutions Principles of Econometrics 46 438 Exercise 128 continued b To determine the order of integration of these series we need to examine the timeseries property of the differenced series Graphs of the rst differences of GDP DG and NF DP are shown below DG DP 300 2 200 1 100 0 0 100 1 200 lllllllllllllllllllllllll 2 quotquotquotquotquotquotquotquot39 84 8 90 93 96 99 02 05 08 84 8 90 93 96 99 02 05 08 Figure xr128b Plots of first differences for GDP and INF Since DG is uctuating around a constant we apply the DickeyFuller test with a constant AD7 O429DG1 43869 tau 5228 Since the tau 5228 is less than the 5 critical Value of 341 the null of nonstationarity is rejected The Variable DG is stationary It follows that since GDP has to be differenced once to be stationary that GDP is integrated of order 1 Since the rst difference of NF DP is uctuating around the zero line we apply the DickeyFuller test without an intercept AD7 O631DP O196ADP 0016ADP 0524ADP tau 4627 O186ADP4 O21OADP5 O126ADB6 0412ADP Since the tau 4627 is less than the 5 critical Value of 341 the null of nonstationarity is rejected The Variable DP is stationary It follows that since INF has to be differenced once to be stationary that INF is integrated of order 1 Chapter 12 Exercise Solutions Principles of Econometrics 46 439 Exercise 128 continued 0 We can use the fact that GDP is a random Walk process as a forecasting model Using the unit root test equation from part a reestimated assuming the coef cient of GDP1 is zero We obtain the following estimated forecasting model AG17 30601 0225 0553AGDP1 Given t 104 is 20094 the forecast for GDP for 20101 is G035 GDP104 30601 0225gtlt 105 0553AGDP104 142773 30601 23625 0553 X 1626 1442144 Similarly we can use the fact that INF is a random Walk process as a forecasting model Reestimating the model from part a We obtain the following forecasting model AINTV 0023 0563A1NF 0181A1NF 0505A1NF 0712AINE4 0395A1NF OO86AINE6 0284A1NF 0414A1NF The forecast for INF for 20101 is mag INFN4 0023 0563AINF104 0181AINF103 0505AINF10 0712AINF101 0395AINE00 0086AINF99 0284AINF98 0414AINF97 INESS 259 0023 0563 X 027 0181gtlt 023 0505 x 044 0712 x 011 0395 x 004 0086 x 001 0284 x 014 0414 x 040 301 Chapter 12 Exercise Solutions Principles of Econometrics 46 440 EXERCISE 129 8 b A plot of the data CANADA is shown below CANADA 1 J M ivf x 10 llI39llIll Illlllllllllllllllllllll 1975 1989 1985 1999 1995 2999 2995 Figure xr129a Plot of the time series CANADA Over the sample period 1971O11987212 the series appears to be uctuating around a trend The DickeyFuller test with a constant and a trend is shown below OO45CANADAH 0042 0000103 O186ACANADAH tau 2392 Since the tau 2392 is greater than the 5 critical value of 341 the null of nonstationarity is not rejected The variable CANADA is not stationary Over the sample period 1988O12006212 the series appears to be uctuating around a constant The DickeyFuller test with a constant is shown below OOO7CANADAH 0009 O244ACANADA1 tau 0897 Since the tau 0897 is greater than the 5 critical value of 286 the null of nonstationarity is not rejected The variable CANADA is not stationary The results for the two sample periods are consistent despite the appearance of different trendlike behavior Chapter 12 Exercise Solutions Principles of Econometrics 46 441 Exercise 129 continued c For the whole sample period we use a DickeyFuller test with a constant but without the trend term since its effect is insigni cant OOO7CANADAH 0008 O226ACANADA1 tau 1559 Since the tau 1559 is greater than the 5 critical Value of 286 the null of nonstationarity is not rejected The Variable CANADA is not stationary A plot of the rst difference DC CANADA CANADA1 is shown below DC W1 ll 4 06 04 02 0 E 1 U2 U4 05 llllllIlllllllllllllllllllllllllll 105 1080 1085 1000 1005 2000 2005 Figure xr129c Plot of rst difference for CANADA Since DC is uctuating around the zero line we apply the DickeyFuller test without an intercept AEC 0776DC tau 16461 Since the tau 16461 is less than the 5 critical Value of 194 the null of nonstationarity is rejected The Variable DC is stationary It follows that since CANADA has to be differenced once to be stationary that CANADA is integrated of order 1 Chapter 12 Exercise Solutions Principles of Econometrics 46 442 EXERCISE 1210 a Results from the three DickeyFuller tests are 1 Dickey Fuller Test 1 no constant term and no trend term AC71 0001Cs1 tau 0299 Since the tau 0299 is greater than the 5 critical Value of 194 the null of nonstationarity is not rejected The Variable CS is not stationary 2 Dickey Fuller Test 2 constant term but no trend term AC71 0051Cs1 4500 tau 3001 Since the tau 3001 is less than the 5 critical Value of 286 the null of nonstationarity is rejected The Variable CS is stationary 3 Dickey Fuller Test 3 constant term and trend term AC71 0068Cs1 5309 0004 tau 3483 Since the tau 3483 is less than the 5 critical Value of 341 the null of nonstationarity is rejected The Variable CS is stationary The result of the DickeyFuller test without an intercept term is not consistent with the other two results This is because Test 1 assumes that when the alternative hypothesis of stationarity is true the series has a zero mean This assumption is not correct see graph in part b b The graph suggests that we should use the DickeyFuller test with a constant term CSI 120 3 W 90 I s0 I B0 50 I I I I I I I I I I I I I I I I I I I I I I I I I I I I Figure xr1210b Plot of time series CS1 c Since the CS is stationary it suggests that the effect of news is temporary hence consumers remember and retain news information for only a short time Chapter 12 Exercise Solutions Principles of Econometrics 46 443 EXERCISE 1211 a The data for MEXICO and USA are plotted below 120 110 100 90 80 IU 60 50 I I I I I I I I I I I I I I I I I I I I I I I I I I an 32 34 as as an 92 94 96 93 on 02 04 05 mEgtltIc0 usA Figure Xr1211a Plots of MEXICO and USA The 3 tests for cointegration are 1 2 3 Cointegration Test 1 regression model has no intercept and no trend MEXICO O995USA A O062 1 t 195143 tau 1948 Since the tau 1948 is greater than the 5 critical Value of 276 the null of no cointegration is not rejected Variables MEXICO and USA are spuriously related Cointegration Test 2 regression model has an intercept term but no trend MEXICOO852USA l2l35 A OO88 1 z 50751 tatu 2078 Since the tau 2078 is greater than the 5 critical Value of 337 the null of no cointegration is not rejected Variables MEXICO and USA are spuriously related Cointegration Test 3 regression model has an intercept term and a trend MEXICO 1283USA 8166 02681 A 01072 z 9229 tau 2396 Since the tau 2396 is greater than the 5 critical Value of 342 the null of no cointegration is not rejected Variables MEXICO and USA are spuriously related Chapter 12 Exercise Solutions Principles of Econometrics 46 444 Exercise 1211 continued b 0 Since none of the tests supported the existence of cointegration including the one without a constant and a trend the results do not support the theory of convergence in economic growth Note however that the cointegration tests examine the relationship between the levels of the series not their growth rates Cointegration Test 1 is the most straightforward test of the comovement of MEXICO and USA The introduction of a trend in Cointegration Test 3 allows MEXICO to diverge from USA A constant term is unnecessary in this example because the two series have been standardised to the same base value If the variables are not cointegrated the relationship between MEXICO and USA can be examined by working with the stationary form of the variables which in this case is their first differences If USA is exogenously determined then one can estimate a dynamic model for MEXICO using the econometric techniques discussed in Chapter 9 Alternatively if USA is endogenously affected by MEXICO one can estimate a VAR model using the econometric techniques discussed in Chapter 13 Chapter 12 Exercise Solutions Principles of Econometrics 46 445 EXERCISE 1212 A plot of the data in imer2dat is shown below The graph shows the level Y the rst difference DY AY K Y1 and the second difference D2Y A2Yt AK AYH of the data Y DY 02v 20 20 06 15 15 1 04 m 10 02 05 5 m 00 0 05 02 5 5390 100 130 200 23950 30 quot10 39 39 39 39 39 quotquot4 I I I u I 5 quot3quot 395 3 25 3 50 100 150 200 250 300 Figure xr1212 Plots of Y and its rst and second differences The ADF unit root tests are shown below XY 0001Y1 0001 000006t 0991AY1 tau 3371 AA7 OO11AY1 0001 tau 1088 AA2Y 0987A2Y tau 16940 Since Y clearly has a trend the ADF test includes a constant and a trend Since the tau 3371 is greater than the 5 critical Value of 341 the null of nonstationarity is not rejected The variable Y is not stationary Since DY is uctuating around a constant the ADF test includes a constant Since the tau 1088 is greater than the 5 critical Value of 286 the null of nonstationarity is not rejected The Variable DY is not stationary Since D2Y is uctuating around zero the ADF test does not include a constant Since the tau 16940 is less than the 5 critical Value of 194 the null of nonstationarity is rejected The Variable DZY is stationary In other words K has to be differenced twice to achieve stationarity thus K is integrated of order 2 Chapter 12 Exercise Solutions Principles of Econometrics 46 446 EXERCISE 1213 3 A plot of the price indices in the United Kingdom and in the Euro Area is shown below 130 125 EURO 120 115 UK 110 105 100 95 lllllllllllllllllllIlllllllllllllllllllllllllllllllllll 98 9 98 99 DD 01 U2 U3 U4 U5 DB 0 U8 09 Figure xr1213 Plots of UK and Euro price indices The data are clearly not stationary and so We use the ADF test which includes a constant and a trend For UK An ADF test with a constant and a trend and 5 augmentation terms gives a mu value of 0996 which is greater than the 5 critical value of 341 Thus the null of nonstationarity is not rejected The variable UK is not stationary To assess Whether UK is 11 We perform an ADF test on the difference DUK UK UK1 An ADF test on DUK with a constant term and no augmentation terms gives a mu value of 1356 which is less than the 5 critical value of 286 Thus the null of nonstationarity is rejected We conclude that the differenced variable DUK is stationary Thus UK is 11 For EURO An ADF test with a constant and a trend and 6 augmentation terms gives a mu value of 2916 which is greater than the 5 critical value of 341 Thus the null of nonstationarity is not rejected The variable EURO is not stationary To assess Whether EURO is 11 We perform an ADF test on the difference DEURO EURO E URO1 An ADF test on DE URO with a constant term and no augmentation terms gives a mu value of 1 149 which is less than the 5 critical value of 286 Thus the null of nonstationarity is rejected We conclude that the differenced variable DE URO is stationary Thus EURO is 11 Chapter 12 Exercise Solutions Principles of Econometrics 46 447 Exercise 1213 continued b The least squares equation relating UK and EURO is 711 O799EURO 20051 Testing the residuals from this equation for stationarity using an ADF test equation with no constant or trend and no augmentation terms We obtain a mu Value of 0179 Since 0179 is greater than the 5 critical Value of 337 the null hypothesis of no cointegration is not rejected The Variables UK and EURO are spuriously related This conclusion is supported by the results from an error correction model Estimating the error correction model directly using nonlinear least squares We obtain AU7 000643U1lt 1027 00385EUR0 O8367AEURO 1 0349 0428 0017 11052 Estimating the error correction model using the residuals from the longrun equation We obtain AU7lt 000644 O8706AEURO 1 0348 1316 In both cases the residuals from the longrun equation have low IValues implying they are not signi cantly different from zero CHAPTER 1 3 Exercise Solutions 448 Chapter 13 Exercise Solutions Principles of Econometrics 4e 449 EXERCISE 131 For the firstorder VAR model below z 51111 512391 51 xx 52111 522391 52 a amp c Effects of a shock to y of size csy on y and x t 2 2 5111512x1 5115y 5120 Z 5115y X2 5201 522351 521Gy 522 O 521Gy t 3 3 Z 5112 512x2 5115115125215y X3 5212 522352 521511 52252lGy t 4 4 5113 512353 511511511 8l282lGy 512 521511 5225215y X4 5213 522353 521511511 5l252lGy 522 521511 622621Gy b amp d Effects of a shock to x of size ox on y and x i1 y1O x1o t 2 2 5111 512351 5110 512Gx 512Gx X2 5211 522x1 5210 522Gx 5225 t 3 J3 5112 512752 511512 512522 5x X3 5202 522x2 521512 522522 5x t 4 4 5103 512x3 2 511511512 512522 6x 512 521512 522522 5x X4 5203 522x3 2 521511512 512522 6x 522 521512 522522 5x Chapter 13 Exercise Solutions Principles of Econometrics 46 450 EXERCISE 132 1 step ahead forecasts J111 E 511yz 512 811 Z 511 512 2 Since E 511 Z O F xt1 E1 521yz 5229 82z1 Z 521 5229 2 Since E52z1 Z O 2 step ahead forecasts F J12 Er 511311 512xz1 81r2 Ez511511yz 51239 51z1 512 52130 52239 82r1 81z2 511511J1 51235 512521J1 52235 F x2 Er 521311 522xx1 322 Et5215111 51279 311 522 52130 52239 821 322 5215111 51235 522 52130 52235 3 step ahead forecasts J113 E 51012 512 81113 E au au auy aux suu au auy aux suu suu au auauy aux suu au auy aux suu suu suu au au auy aux au auy aux 512 521511y 5ux 522 521 5u2x F x3 Z E521J12 52295 83 12 E 5u5u Buy Bux 8u1 8u Buy Bux suu suu au auauy aux euu au auy aux suu auu auu auauauy aux au auy aux au auauy aux auauy aux Chapter 13 Exercise Solutions Principles of Econometrics 46 451 Exercise 132 continued 1step ahead forecast errors and Variances FEE J11 Etit1 81t1 VarFE1 5 FE xz1 Exz1 52z1 VarFE1x G 2step ahead forecast errors and Variances FE2y J12 E J12 51131z1 512521 812 VarFE2y 512155 512252c 55 FE x2 Ezx12521511 52252z1 522 Va139FE 55155 55253 53 3step ahead forecast errors and Variances FE3y J13 Eri13 511lt511511 512521 312 512 lt521511 52252z1 522 513 VarFE3y 516 512151226 51216 52516 52526 51226 6 3 X t3 Etxt3 52151181t1 51282t1 81t2 522 lt52181t1 52282t1 82t2 82t3 VarFE3quot 51516 51526 516 52516 5226 526 6 a The contribution of a shock to 2 on the 3step forecast error Variance of y is 2 4 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5y 511 511 512521 15115y 5115125x 5115y 5125215y 512525 5125 5y b The contribution of a shock to x on the 3step forecast error Variance of y is 5 51215122 512252 5122 5ii15 512151225 51215 51225515 51225525 51225 5 c The contribution of a shock to 2 on the 3step forecast error Variance of x is 222222 222 222 22 222 42 222 5y 521511 521 5225215215115y 5215125x 5215y 5225215y 5225 5225 5x d The contribution of a shock to x on the 3step forecast error Variance of x is 22242 222 222 22 222 42 222 5x 521512 522 522 15215115y 5215125x 5215y 5225215y 5225x 5225x 5x Chapter 13 Exercise Solutions Principles of Econometrics 46 452 EXERCISE 133 a To rewrite the VEC in VAR form rst expand the terms 01 y1 2 05y1 05 35x1 3 xt1 3 03H 03 21x1 Then rearrange in VAR form 2 05 1 05yt1 35x1 3 03 1 03H 21x1 y 1 Simplifying gives the VAR model f2 25 05y1 35x1 2 27 13y1 21x1 b To rewrite the VAR model in the VEC form first rearrange tenns so that the left hand side is in firstdifferenced fonn 0 y1 y1 07y1 03 024x1 3 xt1 x1 06H 06 052x1 Next recognize that the error correction tenn for the rst equation is the coef cient in front of the lagged Variable JH that is 03 Now factorize out this coef cient to obtain the cointegrating equation Af1 03y1 1 08xt1 Afc 06H 06 1 052x1 For the second equation factorize out the cointegrating equation to obtain the error correction coefficient 06 The VEC model is Af1 03y1 1 08x1 M 06y1 1 08x1 Chapter 13 Exercise Solutions Principles of Econometrics 4e EXERCISE 134 a Consider the following estimated VAR model Y 81111 812391 91 xx 52111 522391 V2 The forecasts for yr and xm are F J11 511 51235 F j x1 521 52235 The forecasts for H2 and xt2 are F F F J12 51111 512x1 F F F x2 52111 522xz1 b Consider the following estimated VEC model A 53511011 lA31x 1 91 Ax 6521011 lA31x 1 92 Rearrange terms as 7 6511 11 6511 1xr 1 911 X 652111 56521lA31 1xz 1 92 The forecasts for yr and xm are J111 6511 1 6511lA31x 3511 6521 652131 D35 The forecasts for H2 and xt2 are J112 6511 1 1 6511lA31xf1 F xz2 53521 652131 1x11 453 Chapter 13 Exercise Solutions Principles of Econometrics 46 454 EXERCISE 135 81 b C The data real GDP of Australia and real GDP of the US are shown in Figure 131 Both series are clearly nonstationary which is con rmed by the DickeyFuller test with an intercept and trend For AUS with no augmentation terms We obtain tau 0400 with corresponding pvalue 09866 For USA with one augmentation term we obtain tau 0265 with corresponding pvalue 09908 The estimated relationship with a constant included is T7 1o7210o1UsA r 266 164 The test for cointegration using the residuals from this equation is A 0139e1 tau 305 The 5 critical value is 337 Given 305 gt337 there is insufficient evidence to conclude that cointegration exists One could argue that a negative intercept is not sensible because the real GDP for Australia will be positive even when the GDP for the US is zero and vice versa The cointegration equation excluding the constant term is in equation 137 of the text The test of stationarity in the residuals is in equation 138 It leads to a reversal of the above test decision The estimated VEC model is reported in equation 139 of POE4 Chapter 13 Exercise Solutions Principles of Econometrics 46 455 EXERCISE 136 8 Output for the DickeyFuller test equations for C and Y with a constant and a trend are shown below The critical value for a 5 signi cance level is 341 Because 341 lt 298 and 341lt 168 We conclude that both C and Y are nonstationary series And in particular they are not trend stationary Note that C is labeled as CN in the output This change was made because the output comes from EVieWs in which C is a reserved name Augmented DickeyFuller Test Equation Dependent Variable DCN Method Least Squares Sample adjusted 1961Ql 2009Q4 Included observations 196 after adjustments Coefficient Std Error tStatistic Prob CN1 0041294 0013871 2977019 00033 DCN1 0184933 0069953 2643680 00089 DCN2 0206760 0069927 2956790 00035 DCN3 0185046 0070465 2626078 00093 C 0316165 0104296 3031421 00028 TREND1960Q1 0000335 0000118 2843330 00050 Augmented DickeyFuller Test Equation Dependent Variable DY Method Least Squares Sample adjusted 1960Q2 2009Q4 Included observations 199 after adjustments Coefficient Std Error tStatistic Prob Y1 0024805 0014761 1680440 00945 C 0201274 0113205 1777962 00770 TREND1960Q1 0000174 0000121 1438159 01520 More light is shed on this issue by examining the residuals from separate regressions of C and Y on a constant and a trend These residuals are displayed in Figure xr136a They appear to be nonstationary indicating that C and Y would not be adequately described as trend stationary variables Given that C and Y are nonstationary the next step is to check Whether they are cointegrated The residual series appear to move in similar directions suggesting that cointegration may be a possibility We test for this possibility in part b Chapter 13 Exercise Solutions Principles of Econometrics 46 456 Exercise 136a continued b ogconsumption Iog ncome 08 oo J liq ki 3f quotquot 39739 D0 fp irqki Elm 39395 04J 35 08 40 0 lllllllllllllllllllllllllllllllllllllllllllllllll I H l I I 60 65 0 5 80 85 90 95 D0 05 so 35 o 5 so 35 90 95 no 05 Residua 39gtctua Fined I I Resign m3 Fined Figure xr136a Trend lines and residuals from those trend lines tted to C and Y Results from the potentially cointegrating equation with a constant and no trend are Dependent Variable CN Method Least Squares Sample l960Q1 2009Q4 Included observations 200 Variable Coefficient Std Error tStatistic Prob C 040416 002505 16132 00000 Y 103529 000295 351305 00000 Testing for a unit root in the residuals from this equation We obtain the following output Augmented DickeyFuller Test Equation Dependent Variable DEHAT Method Least Squares Sample adjusted 1960Q3 2009Q4 Included observations 198 after adjustments Variable Coefficient Std Error tStatistic Prob EHAT1 008765 003051 2873 00045 DEHAT1 029941 006716 4458 00000 The tau Value unit root tValue of 2873 is greater than 337 indicating that the errors are not stationary and hence that We have no cointegration The relationship between C and Y could be a spurious one Chapter 13 Exercise Solutions Principles of Econometrics 46 457 Exercise 136b continued 0 Since both C and Y are trending and the coef cient of the trend in the unitroot test equation for C was signi cant at a 5 level of signi cance it is worth checking for a cointegrating relationship that includes a trend term In this case the estimated potential cointegrating relationship is Dependent Variable CN Method Least Squares Sample 1960Q1 2009Q4 Included observations 200 Variable Coefficient Std Error tStatistic Prob C 191299 018700 10230 00000 TREND 000248 000020 12454 00000 Y 073322 002435 30106 00000 The results from the unitroot test on the residuals from this equation are Augmented DickeyFuller Test Equation Dependent Variable DEHATT Method Least Squares Sample adjusted 1960Q4 2009Q4 Included observations 197 after adjustments Variable Coefficient Std Error tStatistic Prob EHATT1 011248 003471 3241 00014 DEHATT1 0 14301 007229 1978 00493 DEHATT2 022216 007094 3132 00020 The value tau 3241 is greater than the 5 critical value of 342 suggesting the residuals are nonstationary and that C and Y are not cointegrated However at a 10 level of signi cance there is evidence of cointegration The results from estimating a VAR model with lags of order 1 for the pair of I0 variables AC AK are provided in equation 1311 on page 504 of POE4 We now ask whether the model can be improved upon by adding more lags If we only include lags where the coef cients of i lagged variables are individually signi cant then a lag order of 1 is suitable If however we include lags when the lag coef cients of one or more of the variables is signi cant or a joint test of both coef cients at a given lag yields a signi cant result then a VAR with lags of order 3 is suitable Also increasing the lags to 3 eliminates serial correlation in the errors of the equation for AC The results from estimating a VAR3 are as follows Chapter 13 Exercise Solutions Principles of Econometrics 46 458 Exercise 136c continued Vector Autoregression Estimates Sample adjusted 1961Q1 2009Q4 Included observations 196 after adjustments Standard errors in amp tstatistics in DCN DY DcN1 013063 042060 007988 010757 1635 3910 DcN2 016620 001868 008161 010990 2037 0170 DcN3 017263 022013 007908 010650 2183 2067 Dy1 012820 020484 005987 008062 2141 2541 Dy2 001935 002100 006300 008484 0307 0247 Dy3 001834 003343 006048 008145 0303 0410 3 000342 000523 000092 000124 3699 4197 Lags for AC of orders 2 and 3 are signi cant at a 5 level in the consumption equation and lags of AC of orders 1 and 3 are signi cant at a 5 level in the income equation Lags of AY beyond 1 are not signi cant in either equation The following table contains the results from joint Wald tests of both coef cients at each lag in a VAR of lag order 3 Results are provided for each equation separately and both equations jointly When testing within each equation separately the joint test is for whether the two coef cients at a given lag are zero When testing the two equations jointly we are testing whether the four coef cients at a given lag are all zero Chapter 13 Exercise Solutions Principles of Econometrics 46 459 Exercise 136c continued The X2 Wald test for a single equation is that described in Appendix 6A of POE4 The joint test involving two equations uses estimation within a seemingly unrelated regression SUR framework that is discussed in Chapter 15 The SUR framework is needed to get the covariances between coef cient estimates from different equations Least squares and SUR estimates of VAR equations are the same because the same explanatory variables appear in each equation but testing hypotheses involving coef cients in different equations requires the SUR framework The separate equation joint tests suggest that the estimates for coef cients at lags 1 and 3 are signi cant at a 5 level for the C equation but only those at lag 1 are signi cant in the Y equation In the joint test for both equations only lag 1 coef cients are signi cant at the 5 level although coef cients at both lags 1 and 3 are signi cant at a 10 level Adding lags of order 4 did not lead to any signi cant coef cients VAR Lag Exclusion Wald Tests Sample 1960Q1 2009Q4 Included observations 196 Chisquared test statistics for lag exclusion Numbers in are pvalues DCN DY Joint Lag 1 1303622 1605791 3197885 00015 00003 00000 Lag 2 4651329 016349 6563613 00977 09215 01608 Lag 3 6740683 4579234 829717 00343 01013 00813 df 2 2 4 Chapter 13 Exercise Solutions Principles of Econometrics 4e EXERCISE 137 The cointegrating equation between x and y normalised on y is 9 0495x t 37550 3 460 The correlogram up to order 4 for the residuals is shown in the first column of the diagram below None of the autocorrelations exceed the signi cance bounds Also the column labeled Prob shows that the probability values are all greater than 5 and hence that there is no evidence of autocorrelation up to order 4 Sample 1 100 Included obsenrations 100 Autocorrelation Partial Correlation AC PAC QStat Prob 1 0181 0181 2 0120 0090 3 0089 0130 4 0004 0032 3355 4803 55981 56999 0056 0088 012 0223 b The negative error correction coef cient in the first equation 0576 indicates that Ay falls while the positive error correction coefficient in the second equation 0450 indicates that Ax rises when there is a positive cointegrating error rest1 gt0 when yt1gt0495x1 This behavior negative change in y and positive change in x is necessary to correct the cointegrating error Chapter 13 Exercise Solutions Principles of Econometrics 4e EXERCISE 138 3 b 461 The correlogram of the residuals from the Aw equation is shown below Since there are no autocorrelations that exceed the signi cance bounds and the pvalues under Prob are all greater than 5 we can infer that there is no evidence of significant autocorrelation up to order 4 Samplezt 100 Included observations 98 Autocorrelation Partial Correlation AC PAC QStat Prob I I I I 1 0009 0009 0008 0930 I I I I 2 0041 0041 0196 0914 I I I I 3 0008 0009 01856 0980 I I I I 4 0054 0055 04885 095 The correlogram of the residuals from the Az equation are shown below Since there are no autocorrelations that exceed the significance bounds and the pvalues under Prob are all greater than 5 we can infer that there is no evidence of significant autocorrelation up to order 4 Samplezt 100 Included observations 98 Autocorrelation Partial Correlation AC PAC QStat Prob I I I I 1 0016 0016 00253 084 I I I I 2 0015 0015 00480 096 I I I I 3 008 008 08296 0842 I I I I 4 0003 0005 08303 0934 Expressions for the impulse responses were derived in Exercise 131 Effects of a shock to Aw of size SAW on Aw andAz Awl cs A21 O Aw O743oAw Aw A2 0155s Effects of a shock to Az of size SAW on Aw andAz Aw 0 A21 GAZ Aw O2l4GAZ AZ2 06415 Chapter 13 Exercise Solutions Principles of Econometrics 46 462 Exercise 138 continued c Expressions for the Variance decompositions were derived in Exercise 132 1step ahead forecast errors and Variances FEIAW Awm EAwt1 SAW VarFE1AW oiw t19 FE1 Azm EtAzt1 SAZ 39 VarFE1AZ 62 t19 2step ahead forecast errors and Variances A A A2 A FE2 W AWz2 EAWz25113zW1 5123z1 5zW2 VarFE2AW O7432oW 021426 oiw FE2AZ AZH2 EzAZz2 5215zAvi 5223AZ 522 11 VarFE2AZ 01552 csjw 064125 cs The contribution of a shock to Aw on the 1step forecast error Variance of Aw is 61V The contribution of a shock to Az on the 1step forecast error Variance of Aw is 06 The contribution of a shock to Aw on the 1step forecast error Variance of Az is V6 The contribution of a shock to Az on the 1step forecast error Variance of AZ is 616 The contribution of a shock to Aw on the 2step forecast error Variance of Aw is owO7432 1 07432csW 021425 cjw The contribution of a shock to Az on the 2step forecast error Variance of Aw is oiZO2142O7432ow 021425 csjw The contribution of a shock to Aw on the 2step forecast error Variance of Az is ow O1552 O1552ow 064125 5 The contribution of a shock to Az on the 2step forecast error Variance of AZ is oZO6412 1 O1552oW 064125 5 Chapter 13 Exercise Solutions Principles of Econometrics 46 463 EXERCISE 139 3 b C d The cointegrating relationship between P and M is P 1004Mt 0039 The coef cient t of 1004 is consistent with the quantity theory of money The error correction coefficients are 0016 and 0067 They are both signi cant and of the right signs This means that both Variables will errorcorrect to achieve equilibrium The system is stable The cointegrating residuals are obtained as res B 1004Mt 0039 The unit root test con rms that the residuals are stationary Ares 0086rest1 04 1 8Arest1 tau 3663 Since tau 3663 is less than the 5 critical Value of 337 the null hypothesis of no cointegration is rejected The residual series is an 10 Variable The VEC model estimated using the cointegrating residuals is AP 0016res1 O514APH 0005AMH 7 2127 7999 0215 M 0067res1 0336APH 0340AM1 7 3017 1796 4802 Chapter 13 Exercise Solutions Principles of Econometrics 46 464 EXERCISE 1310 8 b C d 6 The coefficients 0046 and 0098 suggest an inverse relationship between a change in the unemployment rate DU and a change in the in ation rate DP The response of DU at time t1 following a unit shock to DU at time I is 0180 The response of DP at time t1 following a unit shock to DU at time I is 0098 The response of DU at time t 2 is DUH2 0180DU1 0046DP 11 0180 X 0180 0046gtlt 0098 0037 The response of DP at time t 2 is DP 0098DU1 0373DP 0098 x 0180 0373 x 0098 0054 t2 t1 These results suggest following a shock to unemployment that DU increases but DP falls Chapter 13 Exercise Solutions Principles of Econometrics 46 465 EXERCISE 1311 a A VEC model is concerned with the shortrun relationship between changes in nonstationary variables and departures from the longrun cointegrating relationship between the levels of those variables Hence for estimating a VEC model we should use the data in the levels EURO and STERLING and their changes once we establish that they are indeed nonstationary and cointegrated A VAR model is concerned with the relationship between stationary variables Those stationary variables could be levels if the variables are 10 or changes if the variables are 11 and not cointegrated In Figure 137 the variables appear to be 11 and so we would use the changes in the data once we establish that the variables are 11 and not cointegrated b The least squares regression between EURO and STERLING is 1 0209 O429EURO R2 0939 r 37973 The unit root test of the regression residuals res is 5 0236res tau 3 5 1 8 Since the tau 3518 is less than the critical value of 337 the null hypothesis of no cointegration is rejected and we infer that STERLING and EURO are cointegrated The estimated VEC model is below 0250res 03 75ASTERLING1 O2O9AEURO1 z 2637 2817 2977 XETUFO o09ores 063 3ASTERLING O347AEURO1 z 043 8 2201 2290 Note that the error correction term for the second equation is not signi cant This suggests that in the event of a disequilibrium between EURO and STERLING that STERLING adjusts to restore equilibrium not EURO Chapter 13 Exercise Solutions Principles of Econometrics 4e 466 Exercise 1311 continued 0 The least squares regression of a VAR model between the change in EURO and the change in STERLING is shown below The intercept terms were not significant and hence not included O283ASTERLINGt1 O484AEUR0t1 r 4278 3700 KEEJF0 O373ASTERLINGt1 O672AEUR0t1 t 2707 2467 The order of the lag is 1 as all the second order terms were not signi cant This is confirmed by the correlograms of residuals Residuals from ASTERLING equation SampIe1999iu1U1 2UU6iu112 Included observations 94 Autocorrelation Partial Correlation AC PAC QStat Prob 1 000 000 00054 0941 2 0003 0003 00086 099 3 0012 0012 00199 0999 I I I I I I I I 4 0089 0090 08218 0938 Residuals from AEURO equation Sample 1999lquot1U1 2006iI112 Included observations 94 Autocorrelation Partial Correlation AC PAC QStat Prob 1 0020 0020 00382 0845 2 0044 0044 02315 0891 3 010 0106 1375 0711 I I I I I I I 4 009 0085 19986 0736 Chapter 13 Exercise Solutions Principles of Econometrics 4e EXERCISE 1312 The results for a rstorder VAR and the ARDL equations are as follows Vector Autoregression Estimates Sample adjusted 1891 1979 Included observations 89 after adjustments Standard errors in amp tstatistics in SP DV SP1 0301399 0357491 012119 008770 248689 407637 DV1 0300147 0016231 015562 011261 192877 014414 C 3434256 2605104 177289 128289 193709 203066 Dependent Variable SP Method Least Squares Sample adjusted 1891 1979 Included observations 89 after adjustments Coefficient Std Error tStatistic Prob C 1627032 1578864 1030508 03057 SP1 0053399 0115169 0463655 06441 DV1 0288887 0135393 2133686 00358 DV 0693724 0129639 5351182 00000 Dependent Variable DV Method Least Squares Sample adjusted 1891 1979 Included observations 89 after adjustments Coefficient Std Error tStatistic Prob C 1357627 1140131 1190763 02371 DV1 0092796 0100057 0927432 03563 SP1 0248009 0078989 3139810 00023 SP 0363245 0067881 5351182 00000 467 Chapter 13 Exercise Solutions Principles of Econometrics 46 468 Exercise 1312 continued 8 Comparing the two sets of estimates we nd the coef cients of corresponding variables in the VAR and ARDL models are quite different with the exception of the coef cient of DK1 in the equations for SP The differences should not be surprising since the coef cients in the VAR and ARDL models have are quite different interpretations The pair of ARDL equations represents two simultaneous equations with endogenous variables SP and DK The VAR equations are the reduced form equations from the simultaneous system These concepts were discussed in Chapter 11 To derive the reduced form coef cients from those in the structural ARDL system we solve the two ARDL equations simultaneously for SB and DK The solution is d X10 0 130 20 X11 0 130 21 X12 0 130 22 ezs X139 SI t1eDK1e SP 101 01 101 01 01 01 101 01 13 23 13 23 13 23 13 23 d X20 a3923X39l0 Q21 X3923X39ll X22 a3923X39l2 236 et DK 1 1 SB11eDK1f O 13O 23 O 13O 23 O 13O 23 O 13O 23 Thus deriving estimates of the reduced form coef cients from the structural coef cients estimates we have A 61 61 61 A 61 61 61 B10 3434 B20 2605 1O 130 23 1O 13O 23 amp11amp13amp21 amp21amp23amp11 l311 1ATA03014 B21 O3575 O 13O 23 O 13O 23 A 61 61 61 A 61 61 61 B12 12e1i22 O3OOl B22 2e2A512 0Ol623 10 130 23 10 130 23 These estimates are identical to those obtained by directly estimating the reduced form equations In this model deriving the reduced form estimates from the structural least squares estimates yields the same results as least squares estimation of the reduced form Note however that we are unable to derive structural estimates from the reduced form estimates There are only 6 reduced form coef cients and 8 structural coef cients There are multiple values of the 01 that will lead to the same reduced form estimates In the language of Chapter 11 the structural equations are unidenti ed Thus although the contemporaneous variables SP and DV appear to be signi cant in the ARDL equations the lack of identi cation means that the ARDL results should not be used to infer the contemporaneous role of dividends on share prices As long as 12 and 12 are serially uncorrelated lagged values of SP and DV will be uncorrelated with V and 12 and least squares estimation of the VAR yields consistent estimates It is important to include suf cient lags to eliminate serial correlation in the errors Chapter 13 Exercise Solutions Principles of Econometrics 46 469 Exercise 1312 continued b In the derivation above we showed that s d s d er X l3et Vd X236 6 I S I and l OL on 13 23 10 130 23 Solving these two equations for ef and etd shows that ef and etd both depend on V ad vtd Since SB and DK depend directly on V and vtd through their reduced form equations ef and eff will both be correlated with SB and DK This correlation leads least squares estimates of the ARDL equations to be inconsistent We also have the bigger problem of structural coef cients that are unidenti ed c Using a 5 signi cance level the VAR results show that the lagged rate of change in dividends has no signi cant in uence on the rate of change in share prices but the lagged rate of change in share prices has a signi cant effect on the rate of change in dividends Chapter 13 Exercise Solutions Principles of Econometrics 4e 470 EXERCISE 1313 3 b Growth in GDP of the two economies appears to move together The long run model with LE URO as the lefthandside variable is LE UROt B1 B2LUS1 e We Wish to investigate Whether e is an 10 variable The results from the least squares residuals and an ADF test on the residuals follow Dependent Variable LEURO Method Least Squares Sample 1995Q1 2009Q4 Included observations 60 Coefficient Std Error tStatistic Prob LUSA 0706170 0010277 6871315 00000 C 1354549 0047538 2849389 00000 Estimated Residuals 025 020 015 010 005 000 005 010 015 o2o 1995 1998 2000 2002 2004 2006 2008 Augmented DickeyFuller Test Equation Dependent Variable DEHAT Method Least Squares Sample adjusted 1995Q2 2009Q4 Included observations 5 9 after adjustments Variable Coefficient Std Err tStatistic Prob EHAT1 011720 006534 1794 00781 Since the tau 1794 is greater than the critical value of 337 the null hypothesis of no cointegration is not rejected and We infer that LUSA and LE URO are not cointegrated their relationship could be spurious The Wandering nature of the residuals in the graph suggests they are nonstationary Chapter 13 Exercise Solutions Principles of Econometrics 4e 471 Exercise 1313 continued C Because we concluded that LE URO and LUSA are not cointegrated for a shortrun relationship we specify a VAR model in rst differences Using lags of order 1 the model is ALEURQ 010 BHAL Us14 BIZALEUROH eh ALUSA 020 B21ALUSA1 BZZALEUROH ez The estimated rstorder VAR and the residuals are shown below Vector Autoregression Estimates Sample adjusted 1995Q3 2009Q4 Included observations 58 after adjustments Standard errors in amp tstatistics in DLEURO DLUSA DLEURO1 0375194 0356554 012837 016419 292280 217164 DLUSA1 0361594 0261045 011914 015238 3035l2 171312 C 0000222 0003352 000082 000105 027045 318565 EURO Residuals USA R9SidU88 02 02 o1 01 oo 00 31 01 U23902IIIIIIIIIIIIII 96 98 no 02 04 06 08 98 98 00 02 04 03 08 These results show that LE URO and LUSA affect each other Via the lagged terms The LE URO residuals are generally small relative to those for LUSA with the exception of those at the end of the period where they are larger It seems reasonable to assume the LUSA residuals have constant Variance but that does not appear to be the case for LE URO CHAPTER 1 4 Exercise Solutions 472 Chapter 14 Exercise Solutions Principles of Econometrics 46 473 EXERCISE 141 a The conditional mean Eet IH 0 because EH e EH 4 hit Where EH 0 is an alternative way of writing E 0 IH EH Zt EH since 2 is independent of hr 0 since EH zt O b The conditional Variance Eef IH h because 24 EH 2 EH ht h t since EH 1 and EH h 2 OL0 ocleil ht c er H N O ht because Zt N 01 and hence ZtJh t N Oht since ht is known at time t 1 Chapter 14 Exercise Solutions Principles of Econometrics 46 474 EXERCISE 142 a If 0 O the conditional mean of yr is E10H1 Er B0 em B0 since Ee1 O b If 0 75 O the conditional mean of ym is Er JH1 Ex 30 r aletz et1 30 95 oclef since Etet1 O The extra information used to forecast returns is the news captured in e Chapter 14 Exercise Solutions Principles of Econometrics 4e EXERCISE 143 a If yO ht 5oL1ef1 and When et1 1 hr 5OL1 12 5OL1 When et1 O ht 5OL1O2 5 when e11 lt5OL1125OL1 b If y 7 0 h 5 oL1et21 yd1ef1 and when et1 1 dt11 gt h 5OL1 12 y 12 5OL1Y when et1 O dt1 O gt h 5 OL1O2 5 When et11 dt1O gt ht5OL1125OL1 475 The key difference between the y O and y 7 0 cases lies with the contribution of the asymmetric factor Chapter 14 Exercise Solutions Principles of Econometrics 46 476 EXERCISE 144 GARCH11 model hr 5 oL1ef1 B1h1 Lag the expression ht1 5 oL1e22 B1h2 And substitute hr 5 119124 B15 319122 B192 51 B1 051931 O 1B1ez2 2 B12 z 2 Continue with the recursive substation hr 51B1 0 1e2 1 0 1B1e2 2 O 5 0 1e2 3 B1 3 51B1Bf 0 1lte2 1 B1et2 2 Bfei3 aim The last term drops out as B becomes negligible While the first term is the sum of a geometric progression 5lt1B1BfBi51 B1 Thus the GARCH11 may be reWritten as an ARCHq Where q is a large number in nity hr 51 s1 0L1ltet21 31ef2 sfef3 Chapter 14 Exercise Solutions Principles of Econometrics 4e 477 EXERCISE 145 3 b The correlogram of returns up to order 12 is presented below There is no evidence of autocorrelation since none of the autocorrelations exceed their signi cance bounds and the pvalues are all greater than 005 In other Words there is no indication of signi cant lagged mean effects Samble1988ru1I31 2004M12 Included observations 204 Autocorrelation Partial Correlation AC PAC QStat Prob 0058 0058 020 0400 0091 008 2414 0299 0089 0100 4062 0254 0058 0056 474664 0312 002 0004 49261 0425 0025 0022 50584 0536 008 0085 66854 0462 0068 0058 6639 0466 0030 0062 869 054 10 0043 0039 8266 0603 11 0112 0082 1093 0445 12 0033 0055 11218 0510 OC039lG39JT II LAJNJ The correlogram of squared returns up to order 12 is given below There is evidence of signi cant autocorrelation since the autocorrelations exceed their signi cance bounds at lags 1 4 5 6 and 8 and the pvalues are all less than 005 In other Words there is indication of signi cant lagged variance effects SampIe1988MIJ1 2004M12 Included observations 204 Autocorrelation Partial Correlation AC PAC QStat Prob 01 01 64592 0011 0088 0059 80quot3982 0018 0049 0025 8539r392 0035 0153 0140 134395 0009 0156 0110 18599 0002 0288 0244 36241 0000 003 0062 36531 0000 0204 0183 45444 0000 9 008 0012 467458 0000 10 0056 0043 474431 0000 11 0050 0128 439982 0000 12 0060 002 483969 0000 O3quotlG IU391lb39JhJ Chapter 14 Exercise Solutions Principles of Econometrics 46 478 EXERCISE 146 8 b C d 6 The time series of returns shows that there were periods of big changes around 1990 1998 2000 and 2002 and periods of small changes notably around 1989 and 1995 The distribution of returns is not normal since it is negatively skewed skewness 051 and the kurtosis is greater than 3 kurtosis 4159 The J arqueBera statistic is a test of normality which jointly tests whether skewness is signi cantly different from zero and whether kurtosis is signi cantly different from 3 The statistic is distributed as a X2 distribution with 2 degrees of freedom Since the calculated Value of 20287 is greater than the 5 critical Value 599 we reject the null hypothesis that the distribution is normal This is the unconditional distribution The Istatistic on the squared residuals term indicates the presence of first order ARCH The Lagrange Multiplier test 11431 is greater than the 5 critical Value of 3841 and hence it also suggests the presence of first order ARCH effects The results show that the mean Value of returns is 0879 The Istatistic on the ARCH effects of 2198 is signi cant The plot of the conditional Variance shows that Volatility is high around 1990 1998 2000 and 2002 and it is especially low around 1989 and 1995 These periods coincide with the periods of big and small changes in returns noted in a Chapter 14 Exercise Solutions Principles of Econometrics 46 479 EXERCISE 147 a The unconditional distribution of the series is not normal It has a kurtosis of 6484 which is Very different from the kurtosis of 3 for normality Furthermore the JarqueBera statistic which tests Whether skewness is signi cantly different from zero and Whether kurtosis is signi cantly different from 3 is Very large The Value of 192221 is signi cantly different from the critical Value of 599 b The results show that the average Value of the change in the exchange rate 5 is 0042 From the Variance equation the signi cance of the coef cient of the lagged squared residual term 0149 indicates that lagged newsshocks affect Volatility The signi cance of the coef cient of ht1 0800 indicates the signi cance of lagged Volatility effects c The forecast for the exchange rate is 0042 The forecast for the conditional Variance is fAz 1007 0615 0149e2201006 08002061 since h201006 2061 0615 014952482 08982061 since e201006 529 0042 23227 Chapter 14 Exercise Solutions Principles of Econometrics 46 480 EXERCISE 148 a The value of the conditional variance when e1 1 is fz 3442 025312 3695 The value of the conditional variance when e1 1 is 2 3442 0253 12 3695 b Results for the TARCH model are given in the text c The value of the conditional variance when e1 1 is h 3437 0123 3560 The value of the conditional variance when e1 1 is hr 3437 0123 0268 3828 d Since the coefficient on the asymmetric tenn 0268 is signi cant it suggests that the asymmetric TARCH model is better than the symmetric ARCH model Since the coefficient on the asymmetric effect is positive it suggests that volatility is greater when the shock is negative which is consistent with nancial economic theory Chapter 14 Exercise Solutions Principles of Econometrics 46 481 EXERCISE 149 a The estimated GARCH model is given in the text b The estimated GARCHinmean model is given in the text The contribution of volatility to the tenn premium is captured in the term 0211hit c The signi cance of the GARCHinmean term O211hit suggests that the GARCHin mean model is better than the GARCH model in a nancial econometric sense The positive sign suggest that returns increase when volatility rises which is consistent with nancial economic theory Chapter 14 Exercise Solutions Principles of Econometrics 46 482 EXERCISE 1410 3 b C A plot of the returns is shown below It shows that volatility of returns changes over time There are periods of big changes for example around June 2006 and periods of small changes for example around December 2005 llllll I I I I I I I I 200501 200BQ2 2005Q3 GOLD Figure xr1410a Plot of returns to gold shares The histogram of retums is given below Since the distribution is negatively skewed skewness is 100 and the kurtosis of 4776 is greater than 3 the distribution of retums is not nonnal The Jarque Bera statistic 59926 is signi cantly different from the 5 critical value of 599 and hence we reject the null hypothesis that the distribution is normal It is the unconditional distribution 50 Series GOLD Sample 1211 312005 am 82006 40 Observations 200 Mean IJD2S5D9 30 Median 01 01 649 Maximum 1 25 D98 Minimum 280282 2 Std Dev useavss Skewness 1 EIIII451B 10 Kurtosis 47 B22391 JarqueBera 5992654 0 F I I I I I 5 I I I 1 J I I Probability DDUIJDDU E Figure xr1410b Histogram for returns to gold shares The regression of squared residuals on a constant and the lagged squared residuals is f 03940101 f1 R2 0010 t 1929 The Lagrange Multiplier test statistic for the presence of rstorder ARCH is 2048 It is not signi cant when compared with the 5 critical value of 3841 Note that the tstatistic 1929 is also not signi cant at the 5 level Chapter 14 Exercise Solutions Principles of Econometrics 46 483 Exercise 1410 continued d 6 The estimated GARCH11 model is presented below The coefficients are of the correct sign and magnitude However they are not significant EH 0037 73 0120 020183 053472 7 0822 1259 1775 1887 An estimated GARCH in mean model could improve the forecast of returns 72 0118 O200 f 053973 1232 1837 1903 ED O138 O167 72 7 0655 However these results do not support such a model since the Istatistic on the hi term is not signi cant Chapter 14 Exercise Solutions Principles of Econometrics 46 484 EXERCISE 1411 a The monthly rate of in ation is shown below 01 3 li A IIBIOIIl i lIlbIllIIlgblIIgglllbblllbglllbblllbgl Figure xr1411a Plot of monthly rate of in ation b The estimated TGARCHinmean model is given in the text c The negative asymmetric effect 0221 suggests that negative shocks such as falls in prices reduce volatility in in ation This result is consistent with an economic hypothesis that volatility tends to be low when in ation rates are low d The positive inmean effect 1983 means that in ation in the UK increases when volatility in prices increases Chapter 14 Exercise Solutions Principles of Econometrics 46 485 EXERCISE 1412 a The estimated GARCH11 and ARCH5 models are shown below Dependent Variable RETURN Method ML ARCH Marquardt Normal distribution Sample adjusted 1032008 12312008 Included observations 260 after adjustments Convergence achieved after 45 iterations BollerslevWooldrige robust standard errors amp covariance Presample variance unconditional GARCH C2 C3RESID1 2 C4GARCH1 Coefficient Std Error zStatistic Prob C 0000633 0001507 0420130 06744 Variance Equation C 188E05 142E05 1317690 01876 RESID1 2 0107483 0038693 2777856 00055 GARCH1 0875546 0038095 2298351 00000 Dependent Variable RETURN Method ML ARCH Marquardt Normal distribution Sample adjusted 1032008 12312008 Included observations 260 after adjustments Convergence achieved after 21 iterations BollerslevWooldrige robust standard errors amp covariance Presample variance unconditional GARCH C2 C3RESID1 2 C4RESID2 2 C5RESID 3 2 C6RESID4 2 C7RESID5 2 Coefficient Std Error zStatistic Prob C 0001689 0001300 1299682 01937 Variance Equation C 0000208 657E05 3161446 00016 RESID1 2 0095248 0093286 1021026 03072 RESID2 2 0016531 0035116 0470738 06378 RESID3 2 0118779 0076953 1543519 01227 RESID4 2 0243126 0111862 2173448 00297 RESID5 2 0387344 0164633 2352776 00186 Chapter 14 Exercise Solutions Principles of Econometrics 46 486 Exercise 1412a continued The GARCH11 model is preferred because it is a parsimonious Way of capturing a large order ARCH model especially when the intervening terms in the ARCH model are not signi cant see RESIDlquot2 RESID2quot2 and RESID3quot2 b Based on the GARCH11 model the expected return and volatility next period are Es1 0001 Eh1 0000 0107 f 0875fz c The forecasted return and volatility next period are S 0001 kg 0000 0107st Est2 08750001 0000 0107 0001 00012 08750001 0001 d The estimated TARCH in mean model is shown below Dependent Variable RETURN Method ML ARCH Marquardt Normal distribution Date 100410 Time 1139 Sample adjusted 1032008 12312008 Included observations 260 after adjustments Convergence achieved after 20 iterations BollerslevWooldrige robust standard errors amp covariance Presample variance unconditional GARCH C3 C4RESD1quot2 C5RESD1quot2RESD1lt0 C6GARCH1 Coefficient Std Error zStatistic Prob SQRTGARCH 0037767 0198138 0190609 08488 C 0002399 0004756 0504410 06140 Variance Equation C 290E05 166E05 1750107 00801 RESD1quot2 0023328 0019919 1171177 02415 RESD1quot2RESD1lt0 0228222 0061451 3713889 00002 GARCH1 0877078 0035689 2457561 00000 Chapter 14 Exercise Solutions Principles of Econometrics 46 487 Exercise 1412d continued Since the ARCH term is insigni cant We reestimate the model Dependent Variable RETURN Method ML ARCH Marquardt Normal distribution Sample adjusted 1032008 12312008 Included observations 260 after adjustments Convergence achieved after 41 iterations BollerslevWooldrige robust standard errors amp covariance Presample variance unconditional GARCH C3 C4RESID1 2RESID1lt0 C5GARCH1 Coefficient Std Error zStatistic Prob SQRTGARCH 0043067 0200922 0214346 08303 C 0002513 0004839 0519271 06036 Variance Equation C 260E05 156E05 1671506 00946 RESID1 2RESID1lt0 0195996 0056580 3464072 00005 GARCH1 0873353 0034300 2546199 00000 The inmean effect is not signi cant When news is good the contribution of RESD1quot2 is insignificant While when news is negative the contribution is 0196 Chapter 14 Exercise Solutions Principles of Econometrics 46 488 EXERCISE 1413 8 b C d 6 Model Where only own lagged effects matter here specified as a lagorder 1 model EURO1 51 oc1EURO11 511 511 NOo12 Model Where only own lagged effects matter but with timeVarying Variance EURO1 51 oc1EURO11 811 811 I11 NOh1 hr B0 B1812z 1 B2hz 1 Model Where own lagged growth and lagged USA growth matter EURO1 51 oc1EURO11 y1USA11 511 511 N0cf Model Where shocks affect expected returns EURO1 51 oc1EURO11 911Z1 811 211 111 NOh1 ht B0 131811 52 Model Where shocks from the EURO and USA affect the expected EURO return USA1 52 oc2USA11 821 821 NOo EURO1 51 oc1EURO11 y1USA11 911Z1 811 811 111 N0h1 hr B0 B1312z 1 B2h 1 B3834 CHAPTER 1 5 Exercise Solutions 489 Chapter 15 Exercise Solutions Principles of Econometrics 4e 490 EXERCISE 151 a b C d 6 f The negative coefficient of POP suggests that countries with higher population growth tended to have lower growth in per capita GDP The increasing population has not led to a more than compensating gain in GDP leading to a fall in the ratio of GDP to population A positive coef cient for IN V implies more investment leads to a higher growth rate as one would expect The negative coef cient for GDP suggests that a lower initial level of GDP provides greater scope for growth in per capita GDP a reasonable outcome Finally the positive sign on the human capital variable suggests that a greater level of education leads to a higher growth rate This outcome also conforms with our expectations The coefficient for human capital for the period 1960 is significantly different from zero the Iratio is greater than 2 and the pvalue is less than 005 while those for the periods 1970 and 1980 are not Thus human capital appears to in uence growth rate only for 1960 but not for 1970 and 1980 The null hypothesis is H0 2612 613 623 0 where GU refers to the covariance between the errors in equations 139 and j The test statistic is M M3 133 12 86 x 010842 012872 039872 86gtlt00118 00166 01590 1611 The 5 critical value for a X2 distribution with 3 degrees of freedom is X59593 781 We reject the null hypothesis Thus SUR is preferred over separate least squares estimation The null hypothesis being tested is that the impact of each explanatory variable on the growth rate is the same in each of the three periods The intercepts are left unrestricted The X2 test statistic value is 12309 At a 5 significance level with 8 degrees of freedom the critical value is X89598 1551 Since the test statistic value is not greater than the critical value we do not reject the null hypothesis The pvalue for this test is 01379 The F test statistic value is Fx2J1230981539 where J is the number of equalities in the null hypothesis The corresponding 5 critical value for 8 243 degrees of freedom is 170958 243 1977 Since the test statistic value is less than the critical value we do not reject the null hypothesis The pvalue for this test is 01443 Chapter 15 Exercise Solutions Principles of Econometrics 46 491 EXERCISE 152 a The restrictions are that for each explanatory Variable the coefficients are the same across equations Only the intercept coef cient Varies across equations b The main difference between these results and those in Exercise 151 is the magnitude of the standard errors After imposing the restrictions the standard errors decrease for all coefficients In particular the standard errors for the coefficients of POP and SEC decrease substantially The magnitude of each restricted coefficient estimate lies between the highest and lowest Values for the corresponding unrestricted estimates c The X2 test statistic Value is 93098 At a 1 level of signi cance and 2 degrees of freedom the critical Value is X899 2 921 Since the test statistic Value is greater than the critical Value we reject the null hypothesis The pValue for this test is 000000 Chapter 15 Exercise Solutions Principles of Econometrics 46 492 EXERCISE 153 a b C In Exercise 152 the error variances for the different years were assumed different and correlation between errors for the same country in different years was permitted If the observations are all pooled with dummy variables inserted for each of the years 1960 1970 and 1980 and the model is estimated using least squares implicit assumptions are that the error variance is the same for all observations and all error correlations are zero The estimated equation is G 00315136 00205 D70 00029 D80 04365 POP O1628INV se 00147 00153 00158 01823 00208 143 gtlt10 6 GDP 00149 SEC 942 X 1057 00098 The estimates obtained in this exercise are very similar to those in Exercise 152 They will not be exactly the same because the estimation procedure in Exercise 152 is a generalized least squares one that uses information on different error variances and correlated errors The test statistic value for RESET is 12078 with a pvalue of 03006 The pvalue is greater than a signi cance level of 005 Thus RESET does not suggest the equation is misspecified Chapter 15 Exercise Solutions Principles of Econometrics 4e 493 EXERCISE 154 a b C From equation 1514 in POE4 we have J72 B2562 it where 7 yl 7l 56 it xi 37 and 51 e 51 The fixed effects estimator for B is the least squares estimator applied to this equation It is given by N T xi Tz 3it 139 The random effects estimator for B is the least squares estimator applied to the equation yi E15tB2xZ V where y yl 097 and x xi ocfi This estimator is given by iltx egtlty 7gt 2RE Zlzv T 2 ZZxiix 11t1 Now x lZ1t1xl oLxl x oLx and x 2Tx 6Lx c 3LTx 3L3c c 7 If If I If I Thus The pooled least squares estimator is given by N Z xz39txyz39ty 11t 2 xz39rx The pooled least squares estimator uses variation in xi and y around their overall b s B2PLS Z LM2 LM 1 t means it does not distinguish between variation within and between individuals The fixed effects estimator uses only variation from individual means known as within variation The random effects estimator uses both overall and between variation weighted according to the value of 6c between variation uses 27 36 and 371 7 Chapter 15 Exercise Solutions Principles of Econometrics 46 494 EXERCISE 155 a The three estimates for B2 are i Dummy variable fixed effects estimator I9 00207 seb2 00209 ii Estimator from averaged data A351 00273 seltf3 00075 iiiRandom effects estimator 0 00266 sef3 00070 The estimates from the averaged data and from the random effects model are very similar with the standard error from the random effects model suggesting the estimate from this model is more precise The dummy variable model estimate is noticeably different and its standard error is much bigger than that of the other two estimates b To test H0 3171 512 39 5140 against the alternative that not all of the intercepts are equal We use the usual F test for testing a set of linear restrictions The calculated value is F 3l75 while the 5 critical value is 1709553979 1551 Thus We reject H0 and conclude that the household intercepts are not all equal The F value can be obtained using the equation F SSER SSEUJ l95548l 76l587339 3175 SSEU NT K 7615873120 41 Chapter 15 Exercise Solutions Principles of Econometrics 4e 495 EXERCISE 156 a b Fixed effects estimates of the model are given below Dependent Variable LNPRICE Method Panel Least Squares Fixed Effects Periods included 4 Crosssections included 754 Total panel balanced observations 3016 Variable Coefficient Std Error IValue Prob C 546126 013028 41920 00000 REGULAR 003722 001685 2209 00273 RICH 008264 002053 4025 00001 ALCOHOL 005686 002614 2 175 00297 NOCONDOM 017028 002582 6596 00000 BAR 029846 013445 2220 00265 STREET 045516 013047 3489 00005 i Sex worker characteristics are omitted because they are timeinvariant over the time in which the 4 transactions took place Their effect cannot be separated from the individual effects given by the coefficients of the fixedeffects dummy variables ii All coefficient estimates are signi cantly different from zero at a 5 level iii The estimated risk premium for not using a condom is approximately 17 The exact estimate is 100exp0 170282 1 186 The price is approximately 37 higher for regular customers and approximately 83 higher for rich customers It is 57 lower for customers whio have consumed alcohol The origin of the transaction has a relatively large effect on the price For transactions that originated in a bar there is a 298 premium approximately and for transactions originating in the street the premium is approximately 455 Random effects estimates are presented on the next page Treating the effects as random instead of xed and adding the sex worker characteristics has had a dramatic effect on some of the common coefficients Rich clients are now estimated to pay 116 extra instead of 83 Those who have consumed alcohol are now estimated to pay a higher price instead of a lower price although this coefficient is not significantly different from zero The premium for not using a condom has declined slightly to 139 There have been large changes in the coefficients of BAR and STREET The random effects specification suggests that transactions originating in a bar are much more expensive than those originating on the street whereas the reverse was true with the fixed effects specification The price of commercial sex is lower for older sex workers higher for attractive workers and higher for secondary educated sex workers Chapter 15 Exercise Solutions Principles of Econometrics 4e 496 Exercise 156b continued Other things held constant the extra pecentage premium for having unprotected sex with an attractive secondaryeducated sex Worker compared with protected sex with an unattractive uneduated sex Worker is approximately 100gtlt 013898 027683 021615 632 The exact calculation is 100 x exp063196 1 881 Dependent Variable LNPRICE Method Panel EGLS Crosssection random effects Periods included 4 Crosssections included 754 Total panel balanced observations 3016 Swamy and Arora estimator of component variances Variable Coefficient Std Error Ivalue pvalue C 591037 012782 46240 00000 REGULAR 002363 001587 1488 01367 RICH 011601 001965 5904 00000 ALCOHOL 001489 002448 0608 05430 NOCONDOM 013898 002455 5662 00000 BAR 046425 009798 4738 00000 STREET 010329 009914 1042 02976 AGE 002577 000270 9540 00000 ATTRACTIVE 027683 005908 4685 00000 SCHOOL 021615 004447 4861 00000 Effects Specification SD Rho Crosssection random 6 054163 08602 Idiosyncratic random 68 021839 01398 Note The above results are those computed by EVieWs7 Stata11 gives slightly higher standard errors which lead to smaller tvalues and larger pvalues as presented in the following table Variable Coefficient Std Error tvalue pvalue C 591037 013032 45353 00000 REGULAR 002363 001618 1460 01444 RICH 011601 002003 5790 00000 ALCOHOL 001489 002496 0597 05508 NOCONDOM 013898 002503 5553 00000 BAR 046425 009989 4648 00000 STREET 010329 010108 1022 03069 AGE 002577 000275 9357 00000 ATTRACTIVE 027683 006024 4596 00000 SCHOOL 021615 004534 4767 00000 Chapter 15 Exercise Solutions Principles of Econometrics 46 497 Exercise 156 continued c Results for the Hausman test on each difference between the xed effects and random effects estimates are given in the following table for both EViews and Stata standard errors At a 5 level of signi cance there is a signi cant difference between all coef cients except those for BAR Thus we reject a null hypothesis that the individual random effects are uncorrelated with the variables in the model The xed effects estimates are more reliable in this instance because they are consistent EViews Stata bFE7k bRE7k sebFEDk bRE k tvalue pvalue sebFEDk bREDk tvalue pvalue REGULAR 0013590 0005647 2406 00162 0004684 2901 00037 RICH 0033371 0005939 5619 00000 0004475 7456 00000 ALCOHOL 0071746 0009173 7821 00000 0007777 9225 00000 NOCONDOM 0031298 0007999 3913 00001 0006340 4937 00000 BAR 0165790 0092074 1801 00719 0089992 1842 00655 STREET 0351873 0084809 4149 00000 0082490 4266 00000 d If a sex worker has individual characteristics that make her a risk taker or conversely risk averse then NOCONDOM is likley to be correlated with the individual effect The estimates obtained using the HausmanTaylor estimator assuming NOCONDOM is endogenous are given in the table below The results are very similar to those obtained in part b There have been no dramatic changes in the coef cient estimates and REGULAR ALCOHOL and STREET continue to be insigni cant at a 5 level of signi cance In this case the extra pecentage premium for having unprotected sex with an attractive secondaryeducated sex worker compared with protected sex with an unattractive uneduated sex worker is approximately 100gtlt O16099 028352 O22563 670 The exact calculation is 100 x exp067014 1 955 HausmanTaylor estimates with NOCONDOM endogenous Variable Coef cient Std Error tvalue pvalue C 593145 013894 42691 00000 REGULAR 002640 001585 1666 00959 RICH 010909 001954 5582 00000 ALCOHOL 000315 002442 0129 08975 NOCONDOM 016099 002537 6346 00000 BAR 046510 010263 4532 00000 STREET 015619 010343 1510 01311 AGE 002660 000309 8619 00000 ATTRACTIVE 028352 006770 4188 00000 SCHOOL 022563 005091 4432 00000 6 063373 6 021810 6 Chapter 15 Exercise Solutions Principles of Econometrics 4e 498 EXERCISE 157 a b C The results from estimating the equation with MA T HSCORE as the dependent variable and no xed or random effects are as follows Pooled Least Squares Estimates Coef Std Err Ivalue pvalue C 46970 17476 26877 0000 SMALL 80833 15254 530 0000 AIDE 042210 14692 029 0774 TCHEXPER 065787 01072 614 0000 BOY 78404 12275 639 0000 WHITEASIAN 171241 13177 1300 0000 We nd that being in a small class increases the math score by 81 points other things equal The coef cient for teacher s aide is not signi cant suggesting that having aide does not improve the score Students of experienced teachers score slightly better than those of inexperienced teachers the estimate is signi cant but not very large 066 points Gender and race have a big impact Boys score 78 points Worse than girls and White Asians score 171 points better Including xed effects leads to the following set of estimates Fixed Effects Estimates Coef Std Err Ivalue pvalue C 46617 21579 21603 0000 SMALL 93496 13970 669 0000 AIDE 052689 13491 039 0696 TCHEXPER 042015 01084 388 0000 BOY 66312 11134 596 0000 WHITEASIAN 236509 23109 1023 0000 The general conclusions made in part a when school xed effects were not included remain the same The estimated effect of small classes is slightly larger at 93 points The presence of a teacher s aide continues to be insigni cant Having an experienced teacher has a signi cant but very small effect Boys score 66 points Worse than girls The most dramatic effect is the increase in the coef cient of WHITEASIAN from 171 to 237 points The F value for testing for signi cant school effects is F 18066 Assuming there are no school xed effects it has an F distribution with 78 5682 degrees of freedom Correct to 4 decimal places the corresponding pvalue is 00000 Thus We reject the null hypothesis that there are no school effects Having signi cant school effects that have not changed our general conclusions about the coef cients suggests that the school effects are not highly correlated with the explanatory variables Chapter 15 Exercise Solutions Principles of Econometrics 4e 499 Exercise 157 continued d 6 Random effects estimates are presented in the following table These estimates are those obtained from Stata Version 11 Other software such as EViews may produce a slightly different estimate for 6 and coef cient estimates and standard errors with slight u differences Random Effects Estimates Coef Std Err IValue pValue C 46657 30759 15168 0000 SMALL 93009 13965 666 0000 AIDE 048505 13484 036 0719 T CHEXPER 043742 01076 407 0000 BOY 67145 11135 603 0000 WHITEASIAN 224353 21523 1042 0000 Eu 198714 6 419466 The random effects estimates are Very similar to those obtained using xed effects There are only minor differences and no conclusions change If the Asian students tend to be concentrated in particular schools then WHITEASIAN could be correlated with the school effects Similarly some schools could have a predominance of experienced teachers in which case T CHEXPER would be correlated with the school effects Because of random assignment of SMALL and AIDE and because gender is likely to be random we would not expect the other Variables to be correlated with the school effects Results from the Hausman test for the differences between the xed and random effects estimates are given in the following table That for BOY is not included because in this case sebFEltsebREpk The insignificant differences between the fixed and random effects estimates suggest that the explanatory Variables are not correlated with the school effects We conclude that the random effects estimates are consistent and more ef cient Hausman Test Results bFE7k bREjk S6bFEk bRE3k IValue pValue SMALL 00487 003930 1239 0215 AIDE 00418 004447 0941 0347 TCHEXPER 00173 001302 1327 0185 WHITEASIAN 12156 084122 1445 0148 Chapter 15 Exercise Solutions Principles of Econometrics 46 500 Exercise 157 continued 0 Random effects estimates of a model with AIDE omitted and T CHMASTERS and SCHURBAN included follow Again there are no dramatic changes in the coef cient estimates for the Variables that were in the earlier model Neither T CHMASTERS nor SCHURBAN is signi cant at a 5 level of significance and the effect of a teachers master s degree seems to be negative Fixed effects estimation of this model will break down because of perfect collinearity between SCH URBAN and the school effects Random Effects Estimates of Extended Model Coef Std Err IValue pValue C 46770 35810 13060 0000 SMALL 89455 12218 732 0000 TCHEXPER 048351 01104 438 0000 BOY 66982 11134 602 0000 WHITEISIAN 222880 22167 1005 0000 TCHMASTERS 23960 14264 168 0093 SCHURBAN 11012 52199 021 0833 Eu 197813 6e 419379 Chapter 15 Exercise Solutions Principles of Econometrics 4e 501 EXERCISE 158 a b The coef cient estimates for the different parts of this question are given in the following table with standard errors are in parentheses below the estimated coef cients Part a Part a Part b Parts de Part f Part g Part h Variable 1987 LS 1988 LS Pool LS Fix Eff Diff Dum 88 DifDum Intercept 09348 08993 09482 15468 07346 02010 02407 01506 02522 06050 02688 EXPER 01270 01265 01229 00575 00575 01187 01187 00295 00323 00211 00330 00330 00530 00530 00328 EXPER2 03288 03089 03066 01234 01234 01365 01365 gtlt102 01067 01069 00728 01102 01102 01105 01105 01096 SOUTH 02128 02384 02255 03261 03261 03453 03453 00338 00344 00241 01258 01258 01264 01264 02495 UNION 01445 01102 01274 00822 00822 00814 00814 00382 00387 00272 00312 00312 00312 00312 00367 D88 00774 00774 00524 00524 Clusterrobust standard errors The estimates for this model for the two years 1987 and 1988 are presented in the second and third columns of the table with the coefficients and standard errors for EXPER2 reported as 100 times greater than their actual values The coefficient estimates for the two years and their standard errors are very similar There are no substantial yeartoyear changes in the magnitudes of the coefficients For these individual year estimations we are assuming that all individuals have the same regression parameter values the model does not account for differences that might be attributable to individual heterogeneity Having a separate equation for each year does allow the coefficients to be different in different years however The estimates for this model are presented in the fourth column of the table Again the magnitudes of the coef cients are similar to those obtained for the 1987 and 1988 equations The standard errors are less however re ecting the greater precision from a larger number of observations For this estimation we are assuming that all women have identical coefficients there is no individual heterogeneity and the coef cients are the same in each year We are also assuming the variance of the error term is the same for all individuals and in both years Chapter 15 Exercise Solutions Principles of Econometrics 4e 502 Exercise 158 continued 0 d 6 f The xed effects model accounts for differences in behaviour individual heterogeneity by allowing the intercept to change for each individual In parts a and b differences in the behaviour of individuals have not been accounted for since a single intercept value is estimated for all 139 However this xed effects model assumes that the variance of the error term is the same for both years and that the coefficients are identical in both years assumptions that were not made in part a The estimates of the xed effects model are presented in the fth column of the table To test H 0 13 1312 39 39 39 131716 against the alternative that not all of the intercepts are equal we use the usual F test for testing a set of linear restrictions The calculated value is F 11675 while the 5 critical value is 170957155712 131 Thus we reject H0 and conclude that the intercepts for all women in the sample are not all equal The F value can be obtained using the equation F SSER SSEUJ 2855285 2243925715 11675 SSEU NT K 22439251432 716 4 The existence of individual heterogeneity means the estimates of the remaining coef cients will be biased if such heterogeneity is correlated with explanatory variables such as experience and SOUTH The estimates do suggest some bias could have been present For example the coef cients of EXPER and EXPER2 have more than halved in the xed effects model Clusterrobust standard errors for the xedeffects estimated model are given below the conventional ones in column 5 of the table Without clusterrobust standard errors we are assuming that the error variance is the same for all individuals and in both years and that there is no correlation between errors in the different years for the same individual Using clusterrobust standard errors allows for the variances to be different for different individuals in both 1987 and 1988 and it permits correlation between errors in 1987 and 1988 for the same individual The clusterrobust standard errors are similar to the conventional ones except for the case of SOUTH The clusterrobust standard error for the coef cient of SOUTH is approximately double that of its more restrictive counterpart Writing down the lagged model and subtracting it from the original model yields lnWAGEl 31 BZEXPERU B3EXPER B4SOUTH BSUNIONU em lnWAGEp1 31 B2EXPERlJ1 B3EXPERf1 B 4SOUT H 1 BSUNION e1 zt 1 DL WA GE BZDEXPERI B3DEXPERj 13050 UTH BSDUNIONZ De By taking the rst differences we remove the heterogeneity term The estimates for this model are presented in the sixth column of the table They are identical to the xed effects estimates obtained in part d Chapter 15 Exercise Solutions Principles of Econometrics 46 503 Exercise 158 continued g h The estimates for this model are presented in the nexttolast column of the table The coefficient for the dummy variable D88 is not signi cant at a 5 level of significance since its p value 01402 is greater than 005 This dummy variable describes the growth rate of real wages averaged over all individuals Thus this model estimates that the average growth rate was 774 from 1987 to 1988 The estimates for this model are presented in the last column of the table Subtracting D88 0 from D88 1 yields the constant term 1 Thus in this model the intercept term represents the average growth rate of real wages and is identical to the estimate found in part g Chapter 15 Exercise Solutions Principles of Econometrics 46 504 EXERCISE 159 a b The coef cient estimates for the different parts of this question are given in the following table with standard errors are in parentheses below the estimated coefficients Part a Part a Part b Part c Part e Part f Variable 1987 LS 1988 LS Pool LS PLS cl se Fix Eff Ran Eff Intercept 02268 02216 02381 02381 15468 03086 01881 02227 01406 01528 02522 01610 EDUC 00762 00778 00771 00771 00776 00063 00064 00045 00066 00060 EXPER 00875 00830 00834 00834 00575 00758 00265 00292 00190 00206 00330 00205 EXPER2 02033 01790 01852 01852 01234 01648 X102 00958 00964 00654 00722 01102 00702 BLACK 01562 01309 01432 01432 01319 00366 00372 00260 00314 00345 SOUTH 01029 01368 01199 01199 03261 01350 00327 00334 00233 00306 01258 00303 UNION 01701 01324 01509 01509 00822 01170 00350 00354 00248 00319 00312 00235 The estimates for this model for the two years 1987 and 1988 are presented in the second and third columns of the table with the coefficients and standard errors for EXPER2 reported as 100 times greater than their actual values The coefficient estimates for the two years and their standard errors are similar There are some changes but no substantial year toyear changes in the magnitudes of the coefficients For these individual year estimations we are assuming that all individuals have the same regression parameter values the model does not account for differences that might be attributable to individual heterogeneity Having a separate equation for each year does allow the coefficients to be different in different years however The estimates for this model are presented in the fourth column of the table Again the magnitudes of the coefficients are similar to those obtained for the 1987 and 1988 equations The standard errors are less however re ecting the greater precision from a larger number of observations For this estimation we are assuming that all women have identical coef cients there is no individual heterogeneity and the coefficients are the same in each year We are also assuming the variance of the error term is the same for all individuals and in both years and that the errors are uncorrelated over individuals and between the two years for each individual Chapter 15 Exercise Solutions Principles of Econometrics 46 505 Exercise 159 continued 0 d 6 Pooled least squares estimates of the coef cients with clusterrobust standard errors are presented in column 5 of the table Without clusterrobust standard errors we are assuming that the error variance is the same for all individuals and in both years and that there is no correlation between errors in the different years for the same individual Using cluster robust standard errors allows for the variances to be different for different individuals in both 1987 and 1988 and it permits correlation between errors in 1987 and 1988 for the same individual The clusterrobust standard errors are slightly larger than the regular ones from least squares estimation suggesting that ignoring heteroskedasticity and within individual correlation can lead to an overstatement of the precision of our estimates The xed effects model accounts for differences in behaviour individual heterogeneity by allowing the intercept to be different for each individual The variables EDUC and BLACK have an i subscript and no I subscript because they do not change over time An individual s level of education and color do not change This characteristic means that the coef cients of EDUC and BLACK cannot be estimated separately from the xed effects B1 In parts a and b where fixed effects are not specified it is implicitly assumed that EDUC and BLACK are the only sources of individual heterogeneity Other sources of heterogeneity are possible in the fixed effects model but the effects of each source cannot be estimated separately Other differences are that the xed effects model assumes that the variance of the error term is the same for both years and that the other coef cients are identical in both years assumptions that were not made in part a The estimates of the xed effects model with EDUC and BLACK omitted are presented in the nexttolast column of the table Omission of EDUC and BLACK is necessary to avoid perfect collinearity To test whether the intercepts are identical for all women in the sample we must be clear about which intercepts we want to test Omitting EDUC and BLACK raises a question about the definition of the intercept To appreciate the issue we rewrite the equation in part d as lnWAGE B B3EXPERl B4EXPER B6SOUTHl B7UNIONl el where B B11 BZEDUCZ BSBLACKI It is the B that are estimated by the xed effects model We can test whether the B are identical for all women in the sample This was the test performed in Exercise 158d In the context of the current model it implies B2 35 0 Alternatively we can test whether the B1 are identical for all women in the sample In this latter case we are testing whether ED UC and BLACK are the only sources of individual heterogeneity We proceed with this test namely H0 BU 3172 PD B17716 against the alternative that not all of the intercepts are equal Note that another way of writing the restriction in the null hypothesis is to say that B 31 BZEDUCZ BSBLACKZ for all i We have dropped the subscript i from B1 Chapter 15 Exercise Solutions Principles of Econometrics 46 506 Exercise 159e continued 0 8 h The restricted model is that estimated in part b Because We are replacing 716 intercepts B with three parameters 31 B2 and B5 the number of restrictions is 713 The value of the F statistic is F SSER SSEU J 2268772 2243925713 9098 SSEU NT K 22439251432 716 4 The 5 critical value is 17095713D7m 1131 Because 9098gtll3l We reject H0 and conclude that ED UC and BLACK are not the only sources of individual heterogeneity The estimates of the random effects model are presented in the last column of the table To test the null hypothesis that there are no random effects We test H0 6 0 against the alternative H1 6 gt 0 where cs is the variance of the random effect u The test statistic is that given in equation 1530 on page 554 of POE4 Its value is eit aw f M NT 6 1 143f40832881j214 7 2gtlt 2268772 This value clearly exceeds the critical value z095 1645 Thus We reject the null hypothesis and conclude that random effects are present The return on an additional year of education in the random effects model is 776 Its p value is 00000 indicating that it is signi cant at a 1 level of significance A 95 interval estimate can be calculated as 0 z X sef3 0077557 1962 x 0005969 00658 00893 It is not possible to estimate a return to education from the fixed effects model in part e because EDUC does not change over time and is therefore perfectly collinear with the dummy variables The fixed effects estimator uses only the variation Within each individual to estimate the slope coefficients When there is no Withinindividual variation as is the case with education it fails On the other hand the random effects estimator in part f uses both variation Within individuals and variation between individuals to obtain estimates of slope coefficients In this case We can nd an estimate of the return to education by using the variation in education across individuals Chapter 15 Exercise Solutions Principles of Econometrics 46 507 Exercise 159 continued i The Itest Values for the Hausman tests on the coefficient differences for EXPER EXPER2 SOUTH and UNION are calculated using the general formula bFEk bREk 12 LsebFE7k 2 sebREk 2 The results with pValues in parentheses are 1 EXPER I 0711 pValue 0477 0025826 EXPER2 1 T 0487 pValue 0626 0000850 SOUTH 1 053 1565 pValue 0118 0122081 UNION 1 T 1692 pValue 0091 0020563 All pValues are greater than 005 leading us to conclude that the difference between the two sets of estimates is not significant We do not reject the null hypothesis that the difference between the estimates is zero Thus there is not evidence that the random effects model is an incorrect specification When its assumptions hold the random effects model is better than the fixed effects model because it allows us to estimate the coefficients for the time invariant Variables and it is more precise in large samples If there was a significant difference between the sets of estimates we would choose the fixed effects estimator because the random effects estimator would be biased Chapter 15 Exercise Solutions Principles of Econometrics 46 508 EXERCISE 1510 a b C i If deterrence increases crime rates should drop ii If Wages in the private sector increase the return to legal activities increases relative to the return to illegal activities Therefore crime rates should drop iii Higher population density is linked with a higher residential crime rate iv Young males are the most likely demographic group to be involved in illegal activities Thus an increase in the percentage of young males should increase the crime rate The estimated equation is LWSMWE 60861 O6566LPRBARR O4466LPRBCONV O2082LPRBPRIS se O3654 00403 00277 00727 00586LA VGSEN O2921L WMFG 00606 00619 i LPRBARR LPRBCON V LPRBPRIS and LA VGSEN are explanatory variables that describe the deterrence effect of the legal system We expect the coef cients of these variables to be negative We nd that all of these coefficients are negative except for the coefficient of LPRBPRIS The variable LWMF G which represents Wages in the private sector has a positive coefficient that is not consistent with our expectations All coef cients are signi cantly different from zero at a 5 level of signi cance except for the coef cient of LA VGSEN ii Since the model is in loglog form all coef cients are elasticities The coefficient of LPRBARR suggests that a 1 increase in the probability of being arrested results in a 066 decrease in the crime rate The estimated equation is ZQTMWE 32288 O2313LPRBARR O1378LPRBCONV O1431LPRBPRIS se 03236 00376 00222 00393 00183LA VGSEN O1666L WMFG 003 10 00553 The reported intercept term is the average of the xed effects i All estimated coefficients have the expected sign except for LA VGSEN Moreover all estimated coefficients are signi cantly different from zero at a 5 level of signi cance except for the coef cient for LA VGSEN ii The coef cient on LPRBARR suggests that a 1 increase in the probability of being arrested results in a 023 decrease in the crime rate This estimated elasticity is less than half of the estimated elasticity from part b Thus once We allow for county heterogeneity the deterrent effect of being arrested is much less Chapter 15 Exercise Solutions Principles of Econometrics 46 509 Exercise 1510c continued c iii The coef cient on LA VGSEN suggests that a 1 increase in the average prison sentence results in a 00183 increase in the crime rate However a two tail Itest on the signi cance this estimate yields a t statistic of 05906 and a pvalue of 0555 Thus a null hypothesis that the coef cient of LA VGSEN is zero is not rejected There is no support for the idea that longer prison sentences are a deterrent to crime d To test H0 2131 3172 3190 against the alternative that not all of the intercepts are equal we use the usual F test for testing a set of linear restrictions The calculated value is F 33749 while the 5 critical value is 17095 89a535 1287 Thus we reject H0 and conclude that the county level effects are not all zero The F value can be obtained using the equation F SSER SSEUJ 1068144 161488189 337494 SSEU NT K 1614881630 90 5 e The coef cient estimates and standard errors from least squares LS and xed effects FE estimation are presented in the following table Estimates Standard Errors Variable LS FE LS FE Intercept 36769 22435 04662 13550 LPRBARR 04245 01952 00419 00367 LPRBCONV 02827 01 113 00288 00217 LPRBPRIS 00877 00977 00694 00384 LA VGSEN 01083 00240 00577 00315 LWMFG 00160 05762 00705 01330 LDENSITY 03052 07694 00274 03377 LPCTYMLE 01591 12460 00840 04346 D82 00176 00253 00574 00273 D83 00669 00216 00579 00352 D84 01 194 00121 00585 00426 D85 01056 00589 00600 00528 D86 00657 01586 00612 00652 D87 00101 02782 00617 00772 i It is apparent that the coef cient estimates obtained by using least squares are very different to those obtained using the xed effects method The magnitudes change considerably and there are some sign reversals Ignoring county effects can lead to misleading conclusions Chapter 15 Exercise Solutions Principles of Econometrics 4e 510 Exercise 1510e continued 6 0 ii The outcome of the test for the joint signi cance of the dummy variables is different for each of the two models In the least squares estimated model with no xed effects the F and p values for the test are 1324 and 02442 respectively leading us to conclude that there is no evidence of time effects On the other hand in the xed effects model the F and p values for the test are 9118 and 00000 respectively leading us to conclude that there are time effects Since we have established the importance of the county effects and this importance is con rmed if we carry out a further test for their inclusion in the model with time effects our nal conclusion is that the least squares test result is misleading and the year effects are important An examination of the coef cients for the year dummies in the xed effects model does show some evidence of a trend effect The coef cients for 1982 1983 and 1984 are all small and not signi cantly different from zero and so there does not appear to be a trend effect in these early years However from a small increase in 1985 there are dramatic increases in the coef cients for 1986 and 1987 suggesting an upward trend in the crime rate iii The coef cient of LWMF G represents the elasticity of the crime rate with respect to the average weekly wage in the manufacturing sector The least squares estimation suggests that a 1 increase the average weekly wage in the manufacturing sector results in a 00160 increase in the crime rate although this estimate is not signi cantly different from zero The xed effects estimation suggests that a 1 increase in average weekly wage will result in a 05762 decrease in the crime rate According to the xed effects estimates the explanatory variables which have the expected signs and a signi cant effect on the crime rate are LPRBARR LPRBCONV LPRBPRIS LWMF G LEDNSIT Y and LPCTYMLE Out of these variables those that have the largest effect on crime rate and are reasonable to implement as public policy will be the most effective in dealing with crime Improving policing and court policies that increase the probability of arrest conviction and imprisonment are likely to be effective but lengthening the term of imprisonment is not Opportunities for higher wages and the avoidance of highdensity population areas are also likely to be productive directions for public policy Chapter 15 Exercise Solutions Principles of Econometrics 46 511 EXERCISE 1511 a b C The estimated equation is ET 03787 O8624LK 01373LL se 00983 000488 000684 Since this is a loglog equation the coef cients represent elasticities The coefficient of LK suggests that a 1 increase in capital is associated with a O8624 increase in GDP The coef cient of LL suggests that a 1 increase in labor is associated with a 01373 increase in GDP Testing the null hypothesis H0 2132 133 1 constant returns to scale against the alternative hypothesis H1B2B3 7amp1 yields F and pvalues of 00042 and 09483 respectively Since this pvalue is much larger than the level of signi cance 005 we do not reject the null hypothesis and conclude that there is no evidence against the hypothesis of constant returns to scale The estimated equation is ET 0299508743LK01351LL 00121z se 00952 00048 000661 000095 The coef cient of t represents the growth rate of GDP expressed in decimal form Because it represents the growth rate not attributable to changes in capital and labor it is often viewed as growth from technological change These estimates suggest that the average growth rate of GDP over the period 19601987 is 121 per year The pvalue for testing the significance of this estimate is 00000 and so we can conclude that the coefficient is signi cantly different from zero at a 1 level of significance However we may question whether a negative growth rate is a realistic outcome The addition of I to the model has very little effect on the estimates of 32 and B3 they are almost identical to those obtained in part a Substituting the restriction B2 133 1 into the model from part b yields LYB1B2LK1 B2LLMe Rearranging this equation LY LLB1B2LK LLMe Converting it into a more familiar form lnY lnL 8 8 lnK 8 lnL M e lnB1 132 ln tte LYL 8 8L1ltLme yields Chapter 15 Exercise Solutions Principles of Econometrics 46 512 Exercise 151 1 c continued d 6 0 The estimated equation is LYE 04530 08731LKL 00119 se 00415000476 000094 The estimate of 32 is identical to the estimate obtained in part b to two decimal places The estimated equation is with the average of the xed effects reported as the intercept ET 83751 05316LK 0133322 000747 se0516400124 00336 000093 To test H0 2131 3172 B132 against the alternative that not all of the intercepts are equal we use the usual F test for testing a set of linear restrictions The calculated value is F 21113 while the 5 critical value is 70958122m 1279 Thus we reject H0 and conclude that the country level effects are not all equal The F value can be obtained using the equation F SSER SSEUJ 2927529 335155781 2111322 SSEU NT K 33515572296 82 3 The xed effects estimates are markedly different from those estimated in part b In particular the coefficient of I has changed sign to positive more in line with our expectations The elasticity of output with respect to capital is much smaller and the standard errors of both elasticities are much larger Testing the null hypothesis H0 2132 133 1 constant returns to scale against the alternative hypothesis H1B2B3 7amp1 yields F and pvalues of 10746 and 00000 respectively Since this pvalue is smaller than the level of signi cance 005 we reject the null hypothesis and conclude there are not constant returns to scale The outcome of this hypothesis test is clearly very sensitive to whether or not we include xed effects The estimated equation is L72 31245 05435LKL 0000327 se 01030 00127 0000551 These results are very different from those in part c All estimates have the same sign However relative to the estimates in part c the intercept is much larger and the coef cient estimates are much smaller Furthermore the standard errors of this model are much larger with the exception of seA The xed effects model without the restriction for constant returns to scale is the preferred speci cation It is preferred because according to our hypothesis tests we should allow for country xed effects and we do not have any evidence to support the presence of constant returns to scale Also the trend coef cient is positive in line with our expectation that technological change should have a positive effect on output Exercise 1511 continued g Chapter 15 Exercise Solutions Principles of Econometrics 46 The estimates are presented in the following table Variable Estimate se Variable Estimate se Intercept 02564 01024 D14 00829 00561 LK 08741 00048 D15 01011 00561 LL 01352 00066 D16 01375 00561 D2 00192 00560 D17 01430 00561 D3 00255 00560 D18 01566 00561 D4 00212 00560 D19 01735 00562 D5 00239 00560 D20 01784 00562 D6 00290 00560 D21 02054 00562 D7 00396 00560 D22 02353 00562 D8 00600 00560 D23 02662 00562 D9 00617 00560 D24 02850 00562 D10 00569 00560 D25 02907 00562 D11 00584 00560 D26 02980 00563 D12 00696 00561 D27 02985 00563 D13 00741 00561 D28 02958 00563 513 The single time trend Variable restricts the yeartoyear growth rate to be the same between all years Using time dummy Variables allows the rate of growth between years to be different for each year Since we have an intercept and then time dummies for all years except the first each coefficient of a time dummy gives the growth rate between the year of the time dummy and the rst year Chapter 15 Exercise Solutions Principles of Econometrics 4e EXERCISE 1512 a b C The percentage return to experience is awAGEWAGE X100 8EXPER 92 WA GE 8EXPER When EXPER 5 this quantity becomes 100B3 10134 gtltlO0 100x 8 2B4EXPER Pooled Least Squares Estimates Variable Coef cient Std Error tValue pValue C 0450940 0061691 731 0000 EDUC 0074821 0002765 2706 0000 EXPER 0063114 0007989 790 0000 EXPER2 0001229 0000323 381 0000 HOURS 0000843 0000840 100 0316 BLACK 0134715 0014922 903 0000 The 95 con dence intervals for I and 0 are e gt z04se0 50823 i19606gtlt 04887 412 604 t0975p3574sel 74821 19606 x 02765 694 802 514 When 1311213 and the errors are homoskedastic and uncorrelated We use pooled least squares Without clusterrobust standard errors The results are as follows Relaxing the assumption that the errors are homoskedastic and that they are uncorrelated We use pooled least squares with clusterrobust standard errors The results are as follows Pooled LS Estimates with ClusterRobust Standard Errors Variable Coef cient Std Error IValue pValue C 0450940 0103035 438 0000 EDUC 0074821 0005526 1354 0000 EXPER 0063114 0009953 634 0000 EXPER2 0001229 0000412 299 0003 HOURS 0000843 0001925 044 0662 BLACK 0134715 0028968 465 0000 lit0975p3574sel 74821i19606gtlt05526 640 857 z04se 50823 i19606gtlt 06100 388 628 The 95 con dence intervals for I and 0 are Chapter 15 Exercise Solutions Principles of Econometrics 46 515 Exercise 1512c continued d 9 Using clusterrobust standard errors has led to wider con dence intervals for both quantities of interest Ignoring the heteroskedasticity and withinindividual correlation leads to an overstatement of the precision with which we are estimating the returns to education and to experience When B1 is a random variable with mean E1 and variance 62 and the el are homoskedastic and uncorrelated we use random effects estimation The results are given in the table below A noticeable difference between these results and the earlier ones is the larger and now signi cant estimate for the coefficient of HOURS The standard errors are those computed by Stata EViews standard errors are slightly smaller Random Effects Estimates Variable Coef cient Std Error Ivalue pvalue C 0629405 0083254 756 0000 EDUC 0076867 0005496 1398 0000 EXPER 0059082 0005561 1062 0000 EXPER2 0001 140 0000219 520 0000 HOURS 0005390 0000695 776 0000 BLACK 0127097 0029818 426 0000 6 034083 6 019394 The 95 con dence intervals for I and 9 are iz4se 76867i19606gtltO5496 661 876 0 t0975p3574selt 47680 19606 x 03476 409 545 Compared to the earlier results the estimated return to education is slightly higher and has similar precision to that from estimation with clusterrobust standard errors The estimated return to experience is much smaller than previously and is estimated with more precision If the individual effects capture characteristics such as motivation and ability then it is likely that EDUC and HOURS will be correlated with the individual effects Those with higher ability and greater motivation are likely to have more years of education and to work longer hours Results for Hausman tests on the coefficients separately appear on the next page Because EDUC and BLACK are time invariant it is not possible to get xed effects estimates of their coef cients and they are omitted There is a signi cant difference between the xed and random effects estimates of the coef cient of HOURS but not for EXPER and EXPER2 Chapter 15 Exercise Solutions Principles of Econometrics 46 516 Exercise 1512e continued 0 The overall X2 test yields a value of x 158 with pvalue of 00012 We conclude therefore that the random effects estimates and the xed effects estimates are signi cantly different and hence there is correlation between the B1 and the variables in the model These results use the random effects standard errors from Stata Those from EViews yield results with slight differences Hausman Test Results bFEk bREk SebFEk bREC Value P39Va1u9 EXPER 0000677 0001440 0471 0638 EXPER2 0000012 0000053 0221 0825 HOURS 0000940 0000249 3783 0000 To accommodate the fact that EDUC and HOURS are correlated with the random effects we use the HausmanTaylor estimator The results are presented in the table below Compared to the random effects estimates that did not allow for endogeneity we nd that the estimated return to education has increased dramatically but so has its standard error The coef cient of BLACK has gone down in absolute value but its standard error has also increased Other coef cient estimates and their standard errors are similar in magnitude HausmanTaylor Estimates Variable Coef cient Std Error Ivalue pvalue C 0215297 0553607 039 0697 EDUC 0110916 0042161 263 0009 EXPER 0058326 0005732 1018 0000 EXPER2 0001110 0000225 494 0000 HOURS 0006318 0000737 858 0000 BLACK 0090999 0052886 172 0085 8 035747 6 019384 The 95 con dence intervals for I and 0 are lit09753574sel 110916 i19606 x 42161 283 1935 GD it 0 47231 i19606gtlt 03595 402 543 09753574Selt Although the point estimate for the return to education is higher than previous estimates the interval estimate is so wide we cannot make any rm conclusion about its value The interval estimate for the return to experience is very similar to that from the random effects estimator Chapter 15 Exercise Solutions Principles of Econometrics 46 517 EXERCISE 1513 a Fixed effects estimates for the slope coef cients are bFE2 011013 and bFE3 031003 The error Variance estimate is 63 50299522 2530042 b The sample means are given in the following table i 1NV V K 1 6848 57545 68022 2 61803 231470 486765 3 86124 693210 121245 4 3085 70921 5942 5 102290 1941325 400160 6 608020 4333845 648435 7 41889 333650 297900 8 55411 419865 104285 9 47595 149790 314945 10 410475 1971825 294855 11 42892 670910 85640 c Results from regressing INV on P j and 1 are INV 73825013460Z 0029691 SSE 5062749 20 as 506273949 6328437 d Substituting in the estimated Values yields OL 1 530042 085862 20gtlt6328437 eampf Pooled least squares applied to the transformed Variables and random effects estimates of the original equation yield identical results They are given by W 53944 01093 V 03080K se 2569800099 00164 Chapter 15 Exercise Solutions Principles of Econometrics 4e EXERCISE 1514 ab Least squares and SUR estimates and standard errors for the demand system appear in the following table Estimates Standard Errors Coef cient LS SUR LS SUR Constant 1017 2501 1354 1092 Price1 0567 0911 0215 0130 Income 1434 1453 0229 0217 Cgnstant p C Price2 0648 0867 0188 0125 Income 1144 1136 0261 0248 Constant 4870 5021 0546 0468 Price3 0964 0999 0065 0034 Income 0871 0870 0108 0103 All price elasticities are negative and all income elasticities are positive agreeing with our a priori expectations Also all elasticity estimates are signi cantly different from zero suggesting that the prices and income are relevant variables Relative to the least squares estimates the SUR estimates for the price elasticities for commodities 1 and 2 have increased in absolute value noticeably There have been no dramatic changes in the income elasticities or in the price elasticity for commodity 3 The S UR standard errors are all less than their least squares counterparts re ecting the increased precision obtained by allowing for the contemporaneous correlation For testing the null hypothesis that the errors are uncorrelated against the alternative that they are correlated We obtain a value for the X53 test statistic LM T0 133 72 30gtlt 00144 03708 02405 1877 Where 30 21 2 6Zee 00213 gt 73 1 0 3 2 00144 30 31 03943 04506 30 2 6Z 00448 gt 73 08448 03708 30 31 03943 01867 1 30 41 2 6 Zee 00413 gt 72 02 3 2 02405 30 31 04506 01867 The 5 critical value for a X2 test with 3 degrees of freedom is x 095p3 781 Thus We reject the null hypothesis and conclude that the errors are contemporaneously correlated Chapter 15 Exercise Solutions Principles of Econometrics 46 519 Exercise 1514 continued c We Wish to test H0 B13 1 B23 1 B33 1 against the alternative that at least one income elasticity is not unity This test can be performed using an Ftest or a X2test Both are largesample approximate tests The test Values are F l895 with a pValue of 014 or X2 5686 with a pValue of 013 Thus We do not reject the hypothesis that all income elasticities are equal to 1 Chapter 15 Exercise Solutions Principles of Econometrics 46 520 EXERCISE 1515 a The least squares LS and SUR estimates are given in the following table with standard errors in parentheses Constant ln m 111 ln CAR POP Pam POP Austria LS 37266 07607 07932 05199 03730 02115 01501 01131 SUR 39170 07939 07008 05264 03119 01739 01218 00931 Belgium LS 30419 08451 00417 06735 04525 01702 01579 00933 SUR 30390 10007 01320 07760 03235 01279 01067 00708 Canada LS 31260 03924 03629 04385 02810 00773 00893 00712 SUR 29890 04338 03738 04826 02398 00666 00751 00615 Denmark LS 02368 00928 01371 05171 03322 02194 01529 01282 SUR 03036 01092 01145 05212 02900 01827 01250 01059 France LS 31920 11193 01943 08447 O2847 01591 00912 01264 SUR 31624 11342 02043 08582 02469 01376 00784 01090 Germany LS 42635 04019 01671 02224 02721 01154 00635 00646 SUR 43475 03618 01226 01878 02045 00848 00433 00465 The signs of the coef cients are consistent across countries and estimation techniques although their magnitudes vary considerably An increase in per capita income leads to an increase in gasoline consumption per car presumably because a higher income leads to more travel andor the purchase of a bigger car An increase in price leads to a decline in gas consumption following the usual laws of demand The negative sign for number of cars per capita is likely to occur because each car gets driven less as the number of cars per person increases Most estimated coefficients are signi cantly different from zero Exceptions are the price coefficient in the equation for Belgium and the income and price coefficients for Denmark The use of SUR has led to a reduction in the standard errors relative to those for least squares Chapter 15 Exercise Solutions Principles of Econometrics 46 521 Exercise 1515 continued b The test statistic for testing for contemporaneous correlation is M 139 1 2 LM T226 i2 11 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 19X712 713 714 715 716 723 724 725 726 734 735 736 745 746 756 19045 The 13 are readily available from the least squares residual correlation matrix presented in the following table l 2 3 4 5 6 l 1 022322 021192 010779 017015 052806 z 1 022654 026571 015686 056349 3 1 019566 014583 017883 4 1 011169 007246 s 1 012216 6 1 The degrees of freedom are 6gtlt6 1 2 15 The 5 critical value with 15 degrees of freedom is xfs 2500 We do not reject the null hypothesis and conclude that there is no evidence of contemporaneous correlation c i The test statistic value for testing the null hypothesis that corresponding slope coefficients in different equations are equal is X2 68603 with a very small pvalue of 00000 We therefore reject the null hypothesis Different countries have different slope coefficients ii The test statistic value for testing the null hypothesis that 34 0 for all equations is X2 25292 with a very small pvalue of 00000 Thus We reject the null hypothesis We cannot conclude that lnCAR POP is an irrelevant variable in all countries Chapter 15 Exercise Solutions Principles of Econometrics 46 522 EXERCISE 1516 a b C d 6 One would expect all the coef cients to have positive signs As average price increases cattle numbers should increase as it would be profitable for the farmers to hold more cattle As rainfall increases the feed situation gets better and the farmer can run more cattle per acre The more cattle that are carried on the property in the previous year the greater the number likely to be carried in the current year Alternatively if the rm cannot adjust cattle numbers immediately to a desired level a partial adjustment model might be appropriate in which case the coef cients of lagged cattle numbers would lie between zero and one The three equations should be estimated jointly as a set rather than individually if the errors 69 i 123 are contemporaneously correlated Least squares and SUR estimates and standard errors are given in the table All estimates have the expected signs and are signi cant at a 5 level except for the intercepts Estimates Standard Errors Coefficient LS SUR LS SUR Constant 16363 77619 37987 22818 Price1 0979 0906 0351 0271 Rainfall1 1424 0907 0626 0465 Lagged cattle1 0662 0457 0136 0081 Constant 17509 42309 13541 8368 Price2 1013 0946 0104 0081 Rainfall2 1361 1194 0188 0144 Lagged cattle2 0527 0406 0077 0047 Constant 16576 45145 45613 30135 Price3 1308 1265 0402 0319 Rainfall3 2051 1536 0706 0535 Lagged cattle3 0600 0548 0126 0084 The relevant hypotheses are H0 2612 G13 G23 0 and H1 at least one covariance is nonzero The test statistic value is LM mg r 12 26gtlt 087622 082232 075112 5221 The 5 critical value for X2 test with 3 degrees of freedom is X89593 781 Hence we reject H0 and conclude that contemporaneous correlation exists Like the least squares results the signs of the SUR coefficients agree with our a priori expectations the magnitudes are feasible and the estimates are generally significant However a comparison of the magnitudes of the LS and SUR estimates does show some differences particularly for equation 1 The standard errors for the SUR estimates are uniformly less than those for LS supporting the theoretical result that the coef cients produced by S UR are more reliable than those from LS Chapter 15 Exercise Solutions Principles of Econometrics 4e EXERCISE 1517 a b C d 9 0 Results for parts a c and d are given in the table below standard errors are in parentheses below the estimated coefficients a LS c SUR d Restricted SUR 71 006063 00541 00528 00759 00719 00731 72 00465 00405 00415 00732 00687 00696 091 10862 09422 09124 01047 00480 00336 992 09483 09047 09124 00838 00384 00336 523 The separate least squares estimates of the elasticity of substitution are 031 10862 and 03209483 respectively These values are reasonably close and the magnitudes of their standard errors suggest that they could be two different estimates of the same parameter For testing H0 2612 0 against the alternative H0 2612 7 0 the value of the chisquare statistic is LMTgtltr122 20gtlt09062 l6417 The 5 critical value for X31 is 384 Hence we reject H0 and conclude that there is contemporaneous correlation between the errors The SUR estimates appear in the second column of the table The estimates of the elasticity of substitution are slightly less than those obtained by least squares In this case the elasticity of substitution estimate is 0912 This value lies between the two unrestricted generalized least squares estimates obtained in part c and is closer to the second one that has the smaller standard error The standard errors obtained in part c are less than their counterparts in part a Also the standard errors obtained in part d are less than those in part c with the exception of that for yl Thus the standard errors suggest more precise estimation from using the generalized least squares method and from imposition of the restriction The tvalue for testing H 0 0 1 against the alternative H1 03 7amp1 is 031 09124 12606 se03 003362 The 5 critical values are t002537 2026 and t097537 2026 Since 2606 lt 2026 we reject H0 and conclude that a CobbDouglas function would not be adequate Chapter 15 Exercise Solutions Principles of Econometrics 46 524 EXERCISE 1518 a b C The estimates and their standard errors are presented in the following table In the models with fixed effects the reported intercept is the average of the fixed effects Variable p 91239 91 199121 B12quot iii B12quot 91 UV B12quot B12quot Intercept 15468 03352 14964 02484 02557 03263 02473 03067 lnARE11 03617 05841 03759 06243 00640 00802 00618 00755 lnLABOR 04328 02586 04221 02412 00669 00703 00663 00682 lnFERTzt 02095 00952 02075 00890 003 83 00432 003 80 00415 SSE 4056536 2666229 3620311 2308242 From the table we see that the estimates in parts i and iii are similar and those in ii and iv are similar but those in i and iii are different to those in ii and iv This suggests that the estimates are not sensitive to assumptions about the intercept changing with time but that they are very sensitive to the assumption made about the intercept changing with farms The fixed effects model from part iv is preferred because it accounts for behavioural differences between the farms and differences over time and hypothesis tests support the inclusion of both farm effects and year effects The results of the tests are given in the table below The critical values are for a 5 level of significance The F values are calculated using the formula F SSER SSEUJ SSEU N T K The values for SSE are provided in the table above The values for J and NT K are given in the degrees of freedom df column as J NT K in the table below Test F value df Critical F pvalue i versus ii 3309 43305 1420 00000 i versus iii 5870 7341 2036 00000 i versus iv 4514 50298 1394 00000 ii versus iv 3939 43298 1421 00000 iii versus iv 8447 7298 2040 00000 Chapter 15 Exercise Solutions Principles of Econometrics 46 525 Exercise 1518 continued d The two sets of interval estimates for the elasticities are in the following table In all cases the clusterrobust standard errors lead to wider intervals suggesting that ignoring the withinfarmer error correlation and heteroskedasticity can lead to unjusti ed con dence in the reliability of the estimates 95 Interval Estimates Coventional se Clusterrobust se lower upper lower upper lnAREA 04757 07729 04219 08267 lnLABOR 01071 03753 00395 04429 lnFERT 00074 01706 00947 02727 Chapter 15 Exercise Solutions Principles of Econometrics 4e EXERCISE 1519 a The estimates and their standard errors are presented in the following table 526 Variable 1995 1996 1997 Intercept 25181 09284 14106 06596 05693 O8405 lnARE17 13165 05051 04745 01751 01442 02162 lnLABOR 02612 04363 03249 01540 01553 02066 lnFERTzt 00591 00353 03058 00566 01152 01483 b The assumption that you make when you estimate the equations in part a is that the error terms are correlated across time for each individual It says that the error terms in the three equations for individual i are correlated It encompasses the idea that the individual farm behaves in a similar manner over time The contemporaneous correlations in part a can be represented as COVez391995 9611996 2 519951996 COVez3919969 911997 2 519961997 COVez391995 2911997 519951997 To test Whether the contemporaneous correlation is signi cant We test the null hypothesis H 0 G19951996 2 519961997 2 519951997 2 O The test statistic is LM N X quot129951996 7129951997 715961997 44gtlt 035802 043782 030372 1813 The 5 critical value for a chisquare distribution with 3 degrees of freedom is 781 We reject the null hypothesis and conclude that this contemporaneous correlation is signi cant c The equality of the elasticities in the different years can be tested using an F test or a X2 test The calculated value for the X2test is X2 2559 the corresponding 5 critical value for 6 degrees of freedom is X89596 1259 The calculated value for the Ftest is F 4265 the corresponding 5 critical value is 17095a6a120 2175 Thus We reject the null hypothesis that all elasticities are the same in all 3 years CHAPTER 1 6 Exercise Solutions 527 Chapter 16 Exercise Solutions Principles of Econometrics 46 528 EXERCISE 161 a The least squares estimation of the linear probability model is f9 04848 00703DTIME se 00714 00l29 The estimated marginal effect of DTIME on fa is constant and does not change with DTIME Therefore at DTIME 2 a 20 minute differential the estimated increase in the probability of a person choosing automobile transport for a 10 minute 1 unit increase in DTIME is 00703 b The predicted probabilities PHAT are dtlme auto phat I I I 1 485 0 143792 2 44 0 6563513 3 828 1 1066961 4 246 0 3118327 5 316 0 2626157 6 91 1 1124615 7 521 1 8511097 8 877 0 1318229 9 17 0 3652682 10 515 0 122699 11 907 0 1529159 12 655 1 945325 13 44 1 1754314 14 7 0 4355781 15 516 1 8475943 16 324 1 7125992 17 618 0 O502798 18 34 1 7238488 19 279 1 6809598 I 20 729 0 O277642 21 499 1 8356416 Some predicted probabilities are greater than 1 and some less than 0 These are not plausible probabilities This problem is inherent in the linear probability model because the marginal effect of the dependent Variable on fa is constant Chapter 16 Exercise Solutions Principles of Econometrics 46 529 Exercise 161 continued 0 d 6 Feasible generalized least squares estimation of the linear probability model yields f9 04953 00602DTIME se00333 000419 Compared to part a these coef cients are similar in magnitude but the standard errors are much smaller False The generalized least squares estimation procedure does not x the basic deficiency of the linear probability model It is still possible to predict probabilities that are greater than 1 or less than 0 using generalized least squares It only accounts for heteroskedasticity thereby producing correct standard errors for the linear probability model The predicted probabilities PGLS are dtime auto pgls I I 1 485 O 2033708 2 44 O 6421104 3 828 1 9935836 I 4 246 0 34721 5 316 O 3050813 6 91 1 1042934 7 521 1 8088195 8 877 O O325496 9 17 0 3929496 10 515 0 1853157 11 907 O O506047 12 655 1 8894657 13 44 1 2304535 I 14 7 0 4531334 15 516 1 8058103 16 324 1 6902574 17 618 0 1233264 18 34 1 6998869 I 19 279 1 6631747 20 729 O O565224 21 499 1 795579 Using the generalized least squares estimates of the linear probability model the percentage of correct predictions is 9048 Exercise 161 continued f Chapter 16 Exercise Solutions Principles of Econometrics 46 530 The percentage of correct predictions using the probit model PPROBIT is 9048 This is identical to the percentage of correct predictions using the linear probability model I dtime auto pproblt I I I 1 I 485 O O643329 I 2 I 244 0 7477868 I 3 I 828 1 9922287 I 4 I 246 0 2111583 I 5 I 316 0 1556731 I 6 I 91 1 996156 I 7 I 521 1 933 I 8 I 877 O OO35158 I 9 I 17 0 282843 I 10 I 515 O O537665 I 11 I 907 O OO26736 I 12 I 655 1 9713162 I 13 I 44 1 O831197 I 14 I 7 0 3918784 I 15 I 516 1 931031 I 16 I 324 1 8179376 I 17 I 618 O O27532 I 18 I 34 1 8303455 I 19 I 279 1 780102 I 20 I 729 O O121814 I 21 I 499 1 9240018 I A classi cation table is True Classified I AUTO AUTO I 9 BUS I 1 Total I 10 Chapter 16 Exercise Solutions Principles of Econometrics 46 531 EXERCISE 162 3 b C The maximum likelihood estimates of the logit model are 3 f3DT1ME O2376 O5311DTIME se 07505 O2064 The logit estimate 31 is much smaller than the probit estimate whereas 62 and the standard errors These estimates are quite different from the probit estimates on page 593 are larger compared to the probit model The differences are primarily a consequence of the variance of the logistic distribution R23 being different to that of the standard normal 1 Using 1611 and replacing the standard normal density function with the logistic probability density function 1616 gives j ltB1B2xB2 where lB1B2x Given that DTIME 2 the marginal effect of an increase in DTIME using the logit estimates is dz 4 DTIME x 02376 05311 2 0531101125 dDTIME B1B2 B2 X X This estimate is only slightly larger than the probit estimate of the marginal effect Both the logit and probit estimates suggest that a 10 minute increase in DTIME increases the probability of driving by about 10 Using the logit estimates the probability of a person choosing automobile transportation given that DTIME 3 is MB 339DT1ME A O2376 05311 x 3 07951 The prediction obtained from the probit model is 0798 There is little difference in predicted probabilities from the probit and logit models Exercise 162 continued d Chapter 16 Exercise Solutions Principles of Econometrics 46 The predicted probabilities PHAT are KC CD OL nJgtJu l I CDKQCD l OOlOL nJgtUI I dtime auto phat I I I I 485 O O566042 I I 44 0 7423664 I I 828 1 9846311 I I 246 0 1759433 I I 316 0 1283255 I I 91 1 9900029 I I 521 1 9261805 I I 877 O OO74261 I I 17 0 2422391 I I 515 O O486731 I I 907 O OO63392 I I 655 1 9623526 I I 44 1 O708038 I I 7 O 3522088 I I 516 1 9243443 I I 324 1 8150529 I I 618 O O287551 I I 34 1 827521 I I 279 1 7762923 I I 729 O O161543 I I 499 1 9177834 I Using the logit model 9048 of the predictions are correct 532 Chapter 16 Exercise Solutions Principles of Econometrics 46 533 EXERCISE 163 a The least squares estimated model is f9 00708 0l6OFIXRl TE Ol32MIRGIN O793YIELD se 1288 00822 00498 0323 00341MITURITY 00887POINTS 00289NETWORTH 0191 00711 00118 All the signs of the estimates are consistent with expectations The predicted values are between zero and one except those for observations 29 and 48 which are negative b The estimated probit model in tabular form is Probit regression Number of obs 78 LR chi26 2719 Prob gt chi2 00001 Log likelihood 39207128 Pseudo R2 02575 adjust I Coef Std Err z Pgtz 95 Conf Interval fixrate 4987284 2624758 190 0057 0157148 1013172 margin 4309509 1739101 248 0013 7718083 0900934 yield 2383964 1083047 220 0028 4506698 2612297 maturity 0591854 6225826 010 0924 1279425 1161054 points 2999138 2413875 124 0214 7730246 1731971 networth 0838286 037854 221 0027 0096361 1580211 cons 1877266 4120677 046 0649 9953644 6199112 01 f9 CIgt l 877 O499FIXRATE 043 IMARGIN 23 84YIELD se 4121 0262 0174 1083 00591MITURITY O3OOPOINTS 0083 SNETWORTH 0623 0241 00379 All the estimates have the expected signs Ignoring the intercept and using a 5 level of significance and onetail tests We nd that all coef cients are statistically signi cant with the exception of those for MA T URIT Y and POINTS Chapter 16 Exercise Solutions Principles of Econometrics 46 534 Exercise 163 continued c The percentage of correct predictions using the probit model to estimate the probabilities of choosing an adjustable rate mortgage is 7564 Probit model for adjust True Classified Adjust Fixed Total Adjust 21 8 29 Fixed ll 38 49 Total 32 46 78 d The sample means are FIXRA TE 1325 MARGIN 2292 YIELD 1606 MATURITY 1058 POINTS 1498 NETWORTH 3504 The marginal effect of an increase in MARGIN at the sample means is L 0431gtlt 1877 0499FIXR1 TE 0431MIRGIN dMARGIN 23 84YIELD 00591MA TURITY 03 OOPOINTS 0083 SNETWORTH 0164 This estimate suggests that at the sample means a one percent increase in the difference between the Variable rate and the fixed rate decreases the probability of choosing the Variablerate mortgage by 164 percent The deltamethod standard error is 0066 and a 95 interval estimate of this marginal effect is 0294 0034 using standard normal critical Values Chapter 16 Exercise Solutions Principles of Econometrics 4e 535 EXERCISE 164 a 778 of all high school graduates attended college either 2 or 4year b The estimated probit model in tabular form is Probit regression Number of obs 1000 LR chi26 22642 Prob gt chi2 00000 Log likelihood 41621967 Pseudo R2 02138 college I Coef Std Err z Pgtz 95 Conf Interval grades 2945521 0274882 1072 0000 348428 2406761 faminc 005393 0018099 298 0003 0018457 0089404 famsiz 0531059 0374572 142 0156 1265207 0203089 parcoll 4765344 1424817 334 0001 1972755 7557933 female 0237927 1014679 023 0815 1750806 2226661 black 6109028 2176202 281 0005 184375 1037431 cons 2693662 283459 950 0000 2138092 3249231 01 9 CD26937 02946GRADES 00053 9FAMINC 00531FAMSIZ se 0283 5 00275 000181 00375 04765PARCOLL 0023 SFEMALE 06109BLACK 01425 01015 02176 Because students with better grades are more likely to be accepted into college we expect the coefficient of GRADES to be negative Students from wealthier families are more likely to have college funds so We expect the coefficient for FAMINC to be positive Similarly students from smaller households are more likely to go to college because larger families might not have enough money to send all family members to college Therefore We expect the coef cient of FAMSIZ to be negative We expect the coefficient of PARCOLL to be positive however we do not have expectations on the signs of FEMALE and BLACK All coefficients are consistent with our expectations All coef cients are statistically signi cant except for FAMSIZ and FEMALE Chapter 16 Exercise Solutions Principles of Econometrics 416 536 Exercise 164 continued 0 Using the estimates from b the probability of attending college for a black female with GRADES 5 FAMINC equal to the sample mean FAMSIZE 5 and PARCOLL 1 is f9 CD26937 02946 x 5 000539 x 5139 00531 x 5 04765 gtltl 00238 x 1 06109 x 1 0990 When this student has GRADES 10 f9 CIgt26937 02946 x 10 000539 x 5139 0053lgtlt 5 04765 gtltl 00238 x 1 06109 x 1 0808 d i For a white female with GRADES 5 f9 CIgt26937 02946 x 5 000539 x 5139 0053 1 x 5 04765 gtltl 00238 x 1 06109 x 0 0958 For a white female with GRADES 10 f9 CIgt26937 02946 x 10 000539 x 5139 0053lgtlt 5 04765 gtltl 00238 x 1 06109 x 0 0603 ii For a white male with GRADES 5 f9 CIgt26937 02946 x 5 000539 x 5139 0053 1 x 5 04765 gtltl 00238 x 0 06109 x 0 0956 For a white male with GRADES 10 f9 CIgt26937 02946 x 10 000539 x 5139 0053lgtlt 5 04765 gtltl 00238 x 0 06109 x 0 0593 Chapter 16 Exercise Solutions Principles of Econometrics 416 537 Exercise 164 continued e The estimated model excluding PARCOLL BLACK and FEMALE in tabular form is Probit regression Number of obs 1000 LR chi23 20580 Prob gt chi2 00000 Log likelihood 42652689 Pseudo R2 01944 college I Coef Std Err z Pgtz 95 Conf Interval grades 2938452 0259053 1134 0000 3446186 2430718 famine 0073394 001668 440 0000 0040701 0106087 famsiz 064119 0368791 174 0082 1364007 0081627 cons 2793474 2664321 1048 0000 2271277 3315671 0139 9 c1gt27935 0293 SGRADES 000734FAMINC 00641FAMSIZ se 02664 00259 000167 003 69 The signs of the remaining variables are unaffected All coef cients remain signi cant However FAMSIZ becomes statistically signi cant using a one tailed test and a 005 level of signi cance i Testing the joint signi cance of PARCOLL BLACK and FEMALE using a likelihood ratio test yields the test statistic LR 20 ZR 2 41622 426527 2061 The critical chisquared value at a 005 level of signi cance is x20954 949 Since the test statistic is greater than the critical value We reject the null hypothesis and conclude that PARCOLL BLACK and FEMALE are jointly signi cant and should be included in the model The test pvalue is 00001 Chapter 16 Exercise Solutions Principles of Econometrics 416 538 EXERCISE 165 a 6774 of high school graduates who attended college chose a 4year college 5199 of 4year college students are female and 588 are black b The estimated probit model in tabular form is Probit regression Number of obs 778 LR chi23 11907 Prob gt chi2 00000 Log likelihood 42969285 Pseudo R2 01217 fouryr Coef Std Err z Pgtz 95 Conf Interval grades 2279592 0245122 930 0000 2760023 1799161 faminc 0052572 0014891 353 0000 0023386 0081757 famsiz 0092141 0391261 024 0814 0674716 0858997 cons 1581626 2384773 663 0000 1114219 2049033 The estimate signs are as expected Students with better grades are more likely to be accepted into a 4year college therefore We expect the coefficient of GRADES to be negative Students from wealthier families are more likely to have college funds so We expect the coefficient for FAMINC to be positive Similarly students from smaller households are more likely to go to college because larger families might not have enough money to send all the family members to college We expected the coefficient of FAMSIZ to be negative All estimates are statistically significant except for the coefficient of FAMSIZ c The estimated probit model for black students is Probit regression Number of obs 44 LR chi23 2277 Prob gt chi2 00000 Log likelihood 15321768 Pseudo R2 04263 fouryr Coef Std Err z Pgtz 95 Conf Interval grades I 8360657 2762361 303 0002 1377478 2946529 faminc 0208116 0168619 123 0217 0122372 0538604 famsiz 0834497 2135331 039 0696 3350676 501967 cons 6537597 2114599 309 0002 239306 1068213 Chapter 16 Exercise Solutions Principles of Econometrics 416 539 Exercise 165c continued The estimated probit model for white students is Probit regression Number of obs 734 LR chi23 11272 Prob gt chi2 00000 Log likelihood 40608529 Pseudo R2 01219 fouryr Coef Std Err z Pgtz 95 Conf Interval grades I 2306905 025459 906 0000 2805892 1807918 faminc 0054343 0015133 359 0000 0024683 0084004 famsiz 0194625 0403386 048 0629 0595997 0985247 cons 1512805 2435231 621 0000 1035509 1990102 There are large differences between the coef cients of the two models Given identical Values for the Variables the effect of unit changes in both GRADES and FAMINC on the probability of attending a 4year college is larger for black students than it is for white students All coef cient estimates are signi cant with the exception of FAMSIZ The table below summarizes the results Probit models 1 2 3 full sample black only white only grades 0228 0836 0231 0025 0276 0025 faminc 0005 0021 0005 0001 0017 0002 famsiz 0009 0083 0019 0039 0214 0040 cons 1582 6538 1513 0238 2115 0244 N 778 44 734 ll 429693 15322 406085 chi2 119074 22769 112717 Standard errors in parentheses plt005 plt001 plt0001 Chapter 16 Exercise Solutions Principles of Econometrics 416 540 EXERCISE 166 a The probabilities of this multinomial logit model are A 1 19 quot 1 expltz2gt expltz3gt A eXpEn Z 1 explt22gt explt23gt A eXp53 3 1 explt22gt explt23gt Where 1 if they did not attend college j 2 if they attend a 2year college 3 if they attend a 4year college and 2 312 BHGRADESI B32FAMINCl BMFEMALE BSZBLACKI 213 313 BBGRADES BBFAMINCI BBFEMALE B53BLACKl The estimates for this multinomial logit model are presented in the following table Multinomial logistic regression Number of obs 1000 LR chi28 34380 Prob gt chi2 00000 Log likelihood 84675602 Pseudo R2 0l688 psechoice I Coef Std Err z PgtIzI 95 Conf Interval 2 I grades I 308970l 0552l52 560 0000 4l7l899 2007503 faminc I 0ll8943 003928 303 0002 004l956 0l9593l female I ll69483 l949887 060 0549 2652224 499ll9l black I 56798l3 429546l l32 0l86 2739l36 l409876 cons I l937035 49ll35 394 0000 9744285 2899642 3 I grades I 7272638 0566698 12 83 0000 8383346 6l6l929 faminc I 0204678 00383l9 534 0000 0l29574 027978l female I l337l62 l932327 069 0489 5l24453 2450l28 black I l607l27 4079379 394 0000 807583 240667 cons I 4962637 474465l l046 0000 4032702 589257l psechoicel is the base outcome Chapter 16 Exercise Solutions Principles of Econometrics 46 541 Exercise 166a continued b C d The pValues in this table suggest that all estimated coef cients are signi cant at a 5 level of signi cance except for B42 352 and 043 These three coef cients correspond to the Variables FEMALE and BLACK This probability is calculated by rstly nding E12 and E13 212 19370 03090gtlt 664 00119gtlt 425 01170gtlt 0 05680gtlt 0 03910 213 49626 07273 x 664 00205 x 425 01337 x 0 16071 x 0 10035 Therefore A eXpEz393 1 05239 1 eXpZ2 eXpZ3 The probability that a White male with median Values of GRADES 664 and FAMINC 425 will attend a 4year college is 05239 The probability ratio Value is calculated using an expression analogous to 1621 of the text 15 3 XPZz3 1771 Using the Value of 513 from part b the probability ratio is ff 3139 3 exp10035 27278 1771 Therefore the probability ratio of a White male with median Values of GRADES 664 and FAMINC 425 attending a 4year college rather than not attending any college is 273 to one The probability that a White male with median FAMINC and a Value of 4905 for GRADES is calculated by nding 51 and E173 E172 19370 03090gtlt 4905 00119 x 425 01170 x 0 05680gtlt 0 09270 2173 49626 07273 x 4905 00205 x 425 01337 x 0 16071gtlt 0 22653 then quotI eXpE1quot3 p13 1 1 07320 1 eXpZ2 eXpZ3 Therefore the increase in the probability of attending a 4year college of a White male with median FAMINC Whose GRADES change from 664 to 4905 is 02081 This Value is calculated as 15 91 07320 05239 02081 Chapter 16 Exercise Solutions Principles of Econometrics 46 542 Exercise 166 continued e The estimated logit model is Logistic regression Number of obs 749 LR chi24 29134 Prob gt chi2 00000 Log likelihood 3095561 Pseudo R2 03200 fouryr Coef Std Err z Pgtz 95 Conf Interval grades 7272205 0616125 1180 0000 8479787 6064622 famine 0182128 0038987 467 0000 0105716 0258541 female 1313463 2036463 064 0519 5304858 2677932 black 137962 422851 326 0001 5508474 2208393 Cons I 5069643 504986 1004 0000 4079889 6059397 The probability ratio that a White male student with median will attend a 4year college rather than not attend any college is P L34 2759 l 9 7 0266 Where the median Value of grades for this sample is 642 and the median family income is 42500 Chapter 16 Exercise Solutions Principles of Econometrics 4e EXERCISE 167 a The probabilities of this conditional logit model are A eXpEn p eXp51 eXp52 eXp53 A eXpEn P explt24gtexplt24gtexplt24gt A eXpf3 P 39 eXpEz391 eXpE1392 explt23gt Where 1 if Pepsi j 2 if 7Up 3 if Coke and 2 3PRICE 33DISPLA Y 34FEA TURE 2 3PRICE2 33DISPLAY 34FEATURE 2 3PRICE3 33DISPLA Y 34FEA TURE3 543 The estimates for this conditional logit model are presented in the following table Conditional logistic regression Number of obs 5466 35889 00000 00896 l39l793 6448l85 LR chi23 Prob gt chi2 Log likelihood l8222267 Pseudo R2 choice I Coef Std Err z Pgtz 95 Conf price l744454 l799323 970 0000 2097ll5 display I 4624476 O93048l 497 0000 2800767 feature I OlO6038 O799373 013 0894 l67278 l460705 The coef cient 32 PRICE is negative which suggests that an increase in own price decreases the brand s probability of being bought The coef cient of 33 DISPLAY is positive which implies that displaying the brand increases its probability of being bought The coef cient 34 FEATURE is negative suggesting that being featured decreases the brand s probability of being bought The signs of 32 and 33 are as expected however We Would expect the sign of 34 to be positive signi cance The pvalues suggest that 32 and 33 are statistically signi cant and that 34 is not signi cantly signi cant at a 005 level of Chapter 16 Exercise Solutions Principles of Econometrics 46 544 Exercise 167 continued b The probability ratio of choosing Coke relative to Pepsi and 7Up if the price for each is 125 and there is no display or feature are 1 This value is calculated as 7313 Z 033 1 7313 2 033 1 91 033 132 033 In this scenario the alternatives are equally likely so their choice probabilities are equal c The probability ratio of choosing Coke relative to Pepsi and 7Up if the price for each is 125 Coke is on display and there is feature is 1538 This value is calculated as 2 04426 eXpZ13 211538 91 02787 eXpE11 amp 04426 eXpE13 91 39 02787 39 eXpE12 1538 Where 211 17445 x 125 04624 x 0 00106 x 0 21806 512 17445 gtlt125 04624gtlt 0 00106 x 0 21806 213 17445 gtlt125 04624gtlt1 00106gtlt 0 17181 d Under this scenario the probability of choosing either Pepsi or 7up is 02894 compared to 02787 in part c a change of 00107 The probability of choosing Coke is 04212 compared to 04426 in part c a decrease of 00214 These changes are calculated as eXpf eXpf 14 pl 2 eXpE1391 eXpE1392 eXpZ1393 eXpZ11 eXp 12 eXp 13 where 21 211 from c 2172 512 from c 2173 17445 gtlt130 04624 gtlt1 00106 x 0 18053 e Adding the alternative speci c intercept yields the following conditional logit model speci cations and estimates A eXPEz391 p 39 explt2gt explt22gt explt23gt A eXP z392 p 39 explt21gt explt22gt explt23gt A eXPEz393 p 39 explt2gt explt22gt explt23gt Chapter 16 Exercise Solutions Principles of Econometrics 46 545 Exercise 167e continued Where 211 311 B2PRICE11 B3DISPLAY11 B4FEATURE11 212 312 EZPRICE12 33DISPLAY12 LFEATURE12 213 32PRICE13 B3DISPLAY13 B4FEATURE13 The estimation results are Conditional logistic regression Number of obs 5466 LR chi25 38063 Prob gt chi2 00000 Log likelihood 18113543 Pseudo R2 00951 choice I Coef Std Err z Pgtz 95 Conf Interval price 1849186 1886595 980 0000 2218952 147942 feature 0408576 0830752 049 0623 2036821 1219669 display 4726785 0935445 505 0000 2893346 6560225 pepsi 2840865 0625595 454 0000 1614722 4067008 sevenup 0906629 0639666 142 0156 0347094 2160352 0 If the price of each is 125 and a display for Coke is present the odds of choosing Coke relative to Pepsi is 121 and the odds of choosing Coke relative to 7Up is 147 These odds are calculated as E 03983 I eXp213121 amp 03983 eXp213 911 03299 eXp211 912 02718 eXp212 147 Where 211 02841 18492 gtlt125 04727 x 0 00409 x 0 20274 212 00907 18492 x 125 04727 x 0 00409 x 0 22208 213 18492 gtlt125 04727 x 1 00409 x 0 231 15 Under the first scenario all prices 125 Coke display the probabilities are Prchoice Coke 3983 Prchoice Pepsi 3299 Prchoice SevenUp 2718 Under the second scenario Coke price increase to 130 the probabilities are Prchoice Coke 3764 Prchoice Pepsi 3419 Prchoice SevenUp 2818 Chapter 16 Exercise Solutions Principles of Econometrics 46 546 EXERCISE 168 a Using the estimates in Table 165 the probability that a student with median GRADES 664 will choose no college y 1 is Py 1 CD 29456 03066 x 664 01815 The probability that a student with median GRADES chooses to attend a 2year college is Py 2 CD 20900 03066 x 664 CD 29456 03066 x 664 02970 The probability that a student with median GRADES chooses to attend a 4year college is Py 3 1 CD 20900 03066 x 664 05215 Recomputing these probabilities when GRADES 4905 yields Py 1 CD 29456 03066 x 4905 00747 Py 2 CIgt 2090003066 x 4905 CD 29456 03066 x 4905 02042 Py 3 1 CIgt 2090003066 x 4905 07211 These results are as anticipated since We expect the probability of going to a 4year college to increase and the probability of not going to college to decrease for students with better grades b The ordered probit estimates are Ordered probit regression Number of obs 1000 LR chi25 35759 Prob gt chi2 00000 Log likelihood 83986473 Pseudo R2 01755 psechoice I Coef Std Err z PgtIzI 95 Conf Interval grades I 2952923 0202251 1460 0000 3349328 2556518 famine I 0052525 001322 397 0000 0026615 0078435 famsiz I 024l215 0301846 080 0424 0832822 0350391 black I 7l3l3l2 1767871 403 0000 3666348 1059628 parcoll I 4236226 1016424 17 0000 2244o71 6228381 cutl I 2595845 2045863 2996827 2194864 cut2 I 1694591 1971365 2080971 130821 Where I11 25958 se 1 02046 I12 16946 se 2 01971 Chapter 16 Exercise Solutions Principles of Econometrics 46 547 Exercise 168b continued If 2 O2953GRIDES 000525FAMINC 00241FAMSIZ se00202 000132 003 02 0713 IBLA CK 0423 6PIRCOLL O1768 O1016 then to compute probabilities We use Ply 1 lt1gtFt1 3 Ply 2 13 3 NEH 3 Py311 ltIgtltn2 2gt The marginal effects evaluated at the means and using Stata 111 are Expression Prpsechoice1 predictoutcome1 dydx wrt grades at grades 653039 mean faminc 513935 mean famsiz 4206 mean black 056 mean parco11 308 mean De1ta method I I dydx Std Err z Pgtz 95 Conf Interval grades I 0709968 0052913 1342 0000 0606261 0813676 Expression Prpsechoice2 predictoutcome2 dydx wrt grades at grades 653039 mean faminc 513935 mean famsiz 4206 mean black 056 mean parco11 308 mean De1ta method I dydx Std Err z Pgtz 95 Conf Interval grades 0461587 0053038 8 70 0000 0357634 0565541 Chapter 16 Exercise Solutions Principles of Econometrics 46 548 Exercise 168b continued Expression Prpsechoice3 predictoutcome3 dydx wrt grades at grades 653039 mean faminc 513935 mean famsiz 4206 mean black 056 mean parcoll 308 mean Delta method I dydx Std Err z Pgtz 95 Conf Interval grades I ll7l555 0079975 1465 0000 l328304 l0l4807 These estimates suggest that as the student s grades improve or the family income increases the probability of choosing a 4year college increases but the probability of choosing no college decreases As the family size increases the probability of choosing a 4year college decreases and the probability of choosing no college increases Also a black student or a student Whose parents graduated from college or has an advanced degree has a higher probability of choosing a 4year college and a lower probability of choosing no college The pvalues of these estimates indicate that all variables are statistically significant at a 005 level of significance with the exception of FAMSIZE c Testing the joint significance of FAMINC FAMSIZ PARCOLL and BLACK using a likelihood ratio test yields the test statistic LR 20 1 2 X 83986 87582 7191 The critical chisquared value at a 005 level of significance is 209574 949 Since the test statistic is greater than the critical value We reject the null hypothesis and conclude that FAMINC FAMSIZ PARCOLL and BLACK are jointly signi cant and should be included in the model d The probability that a black student from a household of 4 members with 52000 income will attend a 4year college when 1 GRADES 664 is 1333 1 c1gt 16946 02953 x 664 000525gtlt 52 0024lgtlt 4 07l3lgtlt104236gtltl 08525 Chapter 16 Exercise Solutions Principles of Econometrics 46 549 Exercise 168d continued 6 ii GRADES 4905 is 133 1 CIgt 16946 02953gtlt4905 000525gtlt52 00241 x4 07l3lgtltl 04236gtlt1 09406 The probability that a nonblack student from a household of 4 members with 52000 income will attend a 4year college when i GRADES 664 is 1333 1 CD 16946 02953 x 664 000525gtlt 52 00241 x 4 07l3lgtlt0 04236gtlt1 06309 ii GRADES 4905 is 1333 1 CD 16946 02953gtlt4905 000525gtlt52 0024lgtlt4 07l3lgtlt0 04236gtlt1 08013 Given Values of FAMINC FAMSIZ and PARCOLL We nd that the probability of a black or a nonblack student attending a 4year college increases as their Value of GRADES decreases Also at a given Value of GRADES We nd that the probability of a black student going to a 4year college is higher than a nonblack student Chapter 16 Exercise Solutions Principles of Econometrics 46 550 EXERCISE 169 a The Poisson regression predicts that Australia WOI1 10 medals in the 1988 Olympics This Value is calculated as eXp 158875 O1800ln165 gtlt1O6 O5766ln30gtlt 1011 1041 The probability that Australia would Win 10 medals or more in 1988 is 059 b The Poisson regression predicts that Canada Won 16 medals in the 1988 Olympics exp 158875 O1800ln269 gtlt 1O6O5766ln519 x 1011 1559 The probability that Canada would Win 15 medals or less in 1988 is 051 c The estimates are presented in the following table Poisson regression Number of obs 357 LR chi22 377813 Prob gt chi2 00000 Log likelihood l2784853 Pseudo R2 05964 medaltot I Coef Std Err z Pgtz 95 Conf Interval lpop 2054832 O206922 993 0000 l649272 2460392 lgdp 536092l Ol54764 3464 0000 505759 5664252 cons 1526045 3238893 4712 0000 1589526 l462564 These estimates and standard errors are Very similar to those in Table 166 The most noticeable difference is that these standard errors are smaller than those in Table 166 Chapter 16 Exercise Solutions Principles of Econometrics 46 551 Exercise 169 continued d Estimates for the Poisson regression model that adds HOST and SOVIET are Poisson regression Number of obs 357 LR chi24 416283 Prob gt chi2 00000 Log likelihood l086l382 Pseudo R2 06571 medaltot I Coef Std Err z PgtIzI 95 Conf Interval lpop I l52l344 02224ll 684 0000 l085426 l957263 lgdp I 5640386 0l7ll5l 3296 0000 5304936 5975835 soviet I 2083646 0839005 24 83 0000 l9l9205 2248088 host I l6l0607 l005472 l60 0l09 0360083 358l296 cons I l5l6299 347462l 4364 0000 l5844 l448l98 The signs of the all coefficients are as expected Countries with a larger population have a greater pool of talent so We expect the coef cient of lnPOP to be positive Countries with a larger GDP have more money to spend on sports technology and training so We expect the coef cient of lnGDP to be positive Host countries have the advantage of being acclimatized being familiar with the sporting facilities and having the home crowd Therefore We expect the coefficient of HOST to be positive Former Soviet Union countries Win more medals than the average country therefore We expect that the coefficient of SOVIET will be positive All variables are statistically signi cant at a 5 level of signi cance except for HOST e Estimates for the Poisson regression model that adds HOST and PLANNED are Poisson regression Number of obs 357 LR chi24 380432 Prob gt chi2 00000 Log likelihood l265390l Pseudo R2 06005 medaltot I Coef Std Err z PgtIzI 95 Conf Interval lpop I l397755 0244232 572 0000 09l907 l87644l lgdp I 562546l 0l74452 3225 0000 528354l 596738l planned I 62360l9 ll84734 526 0000 39l3984 8558054 host I ll95092 l005053 ll9 0234 0774775 3l64959 cons I l484l63 3437894 4317 0000 l55l544 l4l6782 All estimates and standard errors are similar to those in part d includes SOVIET is preferred because it has a higher loglikelihood value The model which Chapter 16 Exercise Solutions Principles of Econometrics 46 552 Exercise 169 continued f The Poisson regression model from part e predicts that in 2000 Australia would Win 13 medals and Canada would Win 17 medals These predictions were calculated as eXp 14841601398ln19071gtlt106 05625ln322224 X 10 01 195x 1 06236gtlt 0 12528 eXp 14841601398ln30689gtlt106 05625ln641256gtlt10 01195 x 0 06236 x 0 17495 The prediction for Canada is reasonably close to the actual Value That for Australia is a long Way from the actual Value Chapter 16 Exercise Solutions Principles of Econometrics 4e 553 EXERCISE 1610 a Figure xrl6lOa shows the histogram of the variable SHARE 400 I 300 I Frequency 200 I I I I O 05 1 15 2 share Figure xr1610a Histogram of SHARE There is a large number of observations at SHARE O speci cally 6196 of the observations are zero This value can be classi ed as the limit value The variable is an example of censored data b The least squares estimated model is Source SS df MS Number of obs 508 F 4 503 9878 Model 075988344 4 0l8997086 Prob gt F 00000 Residual 0967396l7 503 000l92325 R squared 04399 Adj R squared 04355 Total l7272796l 507 000340686 Root MSE 0l387 share Coef Std Err t Pgtt 95 Conf Interval lpop 0002036 0004342 047 0639 00l0568 0006495 lgdp 00334l5 0003983 839 0000 0025589 004l24l host I 044673 008l264 550 0000 0287072 0606388 soviet 055598l 0044663 1245 0000 0468232 0643729 Cons I 0694966 006l669 1127 0000 08l6l27 0573805 i The coef cients of lnGDP HOST and SOVIET have the expected signs and these variables are statistically signi cant at a 005 level of signi cance The coef cient of lnPOP does not have the expected sign and is not statistically signi cant However We must be careful when interpreting these coef cients because We are using censored data This data yields least squares coef cients that are biased and inconsistent Chapter 16 Exercise Solutions Principles of Econometrics 46 554 Exercise 1610b continued b 0 ii A plot of the residuals against lnGDP is shown in Figure Xr1610b The residuals do not appear random Where lnGDP is less than 21 all residuals are positive and seem to follow a decreasing linear trend Where lnGDP is greater than 21 the majority of residuals are negative and appear to continue along the decreasing linear trend Also the variance of the residuals increases greatly as lnGDP increases 12 08 04 residuals 00 O4 08 16 18 20 22 24 26 28 30 lnGDP Figure xr1610b A scatter plot of residuals versus lnGDP iii The skewness and kurtosis of the residuals is 364 and 2715 respectively These values are very different to the skewness and kurtosis of the normal distribution which are 0 and 3 respectively A JarqueBera test for normality on the residuals rejects the null hypothesis at a 001 level of signi cance Based on the estimates in part b it is predicted that Australia s share of the Olympic medals in 2000 would be 0060 and Canada s share would be 0018 The actual shares of medals won were 0062 and 0015 for Australia and Canada respectively Our predictions are very close to the actual values The predicted shares are calculated as SHARECANADA 00695 0000204ln30689 X 106 0003341ln641256gtlt10 004467gtlt 0 005560gtlt 0 0018 SHAREAUST 00695 0000204ln19071gtlt106 0003341ln322224 X10 004467gtlt1 005560gtlt 0 0060 Chapter 16 Exercise Solutions Principles of Econometrics 46 555 Exercise 1610 continued d The estimated Tobit model Tobit regression Number of obs 508 LR chi24 34073 Prob gt chi2 00000 Log likelihood 38250206 Pseudo R2 08031 share Coef Std Err t Pgtt 95 Conf Interval lpop 0012241 0009766 125 0211 0006947 0031428 lgdp 0086398 0008358 1034 0000 0069978 0102818 host I 0366962 0124336 295 0003 0122682 0611242 soviet 0625879 0068415 915 0000 0491466 0760292 cons 2339405 0159412 1468 0000 2652599 202621 sigma 0211255 0010714 0190206 0232304 Obs summary 314 1eft censored observations at sharelt0 194 uncensored observations Comparing the Tobit estimates to the least squares estimates the coefficient of lnPOP has a different sign and is still statistically insigni cant the coef cient of lnGDP is larger and the coef cients of HOST and SOVIET are similar e The predicted shares of medals WOI1 in the 2000 Olympics using the Tobit model are 0053 and 0028 for Australia and Canada respectively Compared to the predictions in part c these predicted shares are not closer to the true shares Note that the calculations for these predictions require us to use an expression like 1640 but speci c to this model The expression used is ESHARE SHARE gt 0 3 13 1nP0P B 1nGDP B4H0sT Bs0V1ET Pk B2 IHUDOB B3 IIKGDB B41105 B5SOVIEl g c1gtB 51nP01B1nGDzB4H0sz Bs0V1ETcs Chapter 16 Exercise Solutions Principles of Econometrics 4e EXERCISE 1611 3 The probit results are given in the table below in column 2 556 0028 07l9 02l6 4067 Ol58 4238 0l40 0409 Probit models Small 1 boy whiteasian black freelunch tchwhite tchmasters cons 0523 30208 N 5786 lnL 3536323 5786 5786 5786 3519014 t statistics in parentheses plt005 plt00l plt000l Based on the individual t statistics We conclude that BOY WHITEASIAN and BLACK are not statistically significant The joint test of the signi cance of these three variables is based on the likelihood ratio test statistic LR 2lnLU lnLR 2 35346l7 3536323 3411 The value of the restricted loglikelihood function In LR comes from the model including only an intercept in column 1 The critical values for the Chisquare distribution are given in Table 3 The test degrees of freedom is 3 because We are testing 3 joint hypotheses that the coef cients of the selected variables are 0 The 95 percentile of the Chisquare distribution for 3 degrees of freedom is 7815 Since the value of the LR statistic is less than the critical value We fail to reject the null hypothesis that the 3 variables BOY WHITEASIAN and BLACK have coef cients that are 0 Exercise 161 1 a continued b Chapter 16 Exercise Solutions Principles of Econometrics 46 557 If the assignment of students to small classes is random We would expect to nd no signi cant relationship between SMALL and any Variable Our ndings are consistent with the hypothesis of random student assignment The results of probit models for AIDE and REGULAR are in the following 2 tables whiteasian black freelunch tchwhite tchmasters cons o377 22294 0047 1234 Ol36 2662 006O 1675 O745 2070 5786 3757086 5786 5786 5786 3749378 t statistics in parentheses plt005 plt00l plt000l Chapter 16 Exercise Solutions Principles of Econometrics 46 558 Exercise 1611b continued Probit models Regular 1 2 3 4 boy 0 002 0002 0 000 0 045 0045 0 007 whiteasian 0404 0400 0387 1 071 1064 1 026 black 0386 0396 0277 1 023 1052 0 731 freelunch 0028 0022 0741 0572 tchwhite 0332 6579 tchmasters 0087 2418 cons 0395 0791 0778 0491 23285 2102 2070 1290 N 5786 5786 5786 5786 1nL 3733530 3732827 3732552 3709791 t statistics in parentheses plt005 plt001 plt0001 As in part a none of the Variables BOY WHITEASIAN or BLACK is statistically signi cant based on the t Values For AIDE the LR test Value is 2292 and for REGULAR the LR test Value is 1405 Thus the Variables are neither individually or jointly signi cant which is consistent with the notion that students were assigned randomly c The Variable FREEL UNCH is added in column 3 of the table results It is not statistically signi cant in any of the probit model estimations Its inclusion in the model has little effect on the other coef cient estimates d The Variables T CH WHITE and T CHMAST ERS are added in column 4 of the tables They are statistically signi cant in each estimation except T CHMAST ERS is not signi cant in the AIDE model The LR test for their joint signi cance is obtained using the likelihood ratio test statistic Chapter 16 Exercise Solutions Principles of Econometrics 46 559 Exercise 161 1 d continued LR 2lnLU lnLR In each case the unrestricted loglikelihood Value comes from the model in column 4 and the restricted loglikelihood Value comes from the model in column 3 These Values are 3086 1141 and 4552 for the models for SMALL AIDE and REGULAR respectively We are testing 2 joint hypotheses that the coef cients of T CH WHITE and T CHMAST ERS are both 0 The LR test statistic has a Chisquare distribution with 2 degrees of freedom if the null hypothesis is true The 99th percentile of this distribution is 9210 Thus We reject the null hypothesis at the 001 level of significance in all three cases In the STAR program students were randomly assigned Within schools but not across schools It is possible that schools in wealthier or predominately White school districts have more teachers who are White or who has Master s degrees This would explain the significance of these Variables in the probit models Chapter 16 Exercise Solutions Principles of Econometrics 46 560 EXERCISE 1612 a The least squares estimates are reported in the following table Linear regression Number of obs 1000 F 8 991 4339 Prob gt F 00000 R squared 03363 Root MSE 32673 I Robust delinquent I Coef Std Err t PgtItI 95 Conf Interval 1vr I 0016239 0006752 240 0016 0002988 0029489 ref I 0593237 0240256 2 47 0014 1064706 0121768 insur I 4815849 0303694 1586 0000 5411807 4219891 rate I 0343761 0098194 350 0000 0151068 0536454 amount I 023768 0144509 164 0100 0045898 0521259 credit I 0004419 0002073 2 13 0033 0008487 0000351 term I 0126195 003556 355 0000 0195976 0056414 arm I 1283239 0276932 4 63 0000 0739798 1826681 cons I 6884913 2285064 301 0003 2400792 1136903 The outcome variable is whether a borrower was delinquent on a payment The explanatory variables are LVR If the loantovalue ratio increases the estimated probability of a delinquent payment increases holding all else xed If a borrower is trying to obtain a loan that that is large relative to the value of the property this may indicate that their nances are stretched The positive sign is consistent with that notion REF If the loan is for a re nance to take advantage of lower rates or to cash out some of the equity there is an indication the borrower has been reliably paying on time and more history The estimated probability of being delinquent is smaller and signi cantly so for loans for a re nance holding all else constant INSUR Mortgage insurance is required of loans with loantovalue ratio of greater than 80 If a mortgage carries mortgage insurance there is a large and signi cant reduction in the probability of a delinquent payment Those with mortgage insurance who pay an additional fee for it and may go through additional scrutiny and screening which may increase the lending standard and reduce the probability of a delinquent payment The magnitude of the effect estimated is too large INS UR may be picking up other effects not identi ed in the model RATE The higher the interest rate the larger the probability of a delinquent payment Higher rate loans are more of an economic burden to the borrower increasing the monthly payments Also the riskier the loan the higher the rate charged so higher rates may indicate loans that have a higher probability of default Chapter 16 Exercise Solutions Principles of Econometrics 46 561 Exercise 1612a continued 0 AMOUNT The larger the amount of borrowed money the higher the probability of a delinquent payment Larger loans lead to larger monthly payments increasing the chance of a delinquent payment This effect is signi cant at the 10 level 0 CREDIT The larger the borrower s credit score the lower the chance of a delinquent payment The credit score is a history of borrowing and repayments on everything from credit cards to car loans It makes sense that those with higher scores will have less chance of making a late payment based on their history 0 TERM The longer the term of the loan the smaller the monthly payments reducing the probability of a delinquent loan holding all else xed 0 ARM Adjustable rate mortgages can change the monthly interest applied to the loan If the rate is adjusted upwards the borrower has a larger monthly payment which signi cantly increases the probability of a delinquent payment all else held constant b The probit model estimates are reported below Probit regression Number of obs 1000 LR chi28 33243 Prob gt chi2 00000 Log likelihood 33279661 Pseudo R2 03331 delinquent Coef Std Err z Pgtz 95 Conf Interval lvr 0076007 0045911 166 0098 0013977 0165991 ref 2884561 1259446 229 0022 5353029 0416092 insur 1772714 1158088 1531 0000 1999695 1545733 rate 1711988 0438147 391 0000 0853236 2570741 amount 121236 0615491 197 0049 000602 2418701 credit 0019131 0010638 1 80 0072 0039981 0001718 term 0775769 0198396 391 0000 1164618 038692 arm 8091109 2077119 390 0000 402003 1216219 cons 964646 1088121 089 0375 1168033 3097325 The signs and signi cance of the coef cients is much the same as in the linear probability model The variable AMOUNT is now signi cant at the 5 level Chapter 16 Exercise Solutions Principles of Econometrics 4e 562 Exercise 1612 continued c The predicted values and explanatory variable values for the 500th and l000th observations are I LPM delinquent lvr ref insur rate amount credit term arm I I I 500 I 1827828 0 70 1 1 1095 854 509 30 1 I 1000 I 5785297 0 882 1 0 765 291 624 30 1 I I PROBIT delinquent lvr ref insur rate amount credit term arm I I I 500 I 1404525 0 70 1 1 1095 854 509 30 1 I 1000 I 6167872 0 882 1 0 765 291 624 30 1 I Neither of the individuals made a delinquent payment Both models predicted a low probability of a delinquent payment 018 and 014 for linear probability and probit models respectively for the first borrower who has a lower loan to value ratio LVR and a lower loan AMOUNT The predicted probabilities for the second borrower were 058 and 062 for linear probability and probit models respectively This borrower had a high loan to value ratio 882 and borrowed a larger amount 291000 d The histogram for credit score is 008 I 006 I m I I IIII L 4 600 700 800 credit score Density 004 I 002 I Figure xr1612d Histogram for credit score Note that most borrowers have a credit score between 500 and 800 The predicted probabilities for a delinquent payment using the linear probability model are CREDIT I Margin Std Err z PgtIzI 95 Conf Interval 500 I 1407109 0257668 546 0000 090209 1912129 600 I 0965214 0190135 508 0000 0592556 1337872 700 I 0523319 0303051 173 0084 007065 1117287 Chapter 16 Exercise Solutions Principles of Econometrics 4e 563 Exercise 1612d continued 6 0 CREDIT For probit the predicted probabilities of delinquency are De1ta method I CREDIT Margin Std Err z Pgtz 95 Conf Interval 500 0984307 0233882 421 0000 0525907 1442708 600 069189 0136925 505 0000 0423523 0960257 700 0471468 0157545 299 0 003 0162686 078025 Note that higher credit scores reduced the predicted probabilities For the linear probability model these changes are the same 00441895 for each 100 point increase in CREDIT The effect is not equal for the probit model being 00292417 for a credit score change of 500 to 600 and 00220422 for the credit score change from 600 to 700 For the probit model these marginal effects are De1ta method I dydx Std Err z Pgtz 95 Conf Interval 500 I 0003319 0002268 146 0143 0007764 0001126 600 I 0002546 0001403 182 0070 0005295 0000203 700 I 0001883 000073 258 0010 0003313 0000452 These values are small and decreasing Recall that these are the marginal effects of a 1 point increase in credit score which is a relatively small amount That the values decrease in magnitude shows that the most bene t from improved credit is for those will smaller credit scores As an alternative to examining these marginal effects one could look at discrete changes in probabilities as in the previous question part or scale credit to be in units of 10 points or 100 points which would shift the decimal point in the above marginal effects accordingly The histogram of loan to value ratio is given below loan amount to value of property percent Figure xr1612f Histogram for loan to value ratio Chapter 16 Exercise Solutions Principles of Econometrics 46 564 Exercise 1612f continued The most popular amount is 80 though there is a spike at 20 as well For the linear probability model the predicted probabilities of delinquency in the two cases are Delta method LVR Margin Std Err z Pgtz 95 Conf Interval 20 0009097 0434921 002 0983 0861527 0843332 80 0965214 0190135 508 0000 0592556 1337872 For the probit model the predictions are Delta method I I LVR Margin Std Err z Pgtz 95 Conf Interval 20 0263176 018372 143 0152 0096909 062326 80 069189 0136925 505 0000 0423523 0960257 In each case an increase in the loan to Value ratio increases the probability of a delinquent payment g The predictions from the linear probability model are summarized in the following table The upper Values are the frequencies and the lower Values are the cell percentages 1 if payment I late by Linear Probability Model 90 days I 0 1 Total 0 727 74 801 72 70 7 40 8010 1 68 131 199 6 80 13 10 1990 Total 795 205 1000 79 50 2050 100 00 The successful predictions are on the diagonal 727 of those who did not have a late payment were correctly predicted 131 of those with a delinquent payment were predicted correctly Chapter 16 Exercise Solutions Principles of Econometrics 46 565 Exercise 1612g continued h For the probit model the prediction summary is 1 if payment I late by Probit Model 90 days I 0 1 Total 0 735 66 801 73 50 6 60 I 8010 1 79 120 199 790 1200 1990 Total 814 186 1000 8140 1860 10000 The probit model is more successful in predicting those who did not have a delinquent payment but is slightly less successful in predicting those who make a late payment This is an important and difficult question In the full sample only about 20 of borrowers have a late payment We try several thresholds 050 080 and 020 If we count just successful predictions of 0 and 1 then the hit rate for each threshold is relevant These values for the thresholds 050 080 and 020 are 882 87 and 806 respectively However it may be that a focus on the two types of errors is useful For example what the percentages of those who were not delinquent would have been predicted delinquent using the three rules Presumably if a person is predicted delinquent they would not receive the loan creating an opportunity cost for the borrower who forgoes a good loan These percentages for the thresholds 050 080 and 020 are 52 02 and 152 respectively If this is the most costly error then a higher threshold such as 080 may be better On the other hand what is the cost of giving a loan to a person who is delinquent on payments If we use the thresholds 050 080 and 020 the percentages of these miscalculations are 66 128 and 42 respectively If this is the costlier error then the use of a low threshold such as 020 might be best From the lender39s perspective origination fee typically is 1 of the loan amount However if a loan falls into default then goes into the foreclosure process the total loss will be way higher than 1 of the loan amount since foreclosure processes are very expensive for lenders Some estimates show that the loss of foreclosure process could be more than 203 0 of the outstanding loan amount So we presume that lenders concern more about the cost of giving the loan to a person who is likely to default in the future Chapter 16 Exercise Solutions Principles of Econometrics 46 566 Exercise 1612 h continued 1 if I payment I late by I phat gt 050 90 days I 0 1 I Total 0 I 407 26 I 433 I 81 40 5 20 I 8660 1 I 33 34 I 67 I 6 60 6 80 I 1340 Total I 440 60 I 500 I 88 00 1200 I 100 00 1 if I payment I late by I phat gt 080 90 days I 0 1 I Total 0 I 432 1 I 433 I 86 40 0 20 I 8660 1 I 64 3 I 67 I 12 80 0 60 I 1340 Total I 496 4 I 500 I 99 20 0 80 I 100 00 1 if I payment I late by I phat gt 0 20 90 days I 0 1 I Total 0 I 357 76 I 433 I 71 40 15 20 I 8660 1 I 21 46 I 67 I 4 20 920 I 1340 Total I 378 122 I 500 Chapter 16 Exercise Solutions Principles of Econometrics 46 567 EXERCISE 1613 The probit estimates for the alternative models for parts ae are reported in the following table The Value labeled 11 is the loglikelihood function Value The Variables denoted lvri ending in i are interaction Variables such as LVRI LVR gtlt INSUR Probit models 1 2 3 4 pooled 1nsur0 1nsur1 full delinquent lvr 0002 0009 0006 0009 0 589 1 329 0 831 1 329 ref 0237 0451 0116 0451 2 213 2 493 0 641 2 493 rate 0120 0111 0222 0111 3 131 1 765 3 538 1 765 amount 0259 0106 0114 0106 5014 1030 1450 1030 credit 0001 0003 0001 0003 1242 2148 0333 2148 term 0045 0083 0058 0083 2569 3207 1807 3207 arm 0544 0816 0750 0816 3111 3279 2027 3279 insur 4971 2251 lvri 0003 0299 refi 0335 1310 ratei 0110 1 238 amounti 0008 0062 crediti 0003 1227 termi 0026 0626 armi 0067 0149 cons 0736 2434 2537 2434 0 773 1 583 1 600 1 583 N 1000 280 720 1000 11 468315 171677 159326 331002 chi2 61396 42750 28747 336021 t statistics in parentheses plt005 plt001 plt0001 d Comparing the estimates from the pooled observations parta those with INSUR 0 part b and those with INS UR 1 part c We find the signs of the coef cients are consistent across all estimations but their magnitudes and significance Varies In most cases the coefficients in the equation for INSUR 0 are larger in absolute Value than their counterparts in the other two equations exceptions being the coef cients for RATE and AMOUNT Chapter 16 Exercise Solutions Principles of Econometrics 46 568 Exercise 1613d continued Only the coef cient for RATE is signi cant in the equation for INS UR 1 In the other equations more coef cients are significant but there is little consistency across the two equations For a sample of N individuals the loglikelihood function is formed from the probability function in equation 1613 f139 ICDG31 B2xiyi I1 CDG31 B2xi1 yi a 4quot Z O21 The model in question has more than one explanatory variable but the principle is the same The loglikelihood function is the sum of the natural logarithms of the probability function 1nLBZfL1nfyi In the above notation let B represent all the parameters in the probit model one for each variable plus a constant term Now suppose we have two groups of observations among the N those N 0 individuals for whom INSUR O and N1 individuals for whom INS UR 1 Because the loglikelihood is a sum we can rearrange the terms as we like 1nLBZZ11nfyZZ 11nfyZf11nfy1nL0 lnL1 Thus estimating the model separately and summing then is equivalent to estimating the full model with interactions between INS UR and the remaining variables In this estimation example lnLU 331002 lnL0 lnL1 171677 159326 331003 The slight difference is due to rounding error The likelihood ratio test statistic is LR 2ln LU ln LR If the null hypothesis is true the statistic has an asymptotic chisquare distribution with degrees of freedom equal to the number of hypotheses being tested The null hypothesis is rejected if the value LR is larger than the chisquare distribution critical value In this case there are J 8 hypotheses that the coef cients of INSUR and the interaction variables such as LVRI are zero The value of the unrestricted loglikelihood if 331002 from the full model in column 4 of the above table The restricted model is the pooled model in column 1 of the table The restricted loglikelihood function value is 4683 15 Therefore the value of the likelihood ratio test statistic is LR 2ln LU ln LR 2 331002 468315 2137313 274626 The test critical value is the 95th percentile of the X38 distribution which is 15507 Therefore we reject the null hypothesis that the coef cients for the insured and uninsured groups are equal and conclude there is some behavioral differences between these two groups Chapter 16 Exercise Solutions Principles of Econometrics 46 569 EXERCISE 1614 a The variable NET PRICE shows variation across the alternative brands Whereas INCOME is a household variable and is the same for all 4 alternatives on any choice occasion The first two households data are hhid alt netprice income I I 1 1 Skist water 79 475 2 1 Skist oil 79 475 3 1 ChiSea water 58 475 4 1 ChiSea oil 58 475 I I 5 2 Skist water 56 475 6 2 Skist oil 56 475 7 2 ChiSea water 79 475 8 2 ChiSea oil 79 475 Note that the prices of the alternatives change Within each group of 4 observations but that income is constant b The choices among the 1500 cases are Alternatives summary for alt Alternative Cases Frequency Percent value label I present selected selected I I I 1 Skist water 1500 548 3653 I 2 Skist oil 1500 291 1940 3 ChiSea water 1500 475 3167 4 ChiSea oil 1500 186 1240 We observe that this group of consumers has a preference for tuna packed in Water Chapter 16 Exercise Solutions Principles of Econometrics 46 570 Exercise 1614 continued c The probability that individual i chooses alternative j for each of these 4 alternatives is facilitated by using some simplifying notation Let the variables and parameters for each alternative be denoted as follows xb Skistwater 3NETPR1CESSW BD1sPLAxSW B4FEA TURESSW xb Skistoil 3 3NETPR1CESS0 B3DISPLA YSS0 B4FEA TURESkm0 xb ChiSeawater B13 B2NETPRICEChiSeawater P YChz39Seawater Y TUREChiSeawater 355 ChiSea390i B14 B2NETPRICEChiSeaoil B3DISPLAYChiSeaoil B4FEA TUREChz39sea oz39z Each variable should have a subscript i to denote the individual but this has been suppressed to simplify notation Each of the options has an intercept parameter except for StarkistinWater which has none and serves as our base case Then the probabilities that each of the options is chosen are 2 exp xb Skiswater PSWWW exp xb Skiswaler exp xb Skisoil exp xb C1z Seawaler exp xb C1z Seaoil exp xb Skisoil exp xb Skiswaler exp xb Skisoil exp xb C1z Seawaler exp xb C1iSeaoil pSkisl0il exp xb Chiseawater exp xb Skiswaler exp xb Skisoil exp xb C1z Seawaler exp xb C1iSeaoil pChiseawater exp xb Chiseaoil pchiseaml exp xb Skiswaler exp xb Skisoil exp xb C1z Seawaler exp xb C1z Seaoil Chapter 16 Exercise Solutions Principles of Econometrics 46 571 Exercise 1614 continued d The estimates obtained with Stata 111 are Alternativespecific conditional logit Number of obs 6000 Case variable hhid Number of cases 1500 Alternative variable alt Alts per case min 4 avg 40 max 4 Wald chi23 40515 Log likelihood l5372704 Prob gt chi2 00000 choice I Coef Std Err z Pgtz 95 Conf Interval alt netprice 9971961 8628894 1156 0000 1166319 8280729 display 1635486 2425727 674 0000 1160052 2110919 feature 1343511 1366656 983 0000 1075652 1611371 Skistwater base alternative Skistoil cons 5959682 0732714 813 0000 7395775 4523589 ChiSeawater cons 5333423 0816866 653 0000 693445 3732396 ChiSeaoil cons 1439991 1002377 1437 0000 1636453 1243529 We note that the estimated coef cients are all statistically significant with the coefficient of the continuous variable NET PRICE carrying a negative sign and the indicator variables DISPLAY and FEATURE having positive signs The alternativespeci c variables are negative and statistically signi cant From the probabilities in the previous question part We see that all else being equal the Starkist in oil and Chicken of the Sea brands have a lower estimated probability of being selected than Starkist in Water Chapter 16 Exercise Solutions Principles of Econometrics 46 572 Exercise 1614 continued 6 The marginal effects are given in the tables below The rst table gives the marginal effect of a price change for each of the brands on the probability of choosing Starkist in water The own price effect is given using equation 1624 For example given that DISPLAY and FEATURE are zero the probabilities reduce to a dependence on the alternative speci c constants and NET PRICE If we set the NET PRICE at its mean for each brand we can compute the probabilities of each choice being selected For example the first table shows that the probability of Starkist in water being selected is 0406 with the price of each variable at its mean shown in the nal column labeled X The marginal effect of an increase in the net price of Starkist in water on the probability of choosing Startkist in water is 8171 1 405579l8 l 405579l8 997l96l 240409 aPRICEi1 I7z1 Pz1B2 X X The change in probability is for a 100 change in price which is more than the cost of the item If the change is 10 cents then we anticipate a reduction in the probability of purchase of 024 The crossprice effect of a change in the price of one brand on the probability of selecting another brand is given by dplj The marginal effect of an increase in the price of Starkist in water on the probability of choosing Chicken of the Sea in water is as shown in the third table below 81723 4 26646827 40557918 9971961107771 aPR1CE p 3p B2 X X Chapter 16 Exercise Solutions Principles of Econometrics 46 573 Exercise 1614e continued Prchoice Skistwaterll selected 40557918 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 240409 211831 1135 0000 281927 198891 68112 Skistoi1 I 899315 096762 929 0000 709665 108896 68163 ChiSeawater I 107771 102948 1047 0000 875935 127948 66976 ChiSeaoi1 I 427063 046209 924 0000 336495 51763 67167 Prchoice Skistoilll selected 22235943 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 899315 096762 929 0000 709664 108897 68112 Skistoi1 I 172431 165592 1041 0000 204886 139976 68163 ChiSeawater I 590856 060676 974 0000 471934 709778 66976 ChiSeaoi1 I 234138 026839 872 0000 181533 286742 67167 Prchoice Chiseawaterll selected 26646827 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 107771 102948 1047 0000 875935 127948 68112 Skistoi1 I 590856 060675 974 0000 471934 709778 68163 ChiSeawater I 194915 177526 1098 0000 229709 16012 66976 ChiSeaoi1 I 280583 034964 802 0000 212055 349111 67167 Prchoice ChiSeaoil1 selected 10559312 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 427063 046209 924 0000 336495 517631 68112 Skistoi1 I 234138 02684 872 0000 181533 286743 68163 ChiSeawater I 280583 034964 802 0000 212055 349111 66976 ChiSeaoi1 I 941784 099314 948 0000 113644 747131 67167 Chapter 16 Exercise Solutions Principles of Econometrics 46 574 Exercise 1614 continued f Adding the individual speci c variable INCOME to the model adds 3 parameters to estimate Like alternative speci c constants coef cients of individual speci c variables are different for each alternative with Starkist in Water again set as the base case The estimates are Alternativespecific conditional logit Number of obs 6000 Case variable hhid Number of cases 1500 Alternative variable alt Alts per case min 4 avg 40 max 4 Wald chi26 41976 Log likelihood 15293439 Prob gt chi2 00000 choice I Coef Std Err z Pgtz 95 Conf Interval alt netprice 999618 8641534 1157 0000 1168989 8302471 display 1619318 2429992 666 0000 1143048 2095587 feature 1336417 1367137 978 0000 1068463 1604371 Skistwater base alternative Skistoil income 021638 0060061 360 0000 0334097 0098662 cons 0673146 1611304 042 0676 3831243 2484952 ChiSeawater income 0027101 0058179 047 0641 014113 0086928 cons 4607403 1710712 269 0007 7960337 1254469 ChiSeaoil income 012768 0075311 170 0090 0275287 0019926 cons 1117534 2119356 527 0000 153292 7021478 The likelihood ratio test is based on the difference in the loglikelihood values for the two models LR 2lnLU ln LR 2 15293439 15372704 15853 The test critical value is the 95th percentile of the x distribution which is 7815 Thus We reject the hypothesis that the coef cients on INCOME are all zero and conclude that INCOME has an effect on these choices Chapter 16 Exercise Solutions Principles of Econometrics 46 575 Exercise 1614 continued g The marginal effects of NET PRICE using the speci ed Values for DISPLAY FEATURE and INCOME and with NET PRICE at its mean for each brand are Prchoice Skistwaterll selected 41970781 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 24346 213621 1140 0000 285329 201591 68112 Skistoi1 I 855809 095308 898 0000 669008 104261 68163 ChiSeawater I 11472 110758 1036 0000 930117 136428 66976 ChiSeaoi1 I 431595 048578 888 0000 336383 526806 67167 Prchoice Skistoilll selected 20398375 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 855809 095308 898 0000 669008 104261 68112 Skistoi1 I 162312 161558 1005 0000 193977 130648 68163 ChiSeawater I 557554 059582 936 0000 440775 674332 66976 ChiSeaoi1 I 209761 025517 822 0000 159749 259773 67167 Prchoice Chiseawaterll selected 27343699 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 11472 110758 1036 0000 930116 136428 68112 Skistoi1 I 557554 059582 936 0000 440776 674332 68163 ChiSeawater I 198593 181947 1091 0000 234254 162932 66976 ChiSeaoi1 I 281181 036704 766 0000 209243 353119 67167 Prchoice ChiSeaoil1 selected 10287145 variable I dpdx Std Err z PgtIzI 95 C I X netprice I Skistwater I 431595 048578 888 0000 336383 526806 68112 Skistoi1 I 209761 025517 822 0000 159749 259773 68163 ChiSeawater I 281181 036704 766 0000 209243 353119 66976 ChiSeaoi1 I 922537 10131 911 0000 11211 723972 67167 APPENDIX A Exercise Solutions 576 Appendix A Exercise Solutions Principles of Econometrics 46 577 EXERCISE A1 a b The slope is the change in the quantity supplied per unit change in market price The slope here is 15 which represents a 15 unit increase in the quantity supplied of a good due to a one unit increase in market price Recall that elasticity 0 gs slope x S When P 10 Q 315gtlt1012 Thus elasticity 15 gtlt 125 The elasticity shows the percentage change in Q associated with a 1 percent change in P At the point P 10 and Q 12 a 1 percent change in P is associated with a 125 percent change in Q5 When P 50 Q 315gtlt5072 Thus elasticity 15 x g 1042 At the point P 50 and Q 72 a 1 percent change in P is associated with a 104 percent change in Q5 Appendix A Exercise Solutions Principles of Econometrics 46 578 EXERCISE A2 a A sketch of the curve INF 26UNEMP for Values of UNEMP between 1 and 10 appears below INF UNEMP Figure xra2a Curve relating in ation to unemployment b The impact of a change in the unemployment rate on in ation is given by the slope of the function dINF 6 dUNEMP UNEMP2 The absolute Value of this function is largest as UNEMP approaches zero and it is smallest as UNEMP approaches infinity Thus the impact is greatest as the rate of unemployment approaches zero and it is smallest as unemployment approaches in nity This property is confirmed by examining Figure xra2a c The marginal effect of the unemployment rate on in ation when UNEMP 5 is given by dINF 6 O24 dUNEMP UNEMP2 52 Appendix A Exercise Solutions Principles of Econometrics 4e EXERCISE A3 a x12x16 xi216 X23 1 G xy3xw8xy3w8xmp4nmyx5p4 M X 1 X X C X43 12 x4 123 12 x 2y 32 xzy 32 579 Appendix A Exercise Solutions Principles of Econometrics 46 580 EXERCISE A4 a The Velocity of light is 186 000 186 gtlt105 miles per second b The number of seconds in a year is 60x 60x 24gtlt365 31536 000 31536gtlt107 seconds c The distance light travels in a year is 186000gtlt 31536000 186gtlt105gtlt 31536gtlt107 186gtlt31536gtlt105 x107 5865696 X1012 miles per year Appendix A Exercise Solutions Principles of Econometrics 46 581 EXERCISE A5 a The graph of the relationship between average wheat production WHEAT and time t is shown below For example when t 49 WHEAT 05 020lnt 12784 13 12 11 10 I 09 08 quotI 07 06 I 05 04 I I I I I I I I I I II 0 4 81216 2024 2832 3640444852 Figure xra9a Graph of WHEAT 05 020lnt The slope and elasticity for t 49 are Slope QIWHEH 03920 00041 when t 49 d1 I Elasticity OmHEA73 0 41x 4984 01564 when 1 49 00 air WHEAT b The graph of the relationship between average wheat production WHEAT and time t is shown below For example when t 49 WHEAT 08 00004t2 17604 05 1 r r 1 r 1 r 1 1 r 1 r 0 I 8 I2 15 20 24 33 32 35 I0 44 48 52 Figure xra9b Graph of WHEAT 08 0000412 The slope and elasticity for t 49 are dWHEAT Slope 00004 x 2t 00392 when t 49 dWHEAE t OO392X 49 10911 whent49 dz WHEAT 17604 Elasticity Appendix A Exercise Solutions Principles of Econometrics 46 582 EXERCISE A6 a The equation lny 08 O4lnx can be rewritten as y e0398x0394 A graph of this function follows with 2 labeled as Y1 12 10 08 06 Y1 Q4 02 60 00 62 64 66 68 139o 1392 1394 16 The graph of y 15 O2lnx is shown below with 2 labeled as Y2 12 10 68 66 Y2 64 62 60 00 62 64 66 68 i0 i2 l4 16 X The equation lny 175 20x can be rewritten as y eXp 175 20x A graph of this function follows with 2 labeled as Y3 36 32 28 24 20 Y3 16 12 O8 O4 O0 Appendix A Exercise Solutions Principles of Econometrics 46 583 Exercise A6 continued b C 08 04 X For equation 1 y e the slope is given by 04e0398x390396 3544 When x 010 For equation 2 y 15 02lnx the slope is given by 2222 when x010 dx x 17520x For equation 3 y e the slope is given by 20e 1397520quot 256805 When x 010 The slope is the change in arsenic concentration in toenails associated with a oneunit change in the arsenic concentration in the drinking Water For equation 1 when x 010 y 0886 and the elasticity is Q1 3544gtlt 0391 dx y 0886 04 For equation 2 when x 010 y 103948 and the elasticity is 2X 01 103948 01924 dxy For equation 3 when x 01 y 12840 and the elasticity is Q1 256805 gtlt 0391 20 1284 dxy The elasticity is the percentage change in arsenic concentration in toenails associated with a onepercent change in the arsenic concentration in the drinking Water Appendix A Exercise Solutions Principles of Econometrics 4e EXERCISE A7 a x 4573239 4573239gtlt 106 y 5975711 5975711gtlt106 b xy 4573239gtlt106x5975711gtlt106 4573239gtlt 5975711gtlt 106 X104 27328354597929 X1010 27328354597929 X1011 c x y 4573239gtlt1065975711x106 4573239 5975711gtlt 106 106 O7653O458gtlt1O2 76530458 d x y 4573239gtlt106005975711gtlt106 463299611x106 463299611 584 Appendix A Exercise Solutions Principles of Econometrics 4e EXERCISE A8 a b C d The curve is displayed in Figure Xra8 60 50 40 201 1914 tangent at x 2 199 886 Figure xra8 Graph of quadratic function and tangent at x 2 The derivative is d y26x14 When x2 dx The tangent is sketched in Figure Xr a8 The values located on the sketch are y1 f199 32gtlt1993gtlt1992 188603 y2 f201 32gtlt2O13gtlt2O12 191403 The numerical derivative is f201 f199 191403 188603 002 002 14 585 The analytic and numerical derivatives are equal The values should be close because the tangent and the curve are virtually identical for values of x close to 2 APPENDIX B Exercise Solutions 586 Appendix B Exercise Solutions Principles of Econometrics 46 587 EXERCISE B1 a E EEX1X2 Xn EX1EX2EXn uuuu 1 b VarX VarX1X2Xnj VarX1VarX2VarXn l 2 2 2m 72 72 Since X1X2Xn are independent random Variables their covariances are zero This result was used in the second line of the equation which would contain terms like coVXlXJ if these terms were not zero Appendix B Exercise Solutions Principles of Econometrics 46 588 EXERCISE B2 3 3 1 ltagt EltYgtE z Elt z 3HH i1 i1 1 3 1 b VarYVar Kj6VarY1Y2Y3 i1 1 VarY1VarY2 VarY32e0VKI 2c0VKI 2COVY2Y3 3G2 3 X 1 cs cs 3 262 3 Appendix B Exercise Solutions Principles of Econometrics 46 589 EXERCISE B3 3 b C d 6 The probability density function is shown below f x 1391 f X Total area of the triangle is half the base multiplied by the height ie the area is O5x2xll Whenx1 fxf1 Using geometry PX 21 is given by the area to the right of 1 which is PX21 x1x Using integration PX 21j12 x1dx x2 x 12 1 1 13 Using geometry Using integration 1 12 1 1 2 PXS J0 xldx Zx x For a continuous random Variable the probability of observing a single point is zero Thus PX 1 0 Appendix B Exercise Solutions Principles of Econometrics 46 590 Exercise B3 continued f The mean is given by EX Jxfxdx J 02lt x2 xdx x3 333 lt The second moment is given by 4 1 EX2 Jx2fxdx J 02 x3 x2dx x 3x3 The variance is given by 32 3 3 T 9 g The cumulative distribution function is given by uat Va1 XEX2EX2 Fx ftdt j z1dz X x 1j 0 4 Appendix B Exercise Solutions Principles of Econometrics 46 591 EXERCISE B4 When X is a uniform random variable on ab its probability density function is 1 b a asxsb fx a The mean of X is given by b b 1 2 EX Lxfxdx Lbiajdx baquot7j 1 b2 a2 ba b a 2 2 The second moment of X is given by EX2 Jx2fxdx ajdx 1 1 b3 a3 b2a2ab b a 3 3 The variance of X is Va1 X EX2EX2 b2a2abbaj2 3 2 4192 4a2 4ab 3192 3612 6ab 12 12 ltba2 12 b The cumulative distribution function is Fx ajdtba 1 b Cl Appendix B Exercise Solutions Principles of Econometrics 416 592 EXERCISE B5 After setting up a workfile for 41 observations the following EViews program can be used to generate the random numbers series x x179 scalar m100 scalar a263 scalar cee71 for i 2 to 41 scalar qaxi1cee xiqmceiingqmm next series uxm If the random number generator has worked well the observations in U should be independent draws of a uniform random variable on the 01 interval A histogram of these numbers follows 01 02 03 04 05 06 07 08 09 1 0 These numbers are far from random There are no observations in the intervals O10015 O20025 O30035 p Moreover the frequency of observations in the intervals 005010 O25030 O45050 0 is much less than it is in the intervals O15020 O35040 O55060 The random number generator is clearly not a good one Appendix B Exercise Solutions Principles of Econometrics 4e EXERCISE B6 If X N uo2 its probability density function is given by fltx 6 exp2iG2x L02 If Y aX b then X Y I9a and the probability density function for Y is Q dy 1 1 y b 2 cs 27 eXp2o2 a H eXp few ua b2 gy f y 19ct acs 27 593 This probability density function is that of a normal random Variable with mean uab and Variance azcsz Appendix B Exercise Solutions Principles of Econometrics 46 594 EXERCISE B7 Let E XDY be an expectation taken with respect to the joint density for X Y E X and EY are expectations taken with respect to the marginal distributions of X and Y and EM is an expectation taken with respect to the conditional distribution of Y given X Now coxY gX 0 if EM Y X gX Ema gtlt EM gX Using iterated expectations We can Write EXY Y x goo EX EM Y x gltXgt EX gXEYX Y EX gX gtlt EY Y E gX x EXY Y Appendix B Exercise Solutions Principles of Econometrics 46 595 EXERCISE B8 Using un1f0rm1dat a The histogram obtained by combining Z1 and Z2 into one series of 2000 observations and the summary statistics from that series are displayed below The histogram is bellshaped as one would expect from a normal distribution 240 Series Z Sample 1 2000 200 39 Observations 2000 160 Mean 0026558 Median 0042335 120 i Maximum 3546848 Minimum 3643559 Std Dev 0989170 80 Skewness 0008201 Kurtosis 3056003 40 JarqueBera 0283781 Probability 0867716 0 I I I I I I I I I I I I I I I I I I I I I I I 3 2 1 0 1 2 3 Figure xrb8a Histogram for combined observations Z1 and Z2 b The sample mean and variance are close to zero and one respectively and the pvalue from the JarqueBera test for normality is 0868 There is no evidence to suggest the observations are not normally distributed c The scatter diagram in Figure xrb8c does not suggest any correlation between Z1 and Z2 It is a random scatter Z2 O I Z1 Figure xrb8c Scatter diagram for Z and Z2 Appendix B Exercise Solutions Principles of Econometrics 46 596 EXERCISE B8 Using umf0rm2dat a The histogram obtained by combining Z1 and Z2 into one series of 20000 observations and the summary statistics from that series are displayed below The histogram is bell shaped as one would expect from a normal distribution 2400 Series Z Sample 1 20000 1000 39 Observations 20000 1 600 Mean 0000920 Median 0000987 Maximum 3804187 1200 Minimum 4743557 Std Dev 1001389 Skewness 001 3018 800 Kurtosis 3014171 400 JarqueBera 0732266 Probability 0693411 0 I I I l I I I l I I I l I I I l I I I l I I I l I I I l 4 3 2 1 O 1 2 3 4 Figure xrb8a Histogram for combined observations Z1 and Z2 b The sample mean and variance are very close to zero and one respectively and the p value from the J arqueBera test for normality is 0693 There is no evidence to suggest the observations are not normally distributed c The scatter diagram in Figure xrb8c does not suggest any correlation between Z1 and Z2 It is a random scatter Z2 Z1 Figure xrb8c Scatter diagram for Z and Z2 Appendix B Exercise Solutions Principles of Econometrics 4e EXERCISE B9 The cumulative distribution function forX is given by 81 b 3quot 3 Fx dt J 23 13 7 x 597 Appendix B Exercise Solutions Principles of Econometrics 46 598 EXERCISE B10 3 b C 08 06 FX 04 02 00 Figure xrb10a Exponential density function 08 06 FBIGX 04 02 00 Figure xrb10b Exponential distribution function To use the inverse transformation method We use the distribution function to Write U 1 exp X from which We obtain X ln1 U The histograms from 1000 and 10000 observations generated using X lnl U are given below They resemble the density in part a particularly the one from 10000 observations 240 200 120 80 II III III III III I 1 2 3 4 5 Figure xrb10c Histogram for 1000 observations Appendix B Exercise Solutions Principles of Econometrics 46 599 Exercise B10c continued 2400 2000 1000 1200 800 400 0 lllllIlllllllllllllllllllllllllllll 0 1 2 3 4 5 3 7 3 9 Figure xrb10c Histogram for 10000 observations d The sample means and Variances from the two samples are For 1000 observations X 10272 S2 10025 For 10000 observations X 09984 S2 09945 All four of these sample quantities are Very close to 1 Appendix B Exercise Solutions Principles of Econometrics 46 600 EXERCISE B11 After setting up a Workfile for 41 observations the following EVieWs program can be used to generate the random numbers U1 series x x11234567 scalar m2quot32 scalar a1103515245 scalar cee12345 for i 2 to 1001 scalar qaxi1cee xiqmceiingqmm next series u1xm If the random number generator has Worked well the observations on U should be independent draws of a uniform random variable on the 01 interval Histograms of these numbers and those from U2 obtained using the seed value x195992 follow 40 Series U1 35 Sample 1 1001 Observations 1001 30 Mean 0499503 25 Median 0493013 Maximum 0999800 2 Minimum 0000287 Std Dev 0294802 Skewness 0000379 Kurtosis 1761368 5 JarqueBera 6398931 Probability 0000000 0 00 01 02 03 04 05 06 07 08 09 10 Figure xrb11a Histogram and summary statistics for U 40 Series U2 Sample 1 1001 Observations 1001 35 30 Mean 0508945 Median 0508509 Maximum 0999776 Minimum 223e05 Std Dev 0280072 Skewness 0042108 Ku rtosis 1882576 JarqueBera 5237437 Probability 0000000 Figure xrb11b Histogram and summary statistics for U2 The histograms are approximately uniformly distributed implying the random number generator is a good one The sample means standard deviations and correlation are U X 04995 S 02948 corU1U2 00471 U2 X 05089 S 02801 These sample quantities are very close to the population values it 05 6 02887 and p 0 Appendix B Exercise Solutions Principles of Econometrics 46 601 EXERCISE B12 3 b C d 6 D For fx y to be a valid paf We require fx 2 2 0 and m 01 fx yclxdy 1 It is clear that f x y 6x2 y 2 O for all 0 S x S 1 O S y S1 To establish the second condition We consider J2 1 1 2 0 J J 016x2y dxdy JibJ 016x2dxdy J01y2x3Mdy 2 Jolydy 2 x The marginal pafforX is given by 2 fx J 016x2dy 6x2 J 1 J23 2 0 The mean of X is EX 01 xfxdx j013x3dx T The second moment of X is EX2 01 x2 fxdx j013x4dx The variance of X is 2 varltXgt EltX2gt E0012 2 H The marginal paf for Y is given by X3 1 2 30 y fXy 2 fXy y 2y 336 fy jj xzydx yl The conditional pdf f x y is Since f x y f x the conditional mean and variance of X given Y are identical to the mean and variance of X found in part b Yes Xand Yare independent because fx y 6x2y fxfy 3x2 gtlt 2y Appendix B Exercise Solutions Principles of Econometrics 46 602 EXERCISE B13 3 b C d 6 f The Volume under the joint pdf is 1 161 0O j 1 ydy 21dyy 0 4 0 2 0 The marginal paf for X is 2 2 1 Z fltx jx5dy 2x 1 2 The marginal paf for Y is J rltygt1111czx 0 1 w W 1 1 7 P1X gt ltgtczx 2 I 2 X 4 0 216 w The cdf for Y is t2 V J2 Jl 2 4 0 4 The conditional pdf f x y is given by fltx1ygtfquot 1 2i implying fxY3 3 f y y2 y 2 3 The required probability is 12 PltXlt1Ysgt11gtdx11 Xand Yare not independent because PX lt Y 72 PX lt The mean of Y is 2 2 y2 y i EltYgt I0yfydy j02dy 6 0 3 The second moment of Y is EY2 22 d 2y3dJ422 lt I0yfyy 07jy Appendix B Exercise Solutions Principles of Econometrics 46 603 Exercise B13f continued The Variance of Y is 2 VarY EY2 EY2 2 g From part e V X2 3 J EX Y joyxfoc ydx 2 2 1200 EY EltX 1 Io jj jfltygtdy J ay y3 quotE03 We can check this result by using the marginal paf for X to nd EX EX2 dx2 x2 dx X2 X3 22 42 lt I0xfx 10 x 7 3 0 3 APPENDIX C Exercise Solutions 604 Appendix C Exercise Solutions Principles of Econometrics 4e 605 EXERCISE C1 3 b C d A linear estimator is one that can be Written in the form Zailf Where a is a constant Rearranging Y yields 2 Ygt1ltY1Y2 11 2 2 2 12 Thus Y is a linear estimator Where a 12 fori12 and al 0 for i 34N The expected Value of an unbiased estimator is equal to the true population mean Y Y 1 1 1 1 EYE1J 2j EY1 EY2 H uu The Variance of Y is given by VarY VarY1 Y2 J VarlY1 1172 2 The sample mean is a better estimator because it uses more information The Variance of the sample mean is 62 N which is smaller than 62 2 when N gt2 thus making it a better estimator than Y In general increasing sample information reduces sampling Variation Appendix C Exercise Solutions Principles of Econometrics 4e 606 EXERCISE C2 1 1 1 3 a Y 5 Y2 gY3 ZalYl Where al are constants for z 12 and 3 il 1 l l l l l l l l b EYE Y1 Y2 gY3j EY1 EY2gEY3Et1 t1gt1t1 1 l l c VarYVar Y1 Y2 gY3 l l l ZVarlj6Varl Varl S1I1C6COVY1Y2COVY2Y30 2 lG2 152 ics2 76 4 9 36 18 The Variance of the sample mean is G2 G2 662 VarYW 18 which is smaller than the Variance of 7 d Since Varl7 gt Varl7 7 is not as good an estimator as 7 e If 62 9 then Varl7 G2N 93 3 and Varl7 76218 7gtlt918 35 The probability that the estimator 7 is Within one unit on either side of Ll is P 137 1 P 1 s g 1 H H J VarY VarY VarY l l ifgzsfl P O5345 3 Z 3 05345 0407 Appendix C Exercise Solutions Principles of Econometrics 46 607 EXERCISE C3 Let X be the random Variable denoting the hourly sales of fried chicken which is normally distributed X N2OOO5OO2 The probability that in a 9 hour day more than 20000 pieces will be sold is the same as the probability that average hourly sales of fried chicken is greater than 200009 m 2222 pieces Pgt2222jPX 222239 cW gt GM Z 2222 2000 gt 500 J5 P Zgt 500 666 PZ gt 1332 0091 Appendix C Exercise Solutions Principles of Econometrics 46 608 EXERCISE C4 Let the random Variable X denote the starting salary for Economics majors Assume it is normally distributed X N47OOO8O0O2 PPgt50000P X 5000039 new gt W 50000 47000 13 bl 8000xE i PZ gt 237 1 09911 00089 Appendix C Exercise Solutions Principles of Econometrics 46 609 EXERCISE C5 a We set up the hypotheses H0 11 3170 versus H1 u gt170 The alternative is H1 u gt170 because We Want to establish Whether the mean monthly account balance is more than 170 The test statistic given H0 is true is 170 1 39 R399 The rejection region is 121649 The value of the test statistic is 178170 msW 2462 Since t 2462 gt 1649 We reject H0 and conclude that the new accounting system is cost effective b p Pz 2 2462 1 Pz lt 2462 0007 Appendix C Exercise Solutions Principles of Econometrics 46 610 EXERCISE C6 a To decide Whether the students are studying on average at least 6 hours per week we set up the hypotheses H 0 u 6 versus H1 u gt 6 The test statistic given H0 is true is 8 XlZxl1344668l255 8211 8 2 18 2 cs varX1x X 114286 zj5 5396 0598 1142868 At the 005 level of signi cance the rejection region is tgt 1895 Since I 0598 lt 1895 We do not reject H0 and therefore cannot conclude that at the 005 level of signi cance the students are studying more than 6 hours per Week b A 90 con dence interval for the population mean number of hours studied per Week is A2 fire W 55189511386 3235 7765 Appendix C Exercise Solutions Principles of Econometrics 46 611 EXERCISE C7 3 b C To test Whether current hiring procedures are effective We test the hypothesis that H0 11 S 450 against H1 11 gt 450 The manager is interested in Workers who can process at least 450 pieces per day The test statistic when H0 is true is X450 The value of the test statistic is t 460450 3850 1861 Using a 5 significance level at 49 degrees of freedom the rejection region is I gt 1677 Since 1861 gt 1677 We reject H0 and conclude that the current hiring procedures are effective A type I error occurs when We reject the null hypothesis but it is actually true In this example a type I error occurs when We wrongly reject the hypothesis that the hiring procedures are effective This would be a costly error to make because We would be dismissing a cost effective practice pvalue 1 Pt4 lt1861 1 09656 00344 Appendix C Exercise Solutions Principles of Econometrics 46 612 EXERCISE C8 The interval estimate of a normally distributed random variable is given by I7 i 26 x GW where 26 is the corresponding critical value at a 95 level of con dence The length of the interval is therefore 2 x 26 x 5W To ensure that the length of the interval is less than 4 derive N as follows 2x 4 3 jlt4 l JN Z66 lt 2xN 4 flt4N 96x2D2lt4N 423525 lt N A sample size of 424 employees is needed 613 13905 46 es of Econometr inczp 3 cuAmmAAAAAAAAmmAAAAAAAAmmAmmAAAAAAAAmmAAAAAAAAmmAouououououououououououm O66Ob66666066Ob66666066Ob66666066Ob66666066Ob66666066Ob66666066Ob66666066Ob66666066Ob66666066ObObobobobobobobobobobobo coocooocooocoocooocoocooocooocoocooooooooooooooooooooooooooo vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 3iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiid 3vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv mvKvKvvK33333333HvKmuKKvKvvvvu 55555555 amp viiiviiiiiiiiiiiiiiiiiiiiiiiv viiiviiiiiiiiiiiiiiiiiiiiiiiv viiiviiiiiiiiiiiiiiiiiiiiiiiv oooooooooooooooooo ttttttttttttttt h u 4 0 x 4 3 0 x 3 2 0 x 2 1 0 x 1 iiiiiiiiiiiiii 1 Zyl 1 1 3E 23E 3 1 x3 x33 3 Vm2Vm3 1x11x1l 9 9 3 Appendix C Exercise Solutions Pr EXERCKHEC a sketch of the pdf is shown below lt gt2 Y E1 1 3 1 gtlt b C d 2 3 3 i Var j 1 VaI39 1Y2Y3 3 9 1 lgtlt1 9 Appendix C Exercise Solutions Principles of Econometrics 46 614 EXERCISE C11 8 b The sample size sample mean and standard deviation for the Fulton Fish Market data on various days are shown below Monday Tuesday Wednesday Thursday Friday N 21 23 21 23 23 Mean X 8070762 4847739 4367476 7283956 7083305 StdDev 6 5070127 3964039 2838622 3200351 3814711 i The null and alternative hypotheses are H0 u 210000 against H1u lt 10000 ii The test statistic when H0 is true is t 2 10000 Ni ASW N4 The value of the test statistic is 2 8070762 10000 2 47437 5070127E iii Using an oc 005 level of signi cance at 20 degrees of freedom the rejection region is t lt 1725 iv Since 17437 lt 1725 We reject the null hypothesis that the mean quantity sold is greater than or equal to 10000 V pvalue Pt0 lt 17437 00483 i The null and alternative hypotheses are H0 26363 1 against H 63oi gt1 ii The test statistic when H0 is true is F 6 6 N FN2 1N3 1 39 The value of the test statistic is 2 F 39640392 21950 2838622 iii Using an oc 005 level of significance at 22 20 degrees of freedom the rejection region is F gt 210 iv Since 1950 lt 210 We fail to reject the null hypothesis that the variances are equal V pvalue P170 gt1950 0069 Appendix C Exercise Solutions Principles of Econometrics 46 615 Exercise C11 continued 0 d i The hypotheses are H0 u2 u3 against H1 u2 72 u3 ii The test statistic when H0 is true is N 1 2 N 1 2 t 2 3 tNN2 Whe1eG 2 52 3 gt53 21 1 2 3 N2N3 2 op N2 N3 The variance estimate is 8 22gtlt 39640392 20x 28386222 2 1 2321 2 3473399 The value of the test statistic is t 4847739 4367476 34739 ii 23 21 0458 iii Using an oc 005 level of signi cance and degrees of freedom 42 the rejection regions are tgt 2018 and tlt 2018 iv Since 2018 lt 0458 lt 2018 We do not reject the null hypothesis that the means are equal V pvalue Pt4 gt 0458 Pt4 lt 0458 0649 The mean of Wis given by EWEX1X X3 X4 X5 EX1 EX EX3 EX4 EX5 ZH1 142 143 144 115 The variance of W is given by varW varX1X2 X3 X4 X5 varX1 varX2 varX3 varX4 varX5 zcsf 0 53 03 0 To derive the second line for varW We have used the result that covXX 0 for z 74 j because the X are independent I Appendix C Exercise Solutions Principles of Econometrics 46 616 Exercise C11 continued e The mean for ft is given by E 1E1345 EEEE4E H1H2H3H4H5 11 The variance of f1 is given by var var1 3 4 5 var1varvar3var4var5 lillillvillill7vil In deriving this variance We have used the result that cov 1 J 0 because sales on different days are assumed independent Since the X are distributed normally it follows that the ff are normally distributed and that t which is a linear function of the X is also distributed normally fL N uofV Where 2 i 5 2 5 G u Eu and SW 2 11 11N Hence a 95 interval estimator for u is f1 i Z005c5w Because ow is unknown We need to replace it with an estimate 8W Where The resulting 95 interval estimator f1 i Z 005c 5w is an approximate one in large samples For the Fulton Fish data We obtain 63 183552 and p1 31653 and an approximate 95 interval estimate is p1 i Z sw 31653 i 196 X 18355 28055 35251

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