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Answers to Selected Exercises For Principles of Econometrics, Fourth Edition Louisiana State University WILLIAM E. GRIFFITHS University of Melbourne GUAY C. LIM University of Melbourne JOHN WILEY & SONS, INC New York / Chichester / Weinheim / Brisbane / Singapore / Toronto CONTENTS Answers for Selected Exercises in: 1 ProbaPbriter Chapter 2 The Simple Linear Regression Model 3 Chapter 3 Interval Estimation and Hypothesis Testing 12 Chapter 4 Prediction, Goodness of Fit and Modeling Issues 16 Chapter 5 The Multiple Regression Model 22 Chapter 6 Further Inference in the Multiple Regression Model 29 Chapter 7 Using Indicator Variables 36 Ch8aHteteroskedasticity Chapter 9 Regression with Time Series Data: Stationary Variables 51 Chapter 10 Random Regressors and Moment Based Estimation 58 Chapter 11 Simultaneous Equations Models 60 Chapter 15 Panel Data Models 64 Chapter 16 Qualitative and Limited Dependent Variable Models 66 Appendix A Mathematical Tools 69 72AppBendbiaboiyepts Appendix C Review of Statistical Inference 76 29 August, 2011 PROBABILITY PRIMER Exercise Answers EXERCISE P.1 (a) X is a random variable because attendance is not known prior to the outdoor concert. (b) 1100 (c) 3500 (d) 6,000,000 EXERCISE P.3 0.0478 EXERCISE P.5 (a) 0.5. (b) 0.25 EXERCISE P.7 (a) f )c 0.15 0.40 0.45 (b) 1.3 (c) 0.51 (d)f f (f0,0).05 C B) (0) 0.15 0.15 0.0225 1 Probability Primer, Exercise Answers, Principles of Econometrics, 4e 2 (e) A f )a 5000 0.15 6000 0.50 7000 0.35 (f) 1.0 EXERCISE P.11 (a.)0289 (b.)3176 (c) 0.8658 (d.)444 (e) 1.319 EXERCISE P.13 (a.)1056 (b.)0062 (c) (a) 0.1587 (b) 0.1265 EXERCISE P.15 (a) 9 (b) 1.5 (c) 0 (d) 109 (e) −66 (f) −0.6055 EXERCISE P.17 (a) 4(bx x x ) 1 2 3 4 (b) 14 (c) 34 (d) f(4f)(5) (6) (e) f) ,)(f,)fy (f) 36 CHAPTER 2 Exercise Answers EXERCISE 2.3 (a) 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) b 1.514286 2 b110.8 Figure xr2.3 Observations and fitted line 10 8 6 4 2 1 2 3 x 4 5 6 y Fitted values (c) y 5.5 x 3.5 y 5.5 3 Chapter 2, Exercise Answers Principles of Econometrics, 4e 4 Exercise 2.3 (Continued) (d) e i 0.714286 0.228571 −1.257143 0.257143 −1.228571 1.285714 ei 0. (e) xii 0 EXERCISE 2.6 (a) The intercept estimate b 240 is an estimate of the number of sodas sold when the 1 temperature is 0 degrees Fahrenheit. Clearly, it is impossible to sell 240 sodas and so this estimate should not be accepted as a sensible one. The slope estimate b28 is an estimate of the increase in sodas sold when temperature increases by 1 Fahrenheit degree. One would e xpect the number of sodas sold to increase as temperature increases. (b) y 240 8 8 0 400 (c) She predicts no sodas will be sold below 30F. (d) A graph of the estimated regression line: Figure xr2.6 Regression line 600 400 y 200 0 -200 0 20 40 x 60 80 100 Chapter 2, Exercise Answers Principles of Econometrics, 4e 5 EXERCISE 2.9 (a) Figure xr2.9a Occupancy Rates 100 90 80 70 60 50 40 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 month, 1=march 2003,.., 25=march 2005 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. (b) A plot of MOTEL_PCT against COMP_PCT yields: Figure xr2.9b Observations on occupancy 100 90 80 70 60 per50ntage motel occupancy 40 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 competitor(s) face the same demand for motel rooms. Chapter 2, Exercise Answers Principles of Econometrics, 4e 6 Exercise 2.9 (continued) (c) MOTEL_ PCT 2 1.0 0.8646 COMP_PCT . The competitors’ occupancy rates are positivey 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 0.8646 percentage points. (d) 30 Repair period 20 10 0 residuals -10 -20 -30 0 4 8 12 16 20 24 28 month, 1=march 2003,.., 25=march 2005 Figure xr2.9(d) Plot of residuals against time The residuals during the occupancy period arthose between the two vertical lines. All except one are negative, indicating that the model has over-predicted the motel’s occupancy rate during the repair period. (e) We would expect the slope coefficient of a linear regression of MOTEL_PCT on RELPRICE to be negative, as the higher the relativ e price of the damaged motel’s rooms, the lower the demand will be for those rooms, holding other factors constant. MOTEL_ PCT 1 6.66 122.12 RELPRICE (f) The estimated regression is: MOTEL_ PCT 7 9.3500 13.2357 REPAIR In the non-repair period, the damaged motel had an estimated occupancy rate of 79.35%. During the repair period, the estimated occupancy rate was 79.35 −13.24 = 66.11%. Thus, it appears the motel did suffer a loss of occupancy and profits during the repair period. (g) From the earlier regression, we have MOTEL 0 b1 79.35% MOTEL 1 b b 79.35 13.24 66.11% 1 2 Chapter 2, Exercise Answers Principles of Econometrics, 4e 7 Exercise 2.9(g) (continued) For competitors, the estimated regression is: COMP_PCT 6 .4889 0.8825 REPAIR COMP 0 b 62.49% 1 COMP 1 b1 2b 62.9 0.88 63.37% During the non-repair period, the difference between the average occupancies was: MOTEL 0 0OMP 79.35 62.49 16.86% During the repair period it was MOTEL 1 1OMP 66.11 63.37 2.74% This comparison supports the motel’s claim for lost profits during the repair period. When there were no repairs, their occupancy rate was 16.86% higher than that of their competitors; during the repairs it was only 2.74% higher. (h) MOTEL_ PCT COMP_ PCT 16.8611 14.1183 REPAIR The intercept estimate in this equation (1686) is equal to the difference in average occupancies during the non-repair period, MOTEL C0M0 . The sum of the two coefficient estimates .86( 14.12) 2.74 is equal to the difference in average occupancies during the repair period,TEL 1O1P . This relationship exists because averaging the difference between two series is the same as taking the difference between the averages of the two series. EXERCISE 2.12 (a) and (b) SPRICE 30069 9181.7LIVAREA The coefficient 9181.7 suggests that selli ng price increases by approximately $9182 for each additional 100 square foot in living arThe intercept, if taken literally, suggests a house with zero square feet would cost$30,069, a meaningless value. Figure xr2.12b Observations and fitted line 800000 600000 400000 200000 0 10 20 30 40 50 living area, hundreds of square feet selling price of homFitted values Chapter 2, Exercise Answers Principles of Econometrics, 4e 8 Exercise 2.12 (continued) (c) The estimated quadratic equation for all houses in the sample is 2 SPRICE 5 7728 212.611LIVAREA The marginal effect of an additional 100 squa re feet for a home with 1500 square feet of living space is: dCI slope dLIVAREA 22.61 LI REA=2 212.611 15 6378.33 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. (d) Figure xr2.12d Linear and quadratic fitted lines 800000 600000 400000 200000 0 10 living area, hundreds of square feet selling price of homeFitted values Fitted values The quadratic model appears to fit the data better; it is better at capturing the proportionally higher prices for large houses. 2 2 1 SSEof linear model, (b): SSE ei 2.23 10 SSE of quadratic model, (c): SSE e2 12.03 10 i The SSE of the quadratic model is smaller, indicating that it is a better fit. 2 (e) Larots: SPRICE 1 13279 193.83LIVAREA Smta:ll SPRICE 62172 186.86LIVAREA 2 The intercept can be interpreted as the expect ed 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 isAREA . 2 Thus, we estimate that the mean price of large lots is $113,279 and the mean price of small lots is $62,172. The marginal effect of living area on price is $387.LIVAREA for houses on large lots and $373.72 LIVAREA for houses on small lots. Chapter 2, Exercise Answers Principles of Econometrics, 4e 9 Exercise 2.12 (continued) (f) The following figure contai ns the scatter diagram of PRICE and AGE as well as the estimated equation SPRICE 1 37404 627.16AGE . 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 (i.e., a new house) would cost $137,404. Figure xr2.12f sprice vs age regression line 800000 600000 400000 200000 0 0 20 40 60 80 100 age of home at time of sale, years selling price of home,Fitted values The following figure contains the scatter diagram of ln(PRICE) and AGE as well as the estimated equation ln RICE 1 1.746 0.00476AGE . In this estimated model, each extra year of age reduces the selling price by 0.48%. To find an interpretation from the intercept, we setAGE 0 , and find an estimate of the price of a new home as exp ln ICE exp(11.74597) $126,244 Figure xr2.12f log(sprice) vs age regression line 14 13 12 11 10 0 20 age of home at time of sale, years lsprice Fitted values Based on the plots and visual fit of the estimated regression lines, the log-linear model shows much less of problem with under-prediction and so it is preferred. (g) The estimated equation for all houses is SPRICE 1 15220 133797LGELOT . The estimated expected selling price for a house on a large lot ( LGELOT = 1) is 115220+133797 = $249017. The estimated exp ected selling price for a house not on a large lot (LGELOT = 0) is $115220. Chapter 2, Exercise Answers Principles of Econometrics, 4e 10 EXERCISE 2.14 (a) and (b) xr2-14 Vote versus Growth with fitted regression 60 50 40 Incumbent vote 30 -15 -10 Growth rate before election Incumbent share of the two-parFitted valuesal vote There appears to be a positive association between VOTE and GROWTH. The estimated equation for 1916 to 2008 is VOTE 50.848 0.88595GROWTH The coefficient 0.88595 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 0.88595 percentage points. We estimate, based on the fitted regression intercept, that that the incumbent party’s expected vote is 50.848% 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. (c) The estimated equation for 1916 - 2004 is VOTE 51.053 0.877982GROWTH The actual 2008 value for growth is 0.220. Putting this into the estimated equation, we obtain the predicted vote share for the incumbent party: VOTE 1.053 0.877982G ROWTH 51.053 0.877982 0.220 51.246 2008 2008 This suggests that the incumbent party will ma intain the majority vote in 2008. However, the actual vote share for the incumbent party for 2008 was 46.60, which is a long way short of the prediction; the incumbent party did not maintain the majority vote. Chapter 2, Exercise Answers Principles of Econometrics, 4e 11 Exercise 2.14 (continued) (d) xr2-14 Vote versus Inflation 60 50 Incumbent vote 30 0 2 Inflation rate before election Incumbent share of the two-party prFitted valueste There appears to be a negative association between the two variables. The estimated equation is: VOTE= 53.408 0.444312INFLATION We estimate that a 1 percentage point incr ease in inflation during the incumbent party’s first 15 quarters reduces the share of incumbent party’s vote by 0.444 percentage points. The estimated intercept suggests that when in flation is at 0% for that party’s first 15 quarters, the expected share of votes won by the incumbent party is 53.4%; the incumbent party is predicted to maintain the majority vote when inflation, during its first 15 quarters, is at 0%. CHAPTER 3 Exercise Answers EXERCISE 3.3 (a) Reject H 0because tt.78 c 2.819. (b) Reject H because tt.78 2.508. 0 c (c) Do not reject0 becausett .78 c 1.717. Figure xr3.3 One tail rejection region (d) Reject H 0because tt2.32 c 2.074. (e) A 99% interval estimate of the slope is given by (0.079, 0.541) 12 Chapter 3, Exercise Answers, Principles of Econometrics, 4e 13 EXERCISE 3.6 (a) We reject the null hypothesis because the test statistic value t = 4.265 > c = 2.500. The p- value is 0.000145 Figure xr3.6(a) Rejection region and p-value (b) We do not reject the null hypothesis because the test statistic value tt2.093 c 2.500 . The p-value is 0.0238 Figure xr3.6(b) Rejection region and p-value (c) Since t t2.221 c 1.714 , we rejectH 0 at a 5% significance level. (d) A 95% interval estimate for is given by (25.57,0.91). 2 (ei)nce t t3.542 c 2.500 , we reject H 0 at a 5% significance level. (f) A 95% interval estimate for 2 is given by (22.36,5.87). Chapter 3, Exercise Answers, Principles of Econometrics, 4e 14 EXERCISE 3.9 (a) We set up the hypotheses H : 0 2 0 versus H :1 2 0 . Since t = 4.870 > 1.717, we reject the null hypothesis. (b) A 95% interval estimate for 2 from the regression in part (a) is (0.509, 1.263). (c) We set up the hypotheses H : 0 versus H : 0 . Since t 0.741 1.717 , we 0 2 1 2 do not reject the null hypothesis. (d) A 95% interval estimate for from the regression in part (c) is ( 1.688, 0.800). 2 (e) Wtest H00 15 against the alternative H0: 1 1 . Since t 1.515 1.717 , we do not reject the null hypothesis. (f) The 95% interval estimate is 49.40,55.64 . EXERCISE 3.13 (a) 4.5 4.0 3.5 3.0 2.5 LNWAGE 2.0 1.5 1.0 0.5 -30 -20 -10 0 10 20 30 40 EXPER30 Figure xr3.13(a) Scatter plot of ln(WAGE) against EXPER30 2 (b) The estimated log-polynomial model is ln WAGE 2 .9826 0.0007088EXPER30 . Wteest H 0 2 against the alternative H 1 2 . Because t 8.067 1.646 , we reject H : 0 . 0 2 Chapter 3, Exercise Answers, Principles of Econometrics, 4e 15 Exercise 3.13 (continued) (c) me d 0.4215 10 d ER EXPER10 d me 30 0.0 d PER EXPER30 d me 50 0.4215 d PER EXPER (d) 80 70 60 50 WAGE fitted WAGE 30 20 10 0 -30 -20-10 0 10 20 30 40 EXPER30 Figure xr3.13(d) Plot of fitted and actual values of WAGE CHAPTER 4 Exercise Answers EXERCISE 4.1 (a) R 0.71051 (b) R 0.8455 (c) 6.4104 EXERCISE 4.2 y 83 17.38 x (a) (1.23) (2.34) where x 20 (b) y x 0.1166 0.01738 where y y (0.0246) (0.00234) 50 y x 2915 0.869 ˆ (c) where y x y and x (0.0615) (0.117) 20 20 EXERCISE 4.9 (a) Equatio:n y0 069538 0.015025 48 1.417 y0tf(0.975,45)) 1.4166 2.0141 0.25293 (0.907,1.926) Equa:tion y0 0.5231 0.16961 ln(48) 1.219 y0f(0.975,45)) 1.2189 2.0141 0.28787 (0.639,1.799) 2 Equ3a:tion y0 0.79945 0.000337543 (48) 1.577 ˆ y0(0.975,45)) 1.577145 2.0141 0.234544 (1.105, 2.050) The actual yield in Chapman was 1.844. 16 Chapter 4, Exercise Answers, Principles of Econometrics, 4e17 Exercise 4.9 (continued) (b) Equao:n dyt 0.0150 dt dy Equa:tion t 0.0035 dt dy Equa:tion t 0.0324 dt dytt (c) Equation 1: dt y 0.509 t dytt Equa:tion 0.139 dt yt dytt Equa:tion 0.986 dt yt (d) hlopes dy dt and the elasticidy d t give the marginal change in yield and the percentage change in yield, respectiv ely, 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. EXERCISE 4.11 (a) The estimated regression model for the years 1916 to 2008 is: 2 VOTE 0.8484 0.8859GROWTH R 0.5189 (se 125 0.1819 VOTE 51.043 VOTE VOTE 4.443 2008 2008 2008 (b) The estimated regression model for the years 1916 to 2004 is: VOTE 51.0533 0.8780GROWTH R2 0.5243 (se) (1.0379) (0.1825) VOTE 2008 51.246 EfTV 2008 2008 4.646 This prediction error is larger in mthan 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. Chapter 4, Exercise Answers, Principles of Econometrics, 4e 18 Exercise 4.11 (continued) VOTE t se ( f ) 51.2464 2.0796 4.9185 (41.018,61.475) (c) 2008 (0.975,21) The actual 2008 outcome VOTE 2008 46.6 falls within this prediction interval. (d) GROWTH 1.086 EXERCISE 4.13 (a) The regression results are: ln(PRICE) 1 0.5938 0.000596SQFT e 0.029 0.00003 84.84 .30 The coefficient 0.000596 suggests an increase of one square foot is associated with a 0.06% increase in the price of the house. dPRICE 67.23 dSQFT elasticity = SQFT 0.00059596 1611.9 682 0.9607 2 (b) The regression results are: ln(PRICE) 4.1707 1.0066ln(SQFT) se 0.1655 0.0225 5.20 44.65 The coefficient 1.0066 says that an increase in living area of 1% is associated with a 1% increase in house price. The coefficient 1.0066 is the elasticity. dPRICE 70.444 dSQFT 2 (c) From the linear function, 0.672 . 2 From the log-linear function in part (ag, R 0.715. From the log-log function in part (b)g R 0.673. Chapter 4, Exercise Answers, Principles of Econometrics, 4e 19 Exercise 4.13 (continued) (d) 120 100 80 Jarque-Bera = 78.85 60 p -value = 0.0000 40 20 0 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 Figure xr4.13(d) Histogram of residuals for log-linear model 120 100 80 Jarque-Bera = 52.74 60 p -value = 0.0000 40 20 0 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 Figure xr4.13(d) Histogram of residuals for log-log model 200 160 120 Jarque-Bera = 2456 80 p -value = 0.0000 40 0 -100000 0 100000 200000 Figure xr4.13(d) Histogram of residuals for simple linear model All Jarque-Bera values are significantly diffe rent from 0 at the 1% level of significance. We can conclude that the residuals are not compatible with an assumption of normality, particularly in the simple linear model. Chapter 4, Exercise Answers, Principles of Econometrics, 4e 20 Exercise 4.13 (continued) (e) 1.2 1.2 0.8 0.8 0.4 0.4 re0.0ual re0.0ual -0.4 -0.4 -0.8 -0.8 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 SQFT SQFT Residuals of log-linear model Residuals of log-log model 250000 200000 150000 100000 50000 resid0ul -50000 -100000 -150000 0 1000 2000 3000 4000 5000 SQFT Residuals of simple linear model The residuals appear to increase in magnitude as SQFT 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. (f) Prediction for log-linear model: PRICE 203,516 Prediction for log-log model: PRICE 188,221 Prediction for simple linear model: PRICE 201,365 (g) The standard error of forecast for the log-linear model is se( f ) 0.20363. The 95% confidence interval is: (133,683; 297,316). The standard error of forecast for the log-log model is se( f ) 0.20876. The 95% confidence interval is (122,267; 277,454) . The standard error of forecast for the simple linear model is se( f ) 30348.26. The 95% confidence interval is 141,801; 260,928 . Chapter 4, Exercise Answers, Principles of Econometrics, 4e 21 Exercise 4.13 (continued) (h) The simple linear model is not a good choice because the residuals are heavily skewed to the right and hence far from being normally distri buted. It is difficu lt to choose between the other two models – the log-linear and l og-log models. Their re siduals have similar patterns and they both lead to a plausible elas ticity of price with respect to changes in square feet, namely, a 1% change in square f eet leads to a 1% change in price. The log- linear model is favored on the basis of its higher R g value, and its smaller standard deviation of the error, characteristics that suggest it is the model that best fits the data. CHAPTER 5 Exercise Answers EXERCISE 5.1 (a) y 1,x2 3 0, 0 * * * xi2 xi3 yi 0 1 0 1 2 1 2 1 2 2 0 2 1 1 1 2 1 2 0 1 1 1 1 0 1 0 1 (b) 13, x * 16, yx** 4, x 10 ii2 2 i 3 i i i (c) b2 0.8125 b3 0.4 b11 (d) e 0.4, 0.987, 0.025, 0.375, 1.4125, 0.025, 0.6, 0.4125, 0.175 (e) 0.6396 (f) r23 0 (g) se(b ) 0.1999 2 2 (h) SSE 3.8375 SST 16 SSR 12.1625 R 0.7602 22 Chapter 5, Exercise Answers, Principles of Econometrics, 4e 23 EXERCISE 5.2 (a) b2 (0.975,6) 2 (0.3233,1.3017) (b) We do not rejectH because t 0.9377 and 0.9377 2.447 = t . 0 (0.975, 6) EXERCISE 5.4 (a) The regression results are: 2 WTRANS 0.0315 0.0414ln TOTXP 0.001AGE 0.0130 NK R 0.0247 e (0.0322) (0.0071) (0.0004) (0.0055) (b) Tvealue b20.0414 suggests that as lnTOTEXP increases by 1 unit the budget proportion for transport increases by 0.0414. Alternatively, one can say that a 10% increase in total expenditure will incree the budget proportion for transportation by 0.004. (See Chapter 4.3.3.) The positive sign ofis according to our expectation because 2 as households become richer they tend to us e more luxurious forms of transport and the proportion of the budget for transport increases. vTlue b30.0001 implies that as the age of the head of the household increases by 1 year the budget share for transport decreases by 0.0001. The expected sign for3is not clear. For a given level of total expenditure and a given number of children, it is difficult to predict the effect of age on transport share. valuee b 0.0130 implies that an additional ch ild decreases the budget share for 4 transport by 0.013. 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. (c) The p-value for testingH 0 3 against the alternativeH 1 3 where 3 is the coefficient of AGE is 0.869, suggesting that AGE could be excluded from the equation. Similar tests for the coefficients of the other two variables yield p-values less than 0.05. (d) R 0.0247 (e) For a one-child household: WTRANS 0.1420 For a two-child household: WTRANS 0.1290 Chapter 5, Exercise Answers, Principles of Econometrics, 4e 24 EXERCISE 5.8 (a) Equations describing the marginal effects of nitrogen and phosphorus on yield are EYD RO 8.011 3.888NITRO 0.567PHOS EYL 4.800 1.556PHOS 0.567NITRO OS 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 NITRO 1 and PHOS 1 are EYID EEDL 3.556 2.677 TRO OS (ii) The marginal effects when NITRO 2 and PHOS 2 are EYD EDL 0.899 0.554 TRO OS When NITRO 1 and PHOS 1 , the marginal products of both fertilizers are positive. Increasing the fertilizer applications toRO 2 and PHOS 2 reduces the marginal effects of both fertilizers, with that for nitrogen becoming negative. (c) To test these hypotheses, the coefficients are defined according to the following equation YIELD 1 2TRO 3 P4O S N5TRO 26 PHO S NITRO PH OS e Tei)ting H :2 against the alternative :2 0 , the t-value 0 2 4 6 1 2 4 6 is t 7.367. Since t > tc (0.975, 21)080, we reject the null hypothesis and conclude that the marginal effect of nitrogen on yield is not zero when NITRO = 1 and PHOS = 1. Teiit)ing H 0 24 6 against H 1 2 460 , the t-value is 1.660. Since |t| < 2.080t , we do not reject the null hypothesis. A zero marginal yield (0.975,21) with respect to nitrogen cannot be rejected when NITRO = 1 and PHOS = 2. Teiitii)ng H 0 24 6 against H1 26 460 , the t-value is 8.742. Since | t| > 2.080t (0.975,21) reject the null hypothesis and conclude that the marginal product of yield to nitrogen is not zero when NITRO = 3 and PHOS = 1. (d) The maximizing levels are NITRO .701 and PHOS 2.465 . 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 their marginal value product. Chapter 5, Exercise Answers, Principles of Econometrics, 4e 25 EXERCISE 5.15 (a) The estimated regression model is: VOTE5 2.16 0.6434 GROWTH 0.1721INFLATION (se) (1.46) (0.1656) (0.4290) The hypothesis test results on the significance of the coefficients are: H 0 2: 01 2 p-value = 0.0003 significant at 10% level H 0 3: 01 3 p-value = 0.3456 not significant at 10% level One-tail tests were used because more growth is considered favorable, and more inflation is considered not favorable, for re-election of the incumbent party. (b) (i) For INFLATION 4 and GROWTH 3 VO,E 49.54 . 0 (ii)For INFLATION 4 and GROWTH 0 VOT, 51.07 . (iii) For INFLATION 4 and GROWTH 3 VOT, 53.40 . 0 (c) (i)When INFLATION 4 and GROWTH 3 , the hypotheses are H :345:304 5 0 1 2 3 1 1 2 3 The calculated t-value ist 0.39. Since 0.399 2.457 t (0.99,30)e do not rejectH 0. There is no evidence to suggest tat the incumbent part will get the majority of the vote whenNFLATION 4 and GROWTH 3 . (ii) When INFLATION 4 and GROWTH 0 , the hypotheses are H0 1 34 501 1 3: 4 50 The calculated t-value is1.408 . Since1.408 2.457 t(0.99,30)e do not reject H 0. There is insufficient evidence to suggthat the incumbent part will get the majority of the vote whenNFLATION 4 and GROWTH 0 . (iii) When INFLATION 4 and GROWTH 3 , the hypotheses are H1 2 35:3011 2 3 The calculated t-value is 2.950. Since2.950 2.457 t , we reject . We (0.99,30) 0 conclude that the incumbent part w ill get the majority of the vote when INFLATION 4 and GROWTH 3 . As a president seeking re-election, you would not want to conclude that you would be re- elected without strong evidence to support such a conclusion. Setting up re-election as the alternative hypothesis with a 1% significance level reflects this scenario. Chapter 5, Exercise Answers, Principles of Econometrics, 4e 26 EXERCISE 5.23 The estimated model is SCORE39.594 47.02 4 AGE 20.222 AGE 2 32.49 AGE (se) (28.153) (27.810) (8.901) (0.925) 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 27. 15 10 5 0 SCORE SCOREHAT -5 -10 -15 20 24 28 32 36 40 44 AGE_UNITS Figure xr5.23 Fitted line and observations (a) Wtest H.0 40 The t-value is 2.972, with corresponding p-value 0.0035. We therefore reject H 0 and conclude that the quadratic fu nction is not adequate. For suitable values of 2 3 a4d , the cubic function can decrease at an increasing rate, then go past a point of inflection 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. (b) (i) Age = 30 (ii) Between the ages of 20 and 25. (iii) Between the ages of 25 and 30. (iv) Age = 36. (v) Age = 40. (c) No. At the age of 70, the predicted score (relative to par) for Lion Forrest is 241.71. To break 100 it would need to be less than 28 ( 10 72) . Chapter 5, Exercise Answers, Principles of Econometrics, 4e 27 Exercise 5.23 (continued) Predicted scores at different ages age prsccred 20 4.4403 21 4.5621 22 4.7420 23 4.9633 24 5.2097 25 5.4646 26 5.7116 27 5.9341 28 6.1157 29 6.2398 30 6.2900 31 6.2497 32 6.1025 33 5.8319 34 5.4213 35 4.8544 36 4.1145 37 3.1852 38 2.0500 39 0.6923 40 0.9042 41 2.7561 42 4.8799 43 7.2921 44 10.0092 Chapter 5, Exercise Answers, Principles of Econometrics, 4e 28 EXERCISE 5.24 (a) The coefficient estimates, standard errors, t-values and p-values are in the following table. Dependent Variable: ln(PROD) Error Std. Coeff t-value p-value C -1.5468 0.2557 -6.0503 0.0000 ln(AREA) 0.3617 0.06405.6550.0000 ln(LABOR) 0.43280.0669 6.47180.0000 ln(FERT) 0.2095 0.03835.4750.0000 All estimates have elasticity interpretations. For example, a 1% increase in labor will lead to a 0.4328% increase in rice output. A 1% incr ease in fertilizer will lead to a 0.2095% increase in rice output. All p-values are less than 0.0001 implying all estimates are significantly different from zero at conventional significance levels. (b) Testing 5 H. :0 against H. :0 , the t-value is 2.16 . 0 2 1 2 Since 2.5 2.16 2.59 t(0.995,348) do not reject H 0. The data are compatible with the hypothesis that the elasticity of production with respect to land is 0.5. (c) A 95% interval estimate of the elasticity of production with respect to fertilizer is given by bt b se( ) (0.134, 0.285) 4 (0.975,348) 4 This relatively narrow interval implies the fertilizer elasticity has been precisely measured. (d) Testing 3 H. :0 against H. :0 , the t-value ist 1.99 . We reject H because 0 3 1 3 0 1.99 1.649 (0.95,348)here is evidence to conclude that the elasticity of production with respect to labor is greater than 0.3. Reversing the hypotheses and testing 3 H.0 3 against3 H.1 3 , leads to a rejection region of t 1.649 . The calculated t-value is t 1.99 . The null hypothesis is not rejected because 1.99 1.649 . CHAPTER 6 Exercise Answers EXERCISE 6.3 (a) Let the total variation, unexplained vari ation and explained variation be denoted by SST, SSE and SSR, respectively. Then, we have SSE 42.8281 SST 802.0243 SSR 759.1962 (b) A 95% confidence interval for 2is b2 (0.975,17)2 (0.2343,1.1639) A 95% confidence interval for 3is bt b se( ) (1.3704, 2.1834) 2 (0.975,17)3 (c) teost H : 1 against the alternativeH : < 1, we calculate t 1.3658 . Since 0 2 1 2 1.3658 1.740 t(0.05,17)e fail to reject H0 . There is insufficient evidence to conclude 21 . (d) Ttost H : 0 against the alternative H :0 and/or 0 , we calculate 0 23 1 2 3 F 151 . Since 151 3.59 F(0.95,2,17) reject 0 and conclude that the hypothesis 2 = 3 0 is not compatible with the data. (e) The t-value for testing H :2 against the alternativH :2 is 0 2 3 1 2 3 2bb t 2 3 0.37862 0.634 se 2 3 0.59675 Since 2.1 0.634 2.11 t , we do not reject H . There is no evidence to (0.025,17) 0 suggest that 22 3 . 29 Chapter 6, Exercise Answers, Principles of Econometrics, 4e 30 EXERCISE 6.5 (a) The null and alternative hypotheses are: H 0 2 4 35d H 1 2 4 3o5r or both (b) The restricted model assuming the null hypothesis is true is 2 2 ln(WAGE) 1 4UC EXPER 5 ) (EDUC EXPE6) HRSW K e (c) The F-value is F 70.32 .The critical value at a 5% significance level is F(0.95,2,994)005 . Since the F-value is greater than the criti cal value, we reject the null hypothesis and conclude that education and experience have different effects on ln(WAGE) . EXERCISE 6.10 (a) The restricted and unrestricted least squares estimates and their standard errors appear in the following table. The two sets of esti mates are similar except for the noticeable difference in sign for ln(PL). The positive restricted estimate 0.187 is more in line with our a priori views about the cross-price elasticity with respect to liquor than the negative estimate 0.583. Most standard errors for the rest ricted estimates ar e less than their counterparts for the unrestricted estimates, s upporting the theoretical result that restricted least squares estimates have lower variances. CONST ln(PB) ln( PL) ln( PR) ln(I) Unrestricted 3.243 1.020 0.583 0.210 0.923 (0.40.)8.(6.(37)43) Restricted 4.798 1.299 0.187 0.167 0.946 (0.40.)7.(8.(67)14) (b) The high auxiliary R s 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 ln(PL) are a likely consequence of this correlation. Sample Correlation With 2 Variable Auxiliary R ln( PL) ln( PR) ln( I) ln(PB) 0.955 0.967 0.774 0.971 ln(PL) 0.955 0.80.971 ln(PR) 0.694 0.821 ln(I) 0.964 Chapter 6, Exercise Answers, Principles of Econometrics, 4e 31 Exercise 6.10 (continued) (c) Testing H0 2 3 4 5 against H 12 3 4 5 , the value of the test statistic is F = 2.50, with a p-value of 0.127. The critical value is 4.24. We do (0.95,1,25) not rejectH 0. The evidence from the data is consiste nt with the notion that if prices and income go up in the same proportion, demand will not change. (d)(e) The results for parts (d) and (e) appear in the following table. ln(Q) Q ln(Q) se(f ) c lower upperlower upper (d) Restricted 4.55410.14446 2.056 4.257 4.851 70.6 127.9 (e) Unrestricted 4.42390.16285 2.060 4.088 4.759 59.6 116.7 EXERCISE 6.12 The RESET results for the log-log and the linear demand function are reported in the table below. Test F-value df 5%ritical F p -value Log-log 1term 0.0075 (1,24) 4.260 0.9319 2 terms 0.3581 (2,23) 3.422 0.7028 Linear 1term 8.8377 (1,24) 4.260 0.0066 2 terms 4.7618 (2,23) 3.422 0.0186 Because the RESET returns p-values less than 0.05 (0.0066 and 0.0186 for one and two terms respectively), at a 5% level of significan ce, we conclude that the linear model is not an adequate functional form for the beer data. On the other hand, the log-log model appears to suit the data well with relatively higp-values of 0.9319 and 0.7028 for one and two terms respectively. Thus, based on the RESET we conclude that the log-log model better reflects the demand for beer. EXERCISE 6.20 (a) Testing H 0 23 against H 1 23 , the calculated F-value is 0.342. We do not reject H 0 because 0.342 .868 F(0.95,1,348)e p-value of the test is 0.559. The hypothesis that the land and labor elasticities are equal cannot be rejected at a 5% significance level. Uasing t-test, we fail to reject H 0 because t 0.585and the critical values are t(0.025,348).967 and t(0.975,348)967 . The p-value of the test is 0.559. Chapter 6, Exercise Answers, Principles of Econometrics, 4e 32 Exercise 6.20 (continued) (b) Testing H 0 2 34 against H 1 234 , the F-value is 0.0295. The t- value is t 0.172 . The critical values are F 2.72 or t 1.649 and (0.90,1,348) (0.95,348) t 1.649. The p-value of the test is 0.864. The hypothesis of constant returns to (0.05,348) scale cannot be rejected at a 10% significance level. (c) The null and alternative hypotheses are 2 3 0 2 3 0 and/or H 0 H 1 2 3 4 1 2 3 4 1 The critical value iF(0.95,2,348)02. The calculated F-value is 0.183. The p-value of the test is 0.833. The joint null hypothesis of constant returns to scale and equality of land and labor elasticities cannot be rejected at a 5% significance level. (d) The estimates and (standard errors) from the restricted models, and the unrestricted model, are given in the following table. Because the unrestricted estimates almost satisfy the restriction2 3 4 1, imposing this restriction cha nges the unrestricted estimates and their standard errors very little. Imposing the restriction has an impact, 2 3 changing the estimates for both 2 and 3 , and reducing their standard errors considerably. Adding 2 3 4 1 to this restriction reduces the standard errors even further, leaving the coefficient estimates essentially unchanged. Unrestricted 2 3 2 3 4 1 2 3 2 3 4 1 C –1.5468 –1.4095 –1.5381 –1.4030 (0.2557) (0.1011) (0.2502) (0.0913) 0.3617 0.3964 0.3595 0.3941 ln(AREA) (0.0640) (0.0241) (0.0625) (0.0188) ln(LABOR) 0.4328 0.3964 0.4299 0.3941 (0.0669) (0.0241) (0.0646) (0.0188) 0.2095 0.2109 0.2106 0.2118 ln(FERT ) (0.0383) (0.0382) (0.0377) (0.0376) SSE 40.5654 40.6052 40.5688 40.6079 Chapter 6, Exercise Answers, Principles of Econometrics, 4e 33 EXERCISE 6.21 Full FERT LABOR AREA model omitted omitted omitted b2A( ) 0.3617 0.4567 0.6633 b3(L)ABOR 0.4320.5689 0.7084 b4(F)ERT 0.2095 0.3010.2682 RESET(1) p-value 0.5688 0.8771 0.4281 0.1140 RESET(2) p-value 0.2761 0.4598 0.5721 0.0083 (i) With FERT omitted the elasticity for AREA changes from 0.3617 to 0.4567, and the elasticity for LABOR changes from 0.4328 to 0.5689. The RESET F-values (p-values) for 1 and 2 extra terms are 0.024 (0.877) and 0.779 (0.460), respectively. Omitting FERT appears to bias the other elasticities upwards, but the omitted variable is not picked up by the RESET. (ii) With LABOR omitted the elasticity for AREA changes from 0.3617 to 0.6633, and the elasticity for FERT changes from 0.2095 to 0.3015. The RESET F-values (p-values) for 1 and 2 extra terms are 0.629 (0.428) and 0.559 (0.572), respectively. Omitting LABOR also appears to bias the other elasticities upwards, but again the omitted variable is not picked up by the RESET. (iii) With AREA omitted the elasticity for FERT changes from 0.2095 to 0.2682, and the elasticity for LABOR changes from 0.4328 to 0.7084. The RESET F-values (p-values) for 1 and 2 extra terms are 2.511 (0.114) and 4.863 (0.008), respectively. Omitting AREA appears to bias the other elasticities upwards, particularly that for LABOR. In this case the omitted variable misspecification has been picked up by the RESET with two extra terms. EXERCISE 6.22 (a) F 7.40 Fc= 3.26 p-value = 0.002 rejecet H 0nd conclude that age does affect pizza expenditure. (b) Point estimates, standard errors and 95% inte rval estimates for the marginal propensity to spend on pizza for different ages are given in the following table. Point StandardonfiIntnrcveal Age Estimate Error Lowe Urpper 20 4.515 1.5201.437 2.598 30 3.283 0.9051.444 8.731 40 2.050 0.4651.102 7.993 50 0.818 0.710 0.622 2.258 55 0.202 0.991 1.808 2.212 Chapter 6, Exercise Answers, Principles of Econometrics, 4e 34 Exercise 6.22 (continued) (c) This model is given by PIZZA + AGE INC AGE IN AGE2 NC e 1 2 3 4 5 The marginal effect of income is now given by E IZZA AGE + AGE 2 INCOME 3 4 5 If this marginal effect is to increase with age, up to a point, and then declin5, then The results are given in the table below. The sign of the estimated coeffi5ient b = 0.0042 did not agree with our expectation, but, withp-value of 0.401, it was not significantly different from zero. Variable Coefficierrtor Std. t-value p-value C 109.72 135.57 0.809 0.4238 AGE –2.0383 3.5419 –0.575 0.5687 INCOME 14.0962 8.8399 1.595 0.1198 AGE INCOME –0.4704 0.4139 –1.136 0.2635 2 AGE INCOME 0.004205 0.004948 0.850 0.4012 (d) Point estimates, standard errors and 95% inte rval estimates for the marginal propensity to spend on pizza for different ages are given in the following table. Point Standardonfintnveal Age Estimate Error LoweUrpper 20 6.371 2.6640.96 11.779 30 3.769 1.0741.585 9.949 40 2.009 0.4691.052 6.962

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