APPLIED ECONOMETRICS ECON 446
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Chapter 4 Properties of the Least Squares Estimators Assumptions of the Simple Linear Regression Model SR1 SR2 SR3 SR4 SR5 SR6 yt 2 31 B2x1 er Eet 0 lt3 Eyt 31 3sz varet 62 varyt coxepej cox321321 0 x1 is not random and takes at least two values eINA062 lt3 yINBlBzxt62 Optional Slide 41 Undergraduate Econometrics 2 d Edition iChapler 4 41 The Least Squares Estimators as Random Variables o The least squares estimator b2 of the slope parameter Bz based on a sample of T observations is b TszyzszZJz 338 2 f a T Z xZ x o The least squares estimator 1 of the intercept parameter 31 is b1 zf bzf 338b where 7 2 y T and f 2x T are the sample means of the observations on y and x respectively Slide 42 Undergraduate Econometrics 2 Edition iChapter 4 0 When the formulas for b1 and b2 are taken to be rules that are used whatever the sample data turn out to be then b1 and b2 are random variables In this context we call b1 and b2 the least squares estimators 0 When actual sample values numbers are substituted into the formulas we obtain numbers that are values of random variables In this context we call b1 and b2 the least squares estimates Slide 43 Undergraduate Econometrics 2 Edition iChapter 4 42 The Sampling Properties of the Least Squares Estimators 421 The Expected Values of 1 and b2 0 We begin by rewriting the formula in equation 338a into the following one that is more convenient for theoretical purposes b2 32 Zwlel 421 where w is a constant nonrandom given by xZ x W Z W 422 The expected value of a sum is the sum of the expected values see Chapter 251 Eb2 1532 Zwlel 2 Egg ZEwlel 423 32 Zleel 32 since Eel 0 Slide 44 Undergraduate Econometrics 2 d Edition iChapler 4 421a The Repeated Sampling Context Table 41 contains least squares estimates of the food expenditure model from 10 random samples of size T 40 from the same population Table 41 Least Squares Estimates from 10 Random Samples of size T 40 11 b1 b2 1 511314 01442 2 612045 01286 3 407882 01417 4 801396 00886 5 310110 01669 6 543099 01086 7 696749 01003 8 711541 01009 9 188290 01758 10 361433 01626 Undergraduate Econometrics 2 Edition iChapler 4 Slide 45 421b Derivation of Equation 421 20c f2 2x 2 le Tf2 2x 2fTlejTf2 22x3 2Tf2 Tf2 22x3 Tf2 424a Elba ff fo Tf2 fo fzx2xs Zx 42 T To obtain this result we have used the fact that f 2x1 T so 2x1 T f 205fyz7szyzTf722xzy1 Slide 46 Undergraduate Econometrics 2 d Edition iChapter 4 2 in deviation from the mean form is b 2 Zoe my J 2 2 xi if 0 Recall that Z xZ f O 0 Then the formula for b2 becomes b 2 205 fyz Z 205 fyz Zxz f 2 206 Y2 206 f2 ZxZYy xi f Zea W 2 Zea W quotZw y where w is the constant given in equation 422 Undergraduate Econometrics 2 Edition iChapler 4 426 427 428 Slide 47 To obtain equation 421 replace yt by yr 2 B1 3sz er and simplify b2 szyz szBi Bzxz 61 31sz Bzzw2x2 szez 429a Zwl 0 this eliminates the term 812w Zwlxl 1 so 322wle 32 and 429a simpli es to equation 421 b2 82 Zwlel 429b Slide 48 Undergraduate Econometrics 2 Edition iChapler 4 The term 21wZ 0 because 2w 3552 20 f2 2xl xO us1ng 2xl x0 To show that ZZWZxZ 1 we again use 2x f 0 Another expression for 209 if is 209 f2 2209 fxt YZxt YXt f2xt fZxt Yxt Consequently Slide 49 Undergraduate Econometrics 2 Edition iChapler 4 205 fxz Zxz fxz Z le 206 f2 Zxz fxz 1 422 The Variances and Covariance of 1 and b2 V311 Eibz Eb2 2 If the regression model assumptions SRlSRS are correct SR6 is not required then the variances and covariance of 1 and b2 are Slide 410 Undergraduate Econometrics 2 d Edition iChapler 4 2 A varb1 cs TZOCZ fy 2 cs varb2 2 xi if 4210 COVb1b2 62 Z xz if Slide 411 Undergraduate Econometrics 2 Edition iChapter 4 Let us consider the factors that affect the variances and covariance in equation 4210 a 2 The var1ance of the random error term 6 appears 1n each of the eXpress1ons Q The sum of squares of the values of x about their sample mean 209 if appears in each of the variances and in the covariance b The larger the sample size T the smaller the variances and covariance of the least squares estimators it is better to have more sample data than less 4 The term 2x2 appears in varb1 U l The sample mean of the xvalues appears in covb1b2 Slide 412 Undergraduate Econometrics 2 d Edition iChapler 4 Deriving the variance of b2 The starting point is equation 421 varb2 varB2 Zwlel varZwleZ 2 w varel using c0VeZej O 622wf using varel 62 62 Z 2061 if The very last step uses the fact that since 32 is a constant 2 xi ff 1 ZWZ Z XXL Eff 2052 32 Undergraduate Econometrics 2 Edition iChapler 4 4211 4212 Slide 413 423 Linear Estimators o The least squares estimator b2 is a weighted sum of the observations yr b2 21wZ yl o Estimators like 92 that are linear combinations of an observable random variable linear estimators 43 The GaussMarkov Theorem GaussMarkov Theorem Under the assumptions SRlSRS of the linear regression model the estimators b1 and b2 have the smallest variance of all linear and unbiased estimators of 31 and 32 They are the Best Linear nbiased Estimators BLUE of 31 and 32 Slide 414 Undergraduate Econometrics 2 d Edition iChapter 4 a Q b 4 U1 0 The estimators b1 and b2 are best when compared to similar estimators those that are linear d unbiased The Theorem does not say that 1 and b2 are the best of all possible estimators The estimators b1 and b2 are best within their class because they have the minimum variance In order for the GaussMarkov Theorem to hold the assumptions SRlSRS must be true If any of the assumptions 15 are not true then 1 and b2 are not the best linear unbiased estimators of 31 and 32 The GaussMarkov Theorem does not depend on the assumption of normality In the simple linear regression model if we want to use a linear and unbiased estimator then we have to do no more searching The GaussMarkov theorem applies to the least squares estimators It does not apply to the least squares estimates from a single sample Slide 415 Undergraduate Econometrics 2 d Edition iChapter 4 Proof of the GaussMarkov Theorem 0 Let b Zktyt where the kt are constants be any other linear estimator of 32 0 Suppose that kt 2 w ct where ct is another constant and w is given in equation 422 0 Into this new estimator substitute yr and simplify using the properties of wt in equation 429 19 Zkzyz 2042 czyz 2042 ctB1 Bzxz 61 2wl clB1 Zwl 098le Zwl cleZ 431 BlzzwZ 8120 3221wlxZ 8226le Zwl cleZ 8120 82 Bzzczxz 204 czez since Zwt O and Zwtxt 1 Slide416 Undergraduate Econometrics 2 Edition iChapler 4 151 31201 32 22096 20 CzEez 432 2 81262 Bz Bzzczxz o In order for the linear estimator b 21ktyt to be unbiased it must be true that 2010 and 201x120 433 0 These conditions must hold in order for b 21kgt to be in the class of linear and unbiased estimators Slide 417 Undergraduate Econometrics 2 d Edition iChapter 4 0 So we will assume the conditions 433 hold and use them to simplify expression 431 bZktyt 2B2Zwtctet We can now nd the variance of the linear unbiased estimator 1 following the steps in equation 4211 and using the additional fact that clxZ f l Y Zc wquotZiZm fgt2l2ltx rgtzzc x 2062qu 0 Slide 418 Undergraduate Econometrics 2 d Edition iChapler 4 Use the properties of variance to obtain varb varB2 2wl clel 2 w c2 varel 622WZ 012 2 622w12622012 435 varb2 2ch 2 varb2 since 21012 2 0 Slide 419 Undergraduate Econometrics 2 Edition iChapler 4 44 The Probability Distribution of the Least Squares Estimators o If we make the normality assumption assumption SR6 about the error term then the least squares estimators are normally distributed 622x 441 C32 0 If assumptions SRlSRS hold and if the sample size T is suj ciently large then the least squares estimators have a distribution that approximates the normal distributions shown in equation 441 Slide 420 Undergraduate Econometrics 2 d Edition iChapler 4 45 Estimating the Variance of the Error Term The variance of the random variable er is vareIcs2 EeZ Eel2 Ee2 451 if the assumption Eet0 is correct Slnce the expectatlon 1s an average value we mlght cons1der estlmatlng o as the average of the squared errors 2 e 62 2 452 Slide 421 Undergraduate Econometrics 2 d Edition iChapler 4 0 Recall that the random errors are er yl B1 B2xt o The least squares residuals are obtained by replacing the unknown parameters by their least squares estimators 2 ZA 453 0 There is a simple modi cation that produces an unbiased estimator and that is quot2 e 62 2 454 E62 52 455 Slide 422 Undergraduate Econometrics 2 d Edition iChapler 4 451 Estimating the Variances and Covariances of the Least Squares Estimators 0 Replace the unknown error variance 62 in equation 4210 by its estimator to obtain Va bl 62 seb1 lvarb1 T2 xz if 3192 62 seb2 Ivarb2 466 2061 if 7 coVb1b2 62 f Z xz if Slide 423 Undergraduate Econometrics 2 Edition iChapter 4 462 The Estimated Variances and Covariances for the Food Expenditure Example Table 41 Least Squares Residuals for Food Expenditure Data y 7 2 71 bzx y y 5225 739045 216545 5832 847834 264634 8179 952902 135002 11990 1007424 191576 12580 1027181 230819 62 Z 543113315 T 2 38 214292456 Slide 424 Undergraduate Econometrics 2 d Edition iChapler 4 2 x V rb162 14292456 M 4901200 TZxZ x 401532463 seb1 Jvaraal x490 1200 221387 62 14292456 arb V 2 Zoe ff 1532463 00009326 seb2 1v 1rbZ 00009326 00305 698 c vbb 62 1 2 1532463 14292456 06510 i Z xz if Slide 425 Undergraduate Econometrics 2 Edition iChapter 4 453 Sample Computer Output Method Least Squares Sample 1 40 Included observations 40 Dependent Variable FOODEXP Undergraduate Econometrics 2 d Edition iChapler 4 Variable Coefficient Std Error t Statistic Prob C 4076756 2213865 1841465 00734 INCOME 0128289 0030539 4200777 00002 Rsquared 0317118 Mean dependent var 1303130 Adjusted Rsquared 0299148 SD dependent var 4515857 SE of regression 37 80536 Akaike info criterion 1015149 Sum squared resid 5431133 Schwarz criterion 1023593 Log likelihood 2010297 Fstatistic 1764653 DurbinWatson stat 23 70373 ProbF statistic 00001 5 5 Table 43 EViews Regression Output Slide 426 Dependent Variable FOODEXP Analysis of Variance Sum of Mean Source DE Squares Square F Value ProbgtF Model 1 2522122299 2522122299 17647 00002 Error 38 5431133145 142924556 C Total 39 7953255444 Root MSE 3780536 R square 03171 Dep Mean 13031300 Adj R sq 02991 CV 2901120 Parameter Estimates Parameter Standard T for H0 Variable DE Estimate Error Parameter0 Prob gt T INTERCEP 1 40767556 2213865442 1841 00734 INCOME 1 0128289 003053925 4201 00002 Table 44 SAS Regression Output Slide 427 Undergraduate Econometrics 2 Edition 7Chapler 4 VARIANCE OF THE ESTIMATE SIGMA2 14292 VARIABLE ESTIMATED STANDARD NAME COEFFICIENT ERROR X 012829 03054E 01 CONSTANT 40768 2214 Table 45 SHAZAM Regression Output Covariance of Estimates COVB INTERCEP X INTERCEP 49012001955 O650986935 X O650986935 0000932646 Table 46 SAS Estimated Covariance Array Slide 428 Undergraduate Econometrics 2 Edition iChapler 4 Econ 446 MT 1 February 27 2008 R Sickles Answer all of the following questions You have 50 minutes You may use a calculator and an 8 12 X 11 sheet of paper with notes etc on both sides Questions 16 are worth 5 points each question 7 is worth 10 points and questions 810 are worth 20 points each for a total of 100 points The following data were obtained from a survey of college students The variable X represents the number of nonassigned books read during the past siX months Use this for questions 13 X 0 1 2 3 4 5 6 PXx 055 015 010 010 004 003 003 1 Find P X 2 a 010 b 055 c 070 d 080 e 084 2 What is the expected value of X a 000 b 100 c 114 d 169 e 300 3 What is the variance of X a 100 b 262 c 3 61 d 600 a 1302 4 The standard normal distribution has a mean of and a standard deviation of respectively a a 903 OD ID IO 0 l 0 l A null hypothesis can only be rejected at the 5 signi cance level if and only if a a 95 con dence interval includes the hypothesized value of the parameter b a 95 con dence interval does not include the hypothesized value of the parameter the null hypothesis is void the null hypotheses includes sampling error no If a researcher takes a large enough sample she will almost always obtain virtually signi cant results practically signi cant results consequentially signi cant results statistically signi cant results 993 Answer true or false and state why a The power of a test increases as the sample size increases b The size ofatest increases as the sample size increases c The independent variables in a bivariate regression model needs to be constant to estimate the slope coef cient d In large samples the usual standardized ratios follow the tdistribution 8 Consider the bivariate regression model y A zxzyl 1 for il n Explain the key assumptions needed in order to establish the desirable properties of best linear unbiasedness of the ordinary least squares estimates of A and z 9 A new online auction site specializes in selling automotive parts for classic cars The founder of the company believes that the price received for a particular item increases with its age ie the age of the car on which the item can be used in years She collects information on 10000 auctions Use the multiple regression output below to answer the following questions Regression coef cients Coef cient Std Err t vaue pvaue Constant 277 0739 375 00010 Age of part 75017 10647 705 00000 Interpret the estimated regression coefficients a b Is the founder of the company correct in believing that the price received for the item increases with its age c What would be your answer to b if the number of bidders was an important and omitted explanatory variable d Would you recommend that this company examine any other factors to predict the selling price If yes what other factors would you want to consider 10 Consider a simple regression in which the dependent variable MIMmean income of males who are 18 years or older in thousands of dollars The explanatory variable PMHS 3ercent of males 18 or older who are high school graduates The data consist of 51 observations on the 50 states plus the District of Columbia Thus MIM and PMHS are state averages Assume for this problem that the number of observations and degrees of freedom is large enough so that the difference between the tdistribution and the standardized normal is negligible The estimated slope is 0180 with a tstatistic for the hull hypothesis that the coefficient is zero of 5754 The estimated intercept is not given but the standard error for the intercept is 2174 and the corresponding tstatistic is 1257 a What is the estimated equation intercept Show your calculation Sketch the estimated regression function b What is the standard error of the estimated slope Show your calculation c State the economic interpretation of the estimated slope Is the sign of the coefficient what you would expect from economic theory d Construct a 99 confidence interval estimate of the slope of this relationship e Test the hypothesis that the slope of the relationship is 02 against the alternative that it is not State in words the meaning of the null hypothesis in the context of this problem Chapter 12 Autocorrelation 121 The Nature of the Problem 0 The randomness of the sample implies that the error terms for different observations households or rms will be uncorrelated 0 When we have timeseries data where the observations follow a natural ordering through time there is always a possibility that successive errors will be correlated with each other 0 In any one period the current error term contains not only the effects of current shocks but also the carryover from previous shocks This carryover will be related to or correlated with the effects of the earlier shocks When circumstances such as these lead to error terms that are correlated we say that autocorrelation exists Slide 121 Undergraduate Econometrics 2quotd EditionChapter I2 o The possibility of autocorrelation should always be entertained when we are dealing with timeseries data 0 Suppose we have a linear regression model with two explanatory variables That is ylzl31l32xt2 B3xl3el 1211 0 The error term assumptions utilized in Chapters 3 through 9 are EelO varelcs2 1212a covelesO fortis 12121 0 When 1212b does not hold we say that the random errors er are autocorrelated 1211An Area Response Model for Sugar Cane o Letting A denote area planted and P denote output price and assuming a loglog constant elasticity functional form an area response model of this type can be written as Slide 122 Undergraduate Econometrics 2M EditionChapter I2 1nA 131 1321nP 1213 0 We use the model in 1213 to explain the area of sugar cane planted in a region of the SouthEast Asian country of Bangladesh 0 The econometric model is lnAlBllenPlel 1214 0 We can write this equation as yZ 213132xleZ 1215 where yl lnAl and xZ lnPl 1216 Slide 123 Undergraduate Econometrics 2quotd EditionChapter I2 121 1a Least Squares Estimation 0 Application of least squares yields the following estimated equation A y 6111 0971 x R2O706 R121 Ol690lll std errors 0 The least squares residuals appear in Table 122 and are plotted against time in Figure 121 Figure 121 Least squares residuals plotted against time 0 We can see that there is a tendency for negative residuals to follow negative residuals and for positive residuals to follow positive residuals This kind of behavior is consistent with an assumption of positive correlation between successive residuals Slide 124 Undergraduate Econometrics 2quotd EditionChapter I2 0 With uncorrelated errors we would not expect to see any particular pattern If the errors are negatively autocorrelated we would eXpect the residuals to show a tendency to oscillate in sign 122 FirstOrder Autoregressive Errors 0 If the assumption coveles O is no longer valid what alternative assumption can we use to replace it Is there some way to describe how the ez are correlated If we are going to allow for autocorrelation then we need some way to represent it o The most common is model is a rstorder autoregressive model or more simply an AR1 model 6 peH vl 1221 EVz0 varvloi covvlvsO lis 1222 Slide 125 Undergraduate Econometrics 2quotd EditionChapter I2 o The rationale for the AR1 model is that the random component er in time period I is composed of two parts i pet1 is a carry over from the random error in the previous period ii V is a new shock to the level of the economic variable 0 The autoregressive model asserts that shocks to an economic variable do not work themselves out in one period 1221 Properties of an AR1 Error 0 Assume 1 lt p lt1 1223 It can be shown that Ee 0 1224 Slide 126 Undergraduate Econometrics 2quotd EditionChapter I2 2 c3v 2 varel Ge 1p2 Because 6 does not change overtime the error et is also homoskedastic covelek cipk k gt O The error correlation cov e e correljelik z 27k vareZ var e1 k 2 k 599 pk 2 2 6868 1225 1226 1227 0 p is the correlation between two errors that are one period apart it is sometimes called the autocorrelation coef cient Undergraduate Econometrics 2quotd EditionChapter 12 Slide 127 123 Consequences for the Least Squares Estimator o If we have an equation whose errors exhibit autocorrelation but we ignore it or are simply unaware of it what does it have on the properties of least squares estimates 1The least squares estimator is still a linear unbiased estimator but it is no longer best 2The formulas for the standard errors usually computed for the least squares estimator are no longer correct and hence con dence intervals and hypothesis tests that use these standard errors may be misleading Slide 12 8 Undergraduate Econometrics 2quotd EditionChapter I2 Proofs o For the simple regression model yl 31 3le el we wrote the least squares estimator for 32 as b2 82 lelel 1231 where w z 1232 0 We prove b2 is still an unbiased estimator for 32 under autocorrelation by showing that Eb2Bz ZWZE61BZ 1233 o For the variance of b2 we have 2 varb2 2 WI varel Zgwiwj covel e J Slide 129 Undergraduate Econometrics 2M EditionChapter I2 cswa cs 22 wiwjpk where k i j 1i 6 1 1 22x1 fxj fpk 2xzf2k 2xzf2 1 1234 0 When we were proving that varb2 xiZ xZ if in the absence of autocorrelation the terms coveej were all zero This simpli cation no longer holds however 1 Slide 1210 Undergraduate Econometrics 2M EditionChapter I2 0 Return to least squares estimation of the sugar cane example 0 Given estimates for p and 6 it is possible to use a computer to calculate an estimate for varb2 from equation 1234 A similar estimate for varb1 can also be obtained 0 Suppose that we have estimates of p and 6 and that we have used them to estimate varb1 and varb2 o The square roots of these quantities we can call correct standard errors while those we calculated with our least squares estimates and reported in equation 1217 we call incorrect The two sets of standard errors along with the estimated equation are jZ61110971xt O169O111 quotincorrectquot se39s R122 O226O147 quotcorrectquot se39s 0 Note that the correct standard errors are larger than the incorrect ones Slide 121 1 Undergraduate Econometrics 2quotquot EditionChapter I2 o If we ignored the autocorrelation we would tend to overstate the reliability of the least squares estimates The con dence intervals would be narrower than they should be For example using I 2037 we nd the following 95 con dence interval for 32 For 32 0745 1197 incorrect 0672 1269 correct Slide 1212 Undergraduate Econometrics 2quotd EditionChapter I2 124 Generalized Least Squares 124 1 A Transformation 0 Our objective is to transform the model in equation 1215 yZ 81 8le eZ 1241 0 The relationship between el and vi is given by el pet1 vl 1242 Substituting 1242 into 1241 yields yz Bl Bzxz pezil Vz 1243 0 To substitute out eH we note that 124 1 holds for every single observation ezil yH 31 82x24 1244 Slide 1213 Undergraduate Econometrics 2quotd EditionChapter I2 Multiplying 1244 by p yields pezil pyH pBl szxH Substituting 1245 into 1243 yields y 2 Bl Bzxz py241 pBl szxH Vz or after rearranging y pyzil Bl 1 p Bz xz px171 V The transformed dependent variable is y y pyH I 2373T The transformed explanatory variable is x12 The new constant term is xZ pr l23T Undergraduate Econometrics 2quotd EditionChapter 12 1245 1246 1247a 1247b Slide 1214 x 1 p l23T 12470 0 Making these substitutions we have y ZBI xf2B2 vl 1248 0 Thus we have formed a new statistical model with transformed variables y x and x 2 and importantly with an error term that is not the correlated er but the uncorrelated Vt that we have assumed to be distributed O 63 0 There are two additional problems that we need to solve however 1 Because lagged values of y and xZ had to be formed only T 1 new observations were created by the transformation in 1247 We have values yxx2 for l 23T But we have no yfxf 1x12 2 The value of the autoregressive parameter p Slide 1215 Undergraduate Econometrics 2quotd EditionChapter I2 12 41 a Transforming the First Observation The rst observation in the regression model is y1 Bl x1B2 e1 1249 with error variance vare1 o 63 l oz 0 The transformation that yields an error variance of 63 is multiplication by 1 p2 The result is WM 43 mWxn We1 12410 or y x5181 xf 2l32 ef 124lla where Slide 1216 Undergraduate Econometrics 2M EditionChapter I2 yf 1 pzy1 xf1 p2 12411b x1 ll ple e Il pze1 0 Note that 62 Vare1 Z 1 92Varel Z 1 9212 53 Slide 1217 Undergraduate Econometrics 2M EditionChapter I2 Remark We can summarize these results by saying that providing p is known we can nd the best linear unbiased estimator for 31 and 32 by applying least squares to the transformed model y 81x 82sz Vz 12412 where the transformed variables are de ned by xll y 1 pzyl for the rst observation and y y pyH xi 1 p x x pr for the remaining I 23Tobservations Undergraduate Econometrics 2quotd EditionChapter 12 Slide 1218 125 Implementing Generalized Least Squares The remaining problem is the fact that the transformed variables y x and x cannot be calculated without knowledge of the parameter p Consider the equation e peH vl 1251 I o If the er values were observable we could treat this equation as a linear regression model and estimate p by least squares 0 However the er are not observable because they depend on the unknown parameters 31 and 32 through the equation elzyzl31l32xl 1252 0 As an approximation to the ez we use instead the least squares residuals l yl b1 b2xl 1253 Slide 1219 Undergraduate Econometrics 2M EditionChapter I2 where b1 and 2 are the least squares estimates from the untransformed model 0 Substituting the l for the er in 1251 is justi ed providing the sample size T is large Making this substitution yields the model amp peH 91 1254 0 The least squares estimator of p from 1254 has good statistical properties if the sample size T is large it is given by 1255 Slide 1220 Undergraduate Econometrics 2quotd EditionChapter I2 1251The Sugar Cane Example Revisited 0 We obtain 3 2 20342 1156 0 As examples note that y 2 WM x1 03422 33673 R125 231642 Slide 1221 Undergraduate Econometrics 2M EditionChapter I2 and x x32 13x22 22919 0342 2l637 15519 R126 0 Applying least squares to all transformed observations yields the generalized least squares estimated model 11121 6164 l007lnPt 0213 0137 R127 Slide 1222 Undergraduate Econometrics 2quotd EditionChapter I2 126 Testing for Autocorrelation 0 Looking for runs in the least squares residuals gives some indication of whether autocorrelation is likely to be a problem 0 The DurbinWatson test is by far the most important one for detecting AR1 errors 0 Consider again the linear regression model yZ Bl32xleZ 1261 where the errors may follow the rstorder autoregressiVe model e pet1 vl 1262 I o It is assumed that the Vt are independent random errors with distribution NO 63 The assumption of normally distributed random errors is needed to derive the probability distribution of the test statistic used in the DurbinWatson test Slide 1223 Undergraduate Econometrics 2quotd EditionChapter I2 o For a null hypothesis of no autocorrelation we can use H0 p O For an alternative hypothesis we could use H1 p gt O or H1 p lt O orH1p 2 O 0 We choose H1 p gt O in most empirical applications in economics positive autocorrelation is the most likely form that autocorrelation will take 0 Thus we consider testing H0 p 0 against H1 p gt 0 1263 0 The DW statistic is d 212 1264 where the l are the least squares residuals l yl b1 ble 0 To see why dis closely related to p expand 1264 as Slide 1224 Undergraduate Econometrics 2quotd EditionChapter I2 2 g 2 62271 2 271 2 2 2 2 2 T T 2 T 2 243 26 26quot 11 11 11 11 2 0 Thus we have 0 z 20 1265 1266 0 If f 0 then the DurbinWatson statistic d z 2 which is taken as an indication that the model errors are not autocorrelated Undergraduate Econometrics 2quotd EditionChapter 12 Slide 1225 If f 1 then d z 0 and thus a low value for the DurbinWatson statistic implies that the model errors are correlated and p gt O o What is a critical value dc such that we reject H0 when d 3 dc Determination of a critical value and a rejection region for the test requires knowledge of the probability distribution of the test statistic under the assumption that the null hypothesis H0 p O is true If a 5 signi cance level is required nd dc such that Pd 3 d0 005 Then as illustrated in Figure 122 we reject H0 if d 3 dc and fail to reject H0 if d gt d0 For this onetail test the pvalue is given by the area under f d to the left of the calculated value of d Thus if the pvalue is less than or equal to 005 it follows that d 3 dc and H0 is rejected If the pvalue is greater than 005 then d gt dc and H0 is accepted Slide 1226 Undergraduate Econometrics 2quotd EditionChapter I2 Insert Figure 122 here o A dif culty associated with f d and one that we have not previously encountered when using other test statistics is that this probability distribution depends on the values of the explanatory variables It is impossible to tabulate critical values that can be used for every possible problem 0 There are two ways to overcome this problem The rst way is to use software SHAZAM is an example that computes the pvalue for the explanatory variables in the model under consideration Instead of comparing the calculated d value with some tabulated values of dc we get our computer to calculate the pvalue of the test If this pvalue is less than the speci ed signi cance level H 0 p O is rejected and we conclude that autocorrelation does exist Slide 1227 Undergraduate Econometrics 2quotd EditionChapter I2 o In the sugar cane area response model the calculated value for the DurbinWatson statistic is d 1291 Is this value suf ciently close to zero or suf ciently less than 2 to reject H 0 and conclude that autocorrelation exists Using SHAZAM we nd that pvalue Pd g 1291 00098 0 This value is much less than a conventional 005 signi cance level we conclude therefore that the equation39s error is positively autocorrelated 1261a The Bounds Test 0 In the absence of software that computes a pvalue a test known as the bounds test can be used to partially overcome the problem of not having general critical values Durbin and Watson considered two other statistics dL and dU whose probability distributions do not depend on the explanatory variables and which have the property that dL ltdltdU 1267 Slide 1228 Undergraduate Econometrics 2quotd EditionChapter I2 0 That is irrespective of the explanatory variables in the model under consideration d will be bounded by an upper bound dU and a lower bound dL The relationship between the probability distributions f dL f d and f dU is depicted in Figure 123 0 Let ch be the 5 critical value from the probability distribution for dL Similarly let dUc be such that PdU lt dUc 05 Since the probability distributions f dL and f dU do not depend on the explanatory variables it is possible to tabulate the critical values ch and dUc Table 5 at the end of this book 0 In Figure 123 we have three critical values 0 If the calculated value d is such that d lt ch then it must follow that d lt dc and H 0 is rejected 0 If d gt dUc then it follows that d gt dc and H0 is accepted 0 If ch lt d lt dUc then we cannot be sure whether to accept or reject Slide 1229 Undergraduate Econometrics 2M EditionChapter I2 0 These considerations led Durbin and Watson to suggest the following decision rules which are known collectively as the DurbinWatson bounds lest Ifdltde reject HO p O and accept H1 p gt 0 if d gtdUc do not reject H0 p O idecltdltdU c the test is inconclusive To nd the critical bounds for the sugar cane example we consult Table 5 at the end of the book for T 34 and K 2 The values are ch 1393 dUc 1514 Since d 1291 lt ch we conclude that d lt dc and hence we reject H 0 there is eVidence to suggest that autocorrelation exists Slide 1230 Undergraduate Econometrics 2quotd EditionChapter I2 1262 A Lagrange Multiplier Test 0 To introduce this test return to equation 1243 which was written as yl 81 0le peH vl 1268 0 If el1 was observable an obvious way to test the null hypothesis H 0 p 0 would be to regress yZ on xZ and 671 and to use a Z or F test to test the signi cance of the coef cient of eH Because e11 is not observable we replace it by the lagged least squares residuals and then perform the test in the usual way 717 o Proceeding in this way for the sugar can example yields I 2006 F 4022 pvalue 0054 Slide 1231 Undergraduate Econometrics 2quotd EditionChapter I2 o Obtaining a pvalue greater than 005 means that at a 5 signi cance level the LM test does not reject a null hypothesis of no autocorrelation This test outcome con icts with that obtained earlier using the DurbinWatson test Such con icts are a fact of life You should note the following 4 points lWhen estimating the regression in 1268 using the rst observation I 1 requires knowledge of 0 Two ways of overcoming this lack of knowledge are often employed One is to set e0 O The other is to omit the rst observation In our calculations we set eO O The results change very little if the rst observation is omitted instead 2The DurbinWatson test is an exact test valid in nite samples The LM test is an approximate largesample test the approximation occurring because elf1 is replaced by H Slide 1232 Undergraduate Econometrics 2quotd EditionChapter I2 3The DurbinWatson test is not valid if one of the explanatory variables is the lagged dependent variable yH The LM test can still be used in these circumstances This fact is particularly relevant for a distributed lag model studied in Chapter 15 4We have only been concerned with testing for autocorrelation involving one lagged error eH To test for more complicated autocorrelation structures involving higher order lags such as e172 6173 etc the LM test can be used by including the additional lagged errors in 1268 and using an F test to test the relevance of their inclusion Slide 1233 Undergraduate Econometrics 2quotd EditionChapter I2 127 Prediction With AR1 Errors 0 For the problem of forecasting or predicting a future observation yo that we assume yo 8 xOBZe0 1271 where x0 is a given future value of an explanatory variable and 60 is a future error term 0 When the errors are uncorrelated the best linear unbiased predictor for yo is the least squares predictor o b1bzxO 1272 0 When the errors are autocorrelated the generalized least squares estimators denoted by B and 32 are more precise than their least squares counterparts b1 and 92 A better predictor is obtained therefore if we replace 1 and b2 by 31 and A3 Slide 1234 Undergraduate Econometrics 2M EditionChapter I2 oWhen eo is correlated with past errors we can use information contained in the past errors to improve upon zero as a forecast for e0 0 For example if the last error eT is positive then it is likely that the next error eT1 will also be positive 0 When we are predicting one period into the future the model with an ARl error can be written as yTl Bl Bzxr1 eTl Bl 32xT1 peT VT1 1273 where we have used eT1 peT VTH Slide 1235 Undergraduate Econometrics 2quotd EditionChapter I2 Equation 1273 has three distinct components 1Given the explanatory variable xT the best linear unbiased predictor for 31 32x is 17 T1 r r 319 32x 1 where 113132 are generallzed least squares est1mates 2To predict the component peT we need estimates for both p and eT For p we can use the estimator f speci ed in 1255 To estimate eT we use the generalized least squares residual de ned as e M 138sz 1274 3The best forecast of the third component v TH is zero because this component is uncorrelated with past values v1v2vT 0 Collecting all these results our predictor for yT1 is given by A H A yT1 I FBZxTH peT 1275 Slide 1236 Undergraduate Econometrics 2quotd EditionChapter I2 o What about predicting more than one period into the future For h periods ahead it can be shown that the best predictor is A h A y h th zx h pheT 1276 0 Assuming 3 lt 1 the in uence of the term ph T diminishes the further we go into the future the larger h becomes 0 In the Bangladesh sugar cane example A A 31 61641 32 10066 3 0342 and H er yT lgi 39Bsz 1 AT1 39BZIHPT R1210 54596 61641 10066 09374 0239 Slide1237 Undergraduate Econometrics 2quotd EditionChapter I2 0 To predict yT1 and yT2 for a sugar cane price of 04 in both periods T 1 and T 2 we have 21 13 61641 10066 ln04 o342o239 53235 R1211 2112 1 132x111 13 61641 10066 ln04 034220239 R1212 52697 0 Note that these predictions are for the logarithm of area they correspond to areas of 205 and 194 respectively Slide 1238 Undergraduate Econometrics 2quotd EditionChapter I2 Econ 446 Spring 2007 MT 1 R Sickles Answer all of the following questions The questions are weighted equally You have 50 minutes You may use a calculator and an 8 12 X 11 sheet of paper with notes etc on both sides 1 Labor economists study the determination of labor earnings using a sta tistical earnings function A simple example of such a regression estimated using data for 31093 men logYi 758 0070 Xi ei 000160 Here Y denotes earnings and X is years of education log denotes a natural logarithm The value in parentheses is the estimated standard error a Using your knowledge of logarithmic functional forms explain the inter pretation of the coef cient on education Note that M 007 So if years of educattion increase by 1 the earnings will increase about 7 It is essentially the rate of return to education b What does this equation predict would be the earnings of a hypothetical person with no education If Xi0 then Yiexp75821959 c Obtain a 90 con dence interval for the rate of return to education bgiseb2 twin2 007 i 000160 Wme 05 0067368 0072632 2 Answer true or false and state why a The power of a test increases as the sample size increases True The power of the test is given by l Where Paccept Holalternative hypothesis is true For example in the two tailedtest P0392 S Z S 31777 which decreases as n increases so the power of the test increases b The size of a test increases as the sample size increases False The size of a test is given by a and it is determined at the begining and does not change with n c The independent variables in a bivariate regression model needs to be constant to estimate the slope coef cient False Regressors should Change across observations d In large samples the usual standardized ratios follow the tedistribution False They follow the standardized normal distribution e If we rescale the dependent variable in a regression by dividing by 100 the new coef cients and their estimates will be multiplied by 100 False They are divided by 100 see page 85 on HGL f R2 always decreases as we add variables to a regression False They either increse or leave R2 the same 3 Consider the bivariate regression model yi 2xi 5 for il Explain the key assumptions needed in order to establish the desirable properties of best linear unbiasedness of the ordinary least squares estimates of l and Q Assumptions SR1SR5 of the linear regression model see page 16 on HGL 4 In the macroeconomics literature there are two competing theories con cerning consumption behavior According to Keynes aggregate consumption is determined by aggregate income Alternatively the classical economists feel that consumption should be inversely related to interest rates The results below S A 39 r Jquot in billions 0131982 dollars disposable personal income in billions of 1982 dollars and the real interest rate for the years 1955786 Based on the econometric evidence below what are your conclusions relative to the validity of the two hypotheses are based on ob m atiou on Keynes7 Theory An economic model that relates consumption to income using a linear relationship is Ct 1 ZMt The corresponding statistical model is Ct5152Mt 6t where we will assume the et are independent random variables with ef 002e Classical Economist s Theory In this case we specify the linear economic model relating consumption to interest rate as Ct a1o 2R The correspond ing statistical model is Ct 041 agR r V with the vt assumed to be vf0a392v and independent The two least squares estimated equations with standard errors in parenthee ses are C 737744 09085 M C 13707 59859 R 13527 00076 1154 29488 Note that the standard error of 112 in the rst equation is very large so doing the hypothesis testing whether 2 is 0 will be rejected at any level of signi cance a Also the pvalue is almost zero which makes us reject the null hypothesis On the other hand in the second equation the tratio is 203 and we can nd a to reject the alternative hypothesis especially in the small samples In summary the aggra gate consumtion is likely to be determined by the aggragate income and Keynes theory seems to be valid in many cases Econ 446 Spring 2007 MT 1 R Sickles Answer all of the following questions The questions are weighted equally You have 50 minutes You may use a calculator and an 8 12 x 11 sheet of paper with notes etc on both sides 1 Labor economists study the determination of labor earnings using a statistical earnings function A simple example of such a regression estimated using data for 31093 men is long 758 0070 X 61 000160 Here Y denotes earnings and X is years of education log denotes a natural logarithm The value in parentheses is the estimated standard error a Using your knowledge of logarithmic functional forms explain the interpretation of the coefficient on education b What does this equation predict would be the earnings of a hypothetical person with no education 0 Obtain a 90 confidence interval for the rate of return to education 2 Answer true or false and state why The power of a test increases as the sample size increases The size of a test increases as the sample size increases c The independent variables in a bivariate regression model needs to be constant to estimate the slope coefficient In large samples the usual standardized ratios follow the t distribution If we rescale the dependent variable in a regression by dividing by 100 the new coefficients and their estimates will be multiplied by 100 f R2 always decreases as we add variables to a regression 53 09 3 Consider the bivariate regression model y l zxzyl 81 for i1 n Explain the key assumptions needed in order to establish the desirable properties of best linear unbiasedness of the ordinary least squares estimates of l and z 4 In the macroeconomics literature there are two competing theories concerning consumption behavior According to Keynes aggregate consumption is determined by aggregate income Alternatively the classical economists feel that consumption should be inversely related to interest rates The results below are based on observations on US consumption expenditures in billions of 1982 dollars disposable personal income in billions of 1982 dollars and the real interest rate for the years 1955 86 Based on the econometric evidence below what are your conclusions relative to the validity of the two hypotheses Keynes Theory An economic model that relates consumption to income using a linear relationship is Q Bl 82M The corresponding statistical model is C 81 BzMz 32 where we will assume the e are independent random variables with e N 16 Classical Economist39s Theory In this case we specify the linear economic model relating consumption to interest rate as Q 001 00sz The corresponding statistical model is C 001 00sz v with the v assumed to be v N 063 and independent The two least squares estimated equations with standard errors in parentheses are C 737744 09085 M C 13707 59859 R 13527 00076 1154 29488