������ʦ�ֲ�CHAPTER 8

CHAPTER 8 TEACHING NOTES This is a good place to remind students that homoskedasticity played no role in showing that OLS is unbiased for the parameters in the regression equation. In addition, you probably should mention that there is nothing wrong with the R-squared or adjusted R-squared as goodness-of-fit measures. The key is that these are estimates of the population R-squared, 1 – [Var(u)/Var(y)], where the variances are the unconditional variances in the population. The usual R-squared, and the adjusted version, consistently estimate the population R-squared whether or not Var(u|x) = Var(y|x) depends on x. Of course, heteroskedasticity causes the usual standard errors, t statistics, and F statistics to be invalid, even in large samples, with or without normality. By explicitly stating the homoskedasticity assumption as conditional on the explanatory variables that appear in the conditional mean, it is clear that only heteroskedasticity that depends on the explanatory variables in the model affects the validity of standard errors and test statistics. The version of the Breusch-Pagan test in the text, and the White test, are ideally suited for detecting forms of heteroskedasticity that invalidate inference obtained under homoskedasticity. If heteroskedasticity depends on an exogenous variable that does not also appear in the mean equation, this can be exploited in weighted least squares for efficiency, but only rarely is such a variable available. One case where such a variable is available is when an individual-level equation has been aggregated. I discuss this case in the text but I rarely have time to teach it. As I mention in the text, other traditional tests for heteroskedasticity, such as the Park and Glejser tests, do not directly test what we want, or add too many assumptions under the null. The Goldfeld-Quandt test only works when there is a natural way to order the data based on one independent variable. This is rare in practice, especially for cross-sectional applications. Some argue that weighted least squares estimation is a relic, and is no longer necessary given the availability of heteroskedasticity-robust standard errors and test statistics. While I am sympathetic to this argument, it presumes that we do not care much about efficiency. Even in large samples, the OLS estimates may not be precise enough to learn much about the population parameters. With substantial heteroskedasticity we might do better with weighted least squares, even if the weighting function is misspecified. As discussed in the text on pages 288-289, one can, and probably should, compute robust standard errors after weighted least squares. For asymptotic efficiency comparisons, these would be directly comparable to the heteroskedasiticity-robust standard errors for OLS. Weighted least squares estimation of the LPM is a nice example of feasible GLS, at least when all fitted values are in the unit interval. Interestingly, in the LPM examples in the text and the LPM computer exercises, the heteroskedasticity-robust standard errors often differ by only small amounts from the usual standard errors. However, in a couple of cases the differences are notable, as in Computer Exercise C8.7. SOLUTIONS TO PROBLEMS 8.1 Parts (ii) and (iii). The homoskedasticity assumption played no role in Chapter 5 in showing that OLS is consistent. But we know that heteroskedasticity causes statistical inference based on the usual t and F statistics to be invalid, even in large samples. As heteroskedasticity is a violation of the Gauss-Markov assumptions, OLS is no longer BLUE. 8.2 With Var(u|inc,price,educ,female) = (2inc2, h(x) = inc2, where h(x) is the hetero​skedas​ti​city function defined in equation (8.21). Therefore, = inc, and so the transformed equation is obtained by dividing the original equation by inc: Notice that , which is the slope on inc in the original model, is now a constant in the transformed equation. This is simply a consequence of the form of the heteroskedasticity and the functional forms of the explanatory variables in the original equation. 8.3 False. The unbiasedness of WLS and OLS hinges crucially on Assumption MLR.4, and, as we know from Chapter 4, this assumption is often violated when an important variable is omitted. When MLR.4 does not hold, both WLS and OLS are biased. Without specific information on how the omitted variable is correlated with the included explanatory variables, it is not possible to determine which estimator has a small bias. It is possible that WLS would have more bias than OLS or less bias. Because we cannot know, we should not claim to use WLS in order to solve “biases” associated with OLS. 8.4 (i) These coefficients have the anticipated signs. If a student takes courses where grades are, on average, higher – as reflected by higher crsgpa – then his/her grades will be higher. The better the student has been in the past – as measured by cumgpa – the better the student does (on average) in the current semester. Finally, tothrs is a measure of experience, and its coefficient indicates an increasing return to experience. The t statistic for crsgpa is very large, over five using the usual standard error (which is the largest of the two). Using the robust standard error for cumgpa, its t statistic is about 2.61, which is also significant at the 5% level. The t statistic for tothrs is only about 1.17 using either standard error, so it is not significant at the 5% level. (ii) This is easiest to see without other explanatory variables in the model. If crsgpa were the only explanatory variable, H0: = 1 means that, without any information about the student, the best predictor of term GPA is the average GPA in the students’ courses; this holds essentially by definition. (The intercept would be zero in this case.) With additional explanatory variables it is not necessarily true that = 1 because crsgpa could be correlated with characteristics of the student. (For example, perhaps the courses students take are influenced by ability – as measured by test scores – and past college performance.) But it is still interesting to test this hypothesis. The t statistic using the usual standard error is t = (.900 – 1)/.175  (.57; using the hetero​skedasticity-robust standard error gives t (.60. In either case we fail to reject H0: = 1 at any reasonable significance level, certainly including 5%. (iii) The in-season effect is given by the coefficient on season, which implies that, other things equal, an athlete’s GPA is about .16 points lower when his/her sport is competing. The t statistic using the usual standard error is about –1.60, while that using the robust standard error is about –1.96. Against a two-sided alternative, the t statistic using the robust standard error is just significant at the 5% level (the standard normal critical value is 1.96), while using the usual standard error, the t statistic is not quite significant at the 10% level (cv 1.65). So the standard error used makes a difference in this case. This example is somewhat unusual, as the robust standard error is more often the larger of the two. 8.5 (i) No. For each coefficient, the usual standard errors and the heteroskedasticity-robust ones are practically very similar. (ii) The effect is (.029(4) = (.116, so the probability of smoking falls by about .116. (iii) As usual, we compute the turning point in the quadratic: .020/[2(.00026)] 38.46, so about 38 and one-half years. (iv) Holding other factors in the equation fixed, a person in a state with restaurant smoking restrictions has a .101 lower chance of smoking. This is similar to the effect of having four more years of education. (v) We just plug the values of the independent variables into the OLS regression line: Thus, the estimated probability of smoking for this person is close to zero. (In fact, this person is not a smoker, so the equation predicts well for this particular observation.) 8.6 (i) The proposed test is a hybrid of the BP and White tests. There are k + 1 regressors, each original explanatory variable and the squared fitted values. So, the number of restrictions tested is k + 1, and this is the numerator df. The denominator df is n ( (k + 2) = n ( k ( 2. (ii) For the BP test, this is easy: the hybrid test has an extra regressor, , and so the R-squared will be no less for the hybrid test than for the BP test. For the special case of the White test, the argument is a bit more subtle. In regression (8.20), the fitted values are a linear function of the regressors (where, of course, the coefficients in the linear function are the OLS estimates). So, we are putting a restriction on how the original explanatory variables appear in the regression. This means that the R-squared from (8.20) will be no greater than the R-squared from the hybrid regression. (iii) No. The F statistic for joint significance of the regressors depends on , and it is true that this ratio increases as increases. But, the F statistic also depends on the df, and the df are different among all three tests: the BP test, the special case of the White test, and the hybrid test. So we do not know which test will deliver the smallest p-value. (iv) As discussed in part (ii), the OLS fitted values are a linear combination of the original regressors. Because those regressors appear in the hybrid test, adding the OLS fitted values is redundant; perfect collinearity would result. 8.7 (i) This follows from the simple fact that, for uncorrelated random variables, the variance of the sum is the sum of the variances: . (ii) We compute the covariance between any two of the composite errors as where we use the fact that the covariance of a random variable with itself is its variance and the assumptions that are pairwise uncorrelated. (iii) This is most easily solved by writing Now, by assumption, fi is uncorrelated with each term in the last sum; therefore, fi is uncorrelated with . It follows that where we use the fact that the variance of an average of mi uncorrelated random variables with common variance ( in this case) is simply the common variance divided by mi – the usual formula for a sample average from a random sample. (iv) The standard weighting ignores the variance of the firm effect, . Thus, the (incorrect) weight function used is . A valid weighting function is obtained by writing the variance from (iii) as But obtaining the proper weights requires us to know (or be able to estimate) the ratio . Estimation is possible, but we do not discuss that here. In any event, the usual weight is incorrect. When the mi are large or the ratio is small – so that the firm effect is more important than the individual-specific effect – the correct weights are close to being constant. Thus, attaching large weights to large firms may be quite inappropriate. SOLUTIONS TO COMPUTER EXERCISES C8.1 (i) Given the equation the assumption that the variance of u given all explanatory variables depends only on gender is Then the variance for women is simply and that for men is + ; the difference in variances is (1. (ii) After estimating the above equation by OLS, we regress on malei, i = 1,2, ,706 (including, of course, an intercept). We can write the results as = 189,359.2 – 28,849.6 male + residual (20,546.4) (27,296.5) n = 706, R2 = .0016. Because the coefficient on male is negative, the estimated variance is higher for women. (iii) No. The t statistic on male is only about –1.06, which is not significant at even the 20% level against a two-sided alternative. C8.2 (i) The estimated equation with both sets of standard errors (heteroske​dasticity-robust standard errors in parentheses) is = (21.77 + .00207 lotsize + .123 sqrft + 13.85 bdrms (29.48) (.00064) (.013) (9.01) [36.28] [.00122] [.017] [8.28] n  = 88, R2 = .672. The robust standard error on lotsize is almost twice as large as the usual standard error, making lotsize much less significant (the t statistic falls from about 3.23 to 1.70). The t statistic on sqrft also falls, but it is still very significant. The variable bdrms actually becomes somewhat more significant, but it is still barely significant. The most important change is in the significance of lotsize. (ii) For the log-log model, = 5.61 + .168 log(lotsize) + .700 log(sqrft) + .037 bdrms (0.65) (.038) (.093) (.028) [0.76] [.041] [.101] [.030] n  = 88, R2 = .643. Here, the heteroskedasticity-robust standard error is always slightly greater than the corresponding usual standard error, but the differences are relatively small. In particular, log(lotsize) and log(sqrft) still have very large t statistics, and the t statistic on bdrms is not significant at the 5% level against a one-sided alternative using either standard error. (iii) As we discussed in Section 6.2, using the logarithmic transformation of the dependent variable often mitigates, if not entirely eliminates, heteroskedasticity. This is certainly the case here, as no important conclusions in the model for log(price) depend on the choice of standard error. (We have also transformed two of the independent variables to make the model of the constant elasticity variety in lotsize and sqrft.) C8.3 After estimating equation (8.18), we obtain the squared OLS residuals . The full-blown White test is based on the R-squared from the auxiliary regression (with an intercept), on llotsize, lsqrft, bdrms, llotsize2, lsqrft2, bdrms2, llotsize lsqrft, llotsize bdrms, and lsqrft bdrms, where “l ” in front of lotsize and sqrft denotes the natural log. [See equation (8.19).] With 88 observations the n-R-squared version of the White statistic is 88(.109) 9.59, and this is the outcome of an (approximately) random variable. The p-value is about .385, which provides little evidence against the homoskedasticity assumption. C8.4 (i) The estimated equation is = 37.66 + .252 prtystrA + 3.793 democA + 5.779 log(expendA) (4.74) (.071) (1.407) (0.392) ( 6.238 log(expendB) + (0.397) n = 173, R2 = .801, = .796. You can convince yourself that regressing the on all of the explanatory variables yields an R-squared of zero, although it might not be exactly zero in your computer output due to rounding error. Remember, OLS works by choosing the estimates, , such that the residuals are uncorrelated in the sample with each independent variable (and the residuals have a zero sample average, too). (ii) The B-P test entails regressing the on the independent variables in part (i). The F statistic for joint significant (with 4 and 168 df) is about 2.33 with p-value .058. Therefore, there is some evidence of heteroskedasticity, but not quite at the 5% level. (iii) Now we regress on and ( )2, where the are the OLS fitted values from part (i). The F test, with 2 and 170 df, is about 2.79 with p-value .065. This is slightly less evidence of heteroskedasticity than provided by the B-P test, but the conclusion is very similar. C8.5 (i) By regressing sprdcvr on an intercept only we obtain EMBED Equation.DSMT4 .515 se .021). The asymptotic t statistic for H0: µ = .5 is (.515 ( .5)/.021 .71, which is not significant at the 10% level, or even the 20% level. (ii) 35 games were played on a neutral court. (iii) The estimated LPM is = .490 + .035 favhome + .118 neutral ( .023 fav25 + .018 und25 (.045) (.050) (.095) (.050) (.092) n = 553, R2  = .0034. The variable neutral has by far the largest effect – if the game is played on a neutral court, the probability that the spread is covered is estimated to be about .12 higher – and, except for the intercept, its t statistic is the only t statistic greater than one in absolute value (about 1.24). (iv) Under H0: = = = = 0, the response probability does not depend on any explanatory variables, which means neither the mean nor the variance depends on the explanatory variables. [See equation (8.38).] (v) The F statistic for joint significance, with 4 and 548 df, is about .47 with p-value .76. There is essentially no evidence against H0. (vi) Based on these variables, it is not possible to predict whether the spread will be covered. The explanatory power is very low, and the explanatory variables are jointly very insignificant. The coefficient on neutral may indicate something is going on with games played on a neutral court, but we would not want to bet money on it unless it could be confirmed with a separate, larger sample. C8.6 (i) The estimates are given in equation (7.31). Rounded to four decimal places, the smallest fitted value is .0066 and the largest fitted value is .5577. (ii) The estimated heteroskedasticity function for each observation i is , which is strictly between zero and one because 0 < < 1 for all i. The weights for WLS are 1/ . To show the WLS estimate of each parameter, we report the WLS results using the same equation format as for OLS: = .448 ( .168 pcnv + .0054 avgsen ( .0018 tottime ( .025 ptime86 (.018) (.019) (.0051) (.0033) (.003) ( .045 qemp86 (.005) n = 2,725, R2 = .0744. The coefficients on the significant explanatory variables are very similar to the OLS estimates. The WLS standard errors on the slope coefficients are generally lower than the nonrobust OLS standard errors. A proper comparison would be with the robust OLS standard errors. (iii) After WLS estimation, the F statistic for joint significance of avgsen and tottime, with 2 and 2,719 df, is about .88 with p-value .41. They are not close to being jointly significant at the 5% level. If your econometrics package has a command for WLS and a test command for joint hypotheses, the F statistic and p-value are easy to obtain. Alternatively, you can obtain the restricted R-squared using the same weights as in part (ii) and dropping avgsen and tottime from the WLS estimation. (The unrestricted R-squared is .0744.) C8.7 (i) The heteroskedasticity-robust standard error for EMBED Equation.DSMT4 .129 is about .026, which is notably higher than the nonrobust standard error (about .020). The heteroskedasticity-robust 95% confidence interval is about .078 to .179, while the nonrobust CI is, of course, narrower, about .090 to .168. The robust CI still excludes the value zero by some margin. (ii) There are no fitted values less than zero, but there are 231 greater than one. Unless we do something to those fitted values, we cannot directly apply WLS, as will be negative in 231 cases. C8.8 (i) The equation estimated by OLS is 1.36 + .412 hsGPA + .013 ACT ( .071 skipped + .124 PC (.33) (.092) (.010) (.026) (.057) n = 141, R2 = .259, (ii) The F statistic obtained for the White test is about 3.58. With 2 and 138 df, this gives p-value ( .031. So, at the 5% level, we conclude there is evidence of heteroskedasticity in the errors of the colGPA equation. (As an aside, note that the t statistics for each of the terms is very small, and we could have simply dropped the quadratic term without losing anything of value.) (iii) In fact, the smallest fitted value from the regression in part (ii) is about .027, while the largest is about .165. Using these fitted values as the in a weighted least squares regression gives the following: 1.40 + .402 hsGPA + .013 ACT ( .076 skipped + .126 PC (.30) (.083) (.010) (.022) (.056) n = 141, R2 = .306, There is very little difference in the estimated coefficient on PC, and the OLS t statistic and WLS t statistic are also very close. Note that we have used the usual OLS standard error, even though it would be more appropriate to use the heteroskedasticity-robust form (since we have evidence of heteroskedasticity). The R-squared in the weighted least squares estimation is larger than that from the OLS regression in part (i), but, remember, these are not comparable. (iv) With robust standard errors – that is, with standard errors that are robust to misspecifying the function h(x) – the equation is 1.40 + .402 hsGPA + .013 ACT ( .076 skipped + .126 PC (.31) (.086) (.010) (.021) (.059) n = 141, R2 = .306, The robust standard errors do not differ by much from those in part (iii); in most cases, they are slightly higher, but all explanatory variables that were statistically significant before are still statistically significant. But the confidence interval for (PC is a bit wider. C8.9 (i) I now get R2 = .0527, but the other estimates seem okay. (ii) One way to ensure that the unweighted residuals are being provided is to compare them with the OLS residuals. They will not be the same, of course, but they should not be wildly different. (iii) The R-squared from the regression is about .027. We use this as in equation (8.15) but with k = 2. This gives F = 11.15, and so the p-value is essentially zero. (iv) The substantial heteroskedasticity found in part (iii) shows that the feasible GLS procedure described on page 279 does not, in fact, eliminate the heteroskedasticity. Therefore, the usual standard errors, t statistics, and F statistics reported with weighted least squares are not valid, even asymptotically. (v) Weighted least squares estimation with robust standard errors gives = 5.64 + 1.30 log(income) ( 2.94 log(cigpric) ( .463 educ (37.31) (.54) (8.97) (.149) + .482 age ( .0056 age2 ( 3.46 restaurn (.115) (.0012) (.72) n = 807, R2 = .1134 The substantial differences in standard errors compared with equation (8.36) further indicate that our proposed correction for heteroskedasticity did not fully solve the heteroskedasticity problem. With the exception of restaurn, all standard errors got notably bigger; for example, the standard error for log(cigpric) doubled. All variables that were statistically significant with the nonrobust standard errors remain significant, but the confidence intervals are much wider in several cases. [ Instructor’s Note: You can also do this exercise with regression (8.34) used in place of (8.32). This gives a somewhat larger estimated income effect.] C8.10 (i) In the following equation, estimated by OLS, the usual standard errors are in (() and the heteroskedasticity-robust standard errors are in [(]: = (.506 + .0124 inc ( .000062 inc2 + .0265 age ( .00031 age2 ( .0035 male (.081) (.0006) (.000005) (.0039) (.00005) (.0121) [.079] [.0006] [.000005] [.0038] [.00004] [.0121] n = 9,275, R2 = .094. There are no important differences; if anything, the robust standard errors are smaller. (ii) This is a general claim. Since Var(y|x) = , we can write . Written in error form, . In other words, we can write this as a regression model , with the restrictions (0 = 0, (1 = 1, and (2 = -1. Remember that, for the LPM, the fitted values, , are estimates of . So, when we run the regression (including an intercept), the intercept estimates should be close to zero, the coefficient on should be close to one, and the coefficient on should be close to –1. (iii) The White F statistic is about 310.32, which is very significant. The coefficient on is about 1.010, the coefficient on is about (.970, and the intercept is about -.009. These estimates are quite close to what we expect to find from the theory in part (ii). (iv) The smallest fitted value is about .030 and the largest is about .697. The WLS estimates of the LPM are = (.488 + .0126 inc ( .000062 inc2 + .0255 age ( .00030 age2 ( .0055 male (.076) (.0005) (.000004) (.0037) (.00004) (.0117) n = 9,275, R2 = .108. There are no important differences with the OLS estimates. The largest relative change is in the coefficient on male, but this variable is very insignificant using either estimation method. C8.11 (i) The usual OLS standard errors are in ((), the heteroskedasticity-robust standard errors are in [(]: = 1.50 + .774 inc ( 1.60 age + .029 age2 + 2.47 male + 6.98 e401k (15.31) (.062) (0.77) (.009) (2.05) (2.13) [19.09] [.102] [1.08] [.014] [2.06] [2.19] n = 2,017, R2 = .128. (ii) The smallest is about 12.83 and the largest is about 58,059.74. Thus, there is wide variation in the estimated conditional variances. (iii) The usual WLS standard errors are in ((), the standard errors robust to misspecified variance are in [(]: = (2.58 + .456 inc ( .613 age + .013 age2 + 1.42 male + 4.26 e401k (9.94) (.058) (.541) (.007) (1.03) (1.23) [8.19] [.062] [.408] [.005] [0.82] [1.14] n = 2,017, R2 = .062. Interestingly, except for the income coefficient, the robust standard errors are actually smaller than the usual standard error. This could just be sampling variation, or it could be that the variance function is misspecified in such a way that, when it is used in WLS, the usual standard errors overestimate the actual sampling variation. (iv) The robust standard error for the e401k coefficient is 2.19, while that for WLS is 1.14. Thus, the WLS standard error is just over half as large as the OLS standard error. Assuming that the zero conditional mean assumption actually holds – something that is not clear given some nontrivial changes in the WLS estimates as compared with OLS – the smaller robust standard error for WLS suggests it is the more efficient procedure, whether or not we have properly specified the “skedastic” function. 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