1. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS 1 Heteroscedasticity causes OLS standard errors to be biased is finite samples. However it can be demonstrated that they are nevertheless consistent, provided that their variances are distributed independently of the regressors.

2. 2 Even if this is not the case, it is still possible to obtain consistent estimators. We have seen that the slope coefficient in a simple OLS regression could be decomposed as above. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS

3. 3 We have also seen that the variance of the estimator is given by the expression above if u i is distributed independently of u j for j i. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS

4. 4 White (1980) demonstrates that a consistent estimator of is obtained if the squared residual in observation i is used as an estimator of. Taking the square root, one obtains a heteroscedasticity-consistent standard error. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS

5. 5 Thus in a situation where heteroscedasticity is suspected, but there is not enough information to identify its nature, it is possible to overcome the problem of biased standard errors, at least in large samples, and the t tests and F tests are asymptotically valid. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS

6. 6 Two points need to be kept in mind, however. One is that, although the White estimator is consistent, it may not perform well in finite samples (MacKinnon and White, 1985). The other is that the OLS point estimates are not affected and so remain inefficient. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS

7. 7 To illustrate the use of heteroscedasticity-consistent standard errors, the regression of MANU on GDP in the previous sequence is repeated with the ‘robust’ option available in Stata. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS. reg manu gdp Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 1, 26) = 210.73 Model | 1.1600e+11 1 1.1600e+11 Prob > F = 0.0000 Residual | 1.4312e+10 26 550462775 R-squared = 0.8902 -------------+------------------------------ Adj R-squared = 0.8859 Total | 1.3031e+11 27 4.8264e+09 Root MSE = 23462 ------------------------------------------------------------------------------ manu | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.193693.0133428 14.52 0.000.1662665.2211195 _cons | 603.9453 5699.677 0.11 0.916 -11111.91 12319.8. reg manu gdp, robust Regression with robust standard errors Number of obs = 28 F( 1, 26) = 116.39 Prob > F = 0.0000 R-squared = 0.8902 Root MSE = 23462 ------------------------------------------------------------------------------ | Robust manu | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.193693.0179542 10.79 0.000.1567877.2305983 _cons | 603.9453 3542.388 0.17 0.866 -6677.538 7885.429

8. 8 The point estimates of the coefficients are exactly the same. They are not affected by the procedure, and so their inefficiency is not alleviated. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS. reg manu gdp Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 1, 26) = 210.73 Model | 1.1600e+11 1 1.1600e+11 Prob > F = 0.0000 Residual | 1.4312e+10 26 550462775 R-squared = 0.8902 -------------+------------------------------ Adj R-squared = 0.8859 Total | 1.3031e+11 27 4.8264e+09 Root MSE = 23462 ------------------------------------------------------------------------------ manu | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.193693.0133428 14.52 0.000.1662665.2211195 _cons | 603.9453 5699.677 0.11 0.916 -11111.91 12319.8. reg manu gdp, robust Regression with robust standard errors Number of obs = 28 F( 1, 26) = 116.39 Prob > F = 0.0000 R-squared = 0.8902 Root MSE = 23462 ------------------------------------------------------------------------------ | Robust manu | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.193693.0179542 10.79 0.000.1567877.2305983 _cons | 603.9453 3542.388 0.17 0.866 -6677.538 7885.429

9. 9 However, the standard error of the coefficient of GDP rises from 0.13 to 0.18, indicating that it is underestimated in the original OLS regression. HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS. reg manu gdp Source | SS df MS Number of obs = 28 -------------+------------------------------ F( 1, 26) = 210.73 Model | 1.1600e+11 1 1.1600e+11 Prob > F = 0.0000 Residual | 1.4312e+10 26 550462775 R-squared = 0.8902 -------------+------------------------------ Adj R-squared = 0.8859 Total | 1.3031e+11 27 4.8264e+09 Root MSE = 23462 ------------------------------------------------------------------------------ manu | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.193693.0133428 14.52 0.000.1662665.2211195 _cons | 603.9453 5699.677 0.11 0.916 -11111.91 12319.8. reg manu gdp, robust Regression with robust standard errors Number of obs = 28 F( 1, 26) = 116.39 Prob > F = 0.0000 R-squared = 0.8902 Root MSE = 23462 ------------------------------------------------------------------------------ | Robust manu | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gdp |.193693.0179542 10.79 0.000.1567877.2305983 _cons | 603.9453 3542.388 0.17 0.866 -6677.538 7885.429

10. Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 7.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.10