heteroscedasticity-consistent standard errors 1 heteroscedasticity causes ols standard errors to be...

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.

iii uXY 21

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.


iiuab 2OLS2

iii uXY 21

2XX

XXa

j

ii

3

We have also seen that the variance of the estimator is given by the expression above if ui is distributed independently of uj for j i.


iiuab 2OLS2

22222OLS2 iuiiib

auEa

iii uXY 21

2XX

XXa

j

ii

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.

2OLS2b

2

iu


iiuab 2OLS2

22222OLS2 iuiiib

auEa

222OLS2

iibeas

iii uXY 21

2XX

XXa

j

ii

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.

2OLS2b


iiuab 2OLS2

22222OLS2 iuiiib

auEa

222OLS2

iibeas

iii uXY 21

2XX

XXa

j

ii

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.

2OLS2b


iiuab 2OLS2

22222OLS2 iuiiib

auEa

2XX

XXa

j

ii

222OLS2

iibeas

iii uXY 21

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.

2OLS2b


. 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, robustRegression 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

The point estimates of the coefficients are exactly the same. They are not affected by the procedure, and so their inefficiency is not alleviated.

2OLS2b




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.

2OLS2b




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/.

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

or the University of London International Programmes distance learning course

EC2020 Elements of Econometrics

www.londoninternational.ac.uk/lse.

2012.11.10

http://www.oup.com/uk/orc/bin/9780199567089/

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

heteroscedasticity-consistent standard errors 1 heteroscedasticity causes ols standard errors to be...

Documents