Heteroskedasticity-consistent standard errors

The topic of Heteroscedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and also time series analysis. The alternative names of Huber–White standard errors, Eicker–White or Eicker–Huber–White are also frequently used in relation to the same ideas.

In regression and time-series modelling, basic forms of models make use of the assumption that the errors or disturbances u_i have the same variance across all observation points. When this is not the case, the errors are said to be heteroscedastic, or to have heteroscedasticity, and this behaviour will be reflected in the residuals $\scriptstyle {\widehat {u_{i}}}$ estimated from a fitted model. Heteroscedasticity-consistent standard errors are used to allow the fitting of a model that does contain heteroskedastic residuals. The first such approach was proposed by White (1980), and further improved procedures have been produced since for cross-sectional data, time-series data and GARCH estimation.

Definition

Assume that we are regressing the linear regression model

$y=X\beta +u,$

where X is the design matrix and β is a column vector of parameters to be estimated.

The ordinary least squares (OLS) estimator is

${\widehat {\beta }}=(X'X)^{-1}X'y.$

If the residuals all have the same variance σ² and are uncorrelated, then the least-squares estimates of β satisfy the assumption of being BLUE (best linear unbiased estimator). If they are not BLUE, then suppose they have variances σ_i² and the OLS variance estimator is

${\widehat {\sigma }}^{2}={{{\widehat {u}}'{\widehat {u}}} \over {n-k}},$

where $\scriptstyle {\widehat {u}}\,=\,y-X(X'X)^{-1}X'y.$ There are many different kinds of heteroscedasticity, however, and one should use caution when constructing heteroscedastic robust standard errors.

HC estimators are recommended to deal with this problem.

White's heteroscedasticity-consistent estimator

White's (1980) HC estimator, often referred to as HCE, has the estimator

$E({\widehat {u}}{\widehat {u}}')=\operatorname {diag} ({\widehat {u}}_{1}^{2},{\widehat {u}}_{2}^{2},\dots ,{\widehat {u}}_{n}^{2}).$

The estimator can be derived in terms of the generalized method of moments (GMM).

See also

• Generalized estimating equations
• White test — a test for whether heteroscedasticity is present.

References

• Hayes, Andrew F.; Cai, Li (2007), "Using heteroscedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation", Behavior Research Methods, 37: 709–722

• MacKinnon, James G.; White, Halbert (1985), "Some Heteroskedastic-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties", Journal of Econometrics, 29 (29): 305–325, doi:10.1016/0304-4076(85)90158-7

• White, Halbert (1980), "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity", Econometrica, 48 (4): 817–838, doi:10.2307/1912934, MR 0575027, JSTOR 1912934 {{citation}}: More than one of |author1= and |last= specified (help)

• Greene, William. (1998), "Econometric Analysis", Prentice Hall