Heteroscedasticity-consistent standard errors

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In statistics, a frequent assumption in linear regression is that the disturbances u_i have the same variance. When this is not the case, we get heteroskedasticity in the estimated residuals $\scriptstyle\widehat{u_i}$ . Heteroskedasticity-consistent (HC) standard errors are used to dealing with this problem by producing more normally-distributed standard errors. The first model was proposed by White (1980), and further improved models have been produced since for cross-sectional data, time-series data and GARCH estimation.

[edit] Definition

Assume that we are regressing the linear regression model

y = X β + 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. 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 kinds of heteroskedasticity and imagination is the only limit to think of what type is possible.

HC estimators are recommended to deal with this problem.

[edit] White's heteroskedasticity-consistent estimator

White's (1980) HC estimator, often referred to as HC0, has the estimator

$E(\widehat{u} \widehat u') = \operatorname{diag}(\widehat{u}^2_1, \widehat{u}^2_2, \dots , \widehat{u}^2_n).$

The estimator can be derived in terms of GMM.

[edit] 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, <http://www.comm.ohio-state.edu/ahayes/SPSS%20programs/HCSEp.htm>

MacKinnon, James, G. & White, Halbert, “Some Heteroskedastic-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties”, Journal of Econometrics (no. 29): 305-325

White & Halbert (1980), “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity”, Econometrica 48 (4): 817--838, <http://www.jstor.org/stable/1912934>