Heteroscedasticity-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[1] 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 ui 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 heteroscedastic 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.

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Definition

Assume that we are regressing the linear regression model

y = X \beta %2B 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 sample errors have equal variance σ2 and are uncorrelated, then the least-squares estimates of β is called BLUE (best linear unbiased estimator).

However, all the necessary assumptions may not be valid. For example, suppose the errors have distinct variances σi2 and the OLS variance estimator is

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

where

\widehat{u}\, =\, y - X (X'X)^{-1} X'y,

then OLS estimator is not "best" in the sense of having minimum mean square error (but it is unbiased) and the OLS variance estimator does not provide a consistent estimate of the variance of the residuals.

There are many different kinds of heteroscedasticity, however, and one should use caution when constructing heteroscedastic robust standard errors. HC (heteroscedasticity-consistent) 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}^2_1, \widehat{u}^2_2, \dots , \widehat{u}^2_n). \,

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

See also

References

  1. ^ Kleiber, C., Zeileis, A (2006) Applied Econometrics with R, UseR-2006 conference