The topic of heteroscedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression as well as time series analysis. These are also known as Eicker–Huber–White standard errors (also Huber–White standard errors or White standard errors), to recognize the contributions of Friedhelm Eicker, Peter J. Huber, and Halbert White.
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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
Definition
Assume that we are studying the linear regression model
where X is the vector of explanatory variables and β is a k × 1 column vector of parameters to be estimated.
The ordinary least squares (OLS) estimator is
where
If the sample errors have equal variance σ2 and are uncorrelated, then the least-squares estimate of β is BLUE (best linear unbiased estimator), and its variance is easily estimated with
where
When the assumptions of
where
While the OLS point estimator remains unbiased, it is not "best" in the sense of having minimum mean square error, and the OLS variance estimator
For any non-linear model (for instance Logit and Probit models), however, heteroscedasticity has more severe consequences: the maximum likelihood estimates of the parameters will be biased (in an unknown direction), as well as inconsistent (unless the likelihood function is modified to correctly take into account the precise form of heteroscedasticity). As pointed out by Greene, “simply computing a robust covariance matrix for an otherwise inconsistent estimator does not give it redemption.”
Eicker's heteroscedasticity-consistent estimator
If the regression errors
where as above
Note that also often discussed in the literature (including in White's paper itself) is the covariance matrix
where,
and
Thus,
and
Precisely which covariance matrix is of concern should be a matter of context.
Alternative estimators have been proposed in MacKinnon & White (1985) that correct for unequal variances of regression residuals due to different leverage. Unlike the asymptotic White's estimator, their estimators are unbiased when the data are homoscedastic.
Software
vcovHC()
command.robust
option applicable in many pseudo-likelihood based procedures.