Heteroscedasticity-consistent standard errors

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 $scriptstylewidehat\{u\_i\}$. 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.

Definition

Assume that we are regressing the linear regression model

:

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

The ordinary least squares (OLS) estimator is

:

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_"i"² and the OLS variance estimator is

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

where $scriptstyle\; widehat\{u\},\; =,\; y\; -\; X\; (X\text{'}X)^\{-1\}\; X\text{'}y.$ 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.

White's heteroskedasticity-consistent estimator

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

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

The estimator can be derived in terms of GMM.

References

Citation
last = Hayes
first = Andrew F.
last2 = Cai
first2 = Li
title = Using heteroscedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation
journal = Behavior Research Methods
volume = 37
number =
pages = 709--722
url = http://www.comm.ohio-state.edu/ahayes/SPSS%20programs/HCSEp.htm
year = 2007

Citation
last = MacKinnon
first = James, G.
author-link =
last2 = White
first2 = Halbert
author2-link =
title = Some Heteroskedastic-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties
journal = Journal of Econometrics
volume =
issue = 29
pages = 305-325
date =
year = 1985
url =
doi =
id =

Citation
last = White
first =
last2 = Halbert
first2 =
title = A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity
journal = Econometrica
volume = 48
number = 4
pages = 817--838
url = http://www.jstor.org/stable/1912934
year = 1980