Heteroscedasticity-consistent standard errors

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

: y = X eta + 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 eta = (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"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'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.

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.

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


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