The topic of heteroskedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis. These are also known as heteroskedasticity-robust standard errors (or simply robust standard errors), 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. In regression and time-series modelling, basic forms of models make use of the assumption that the errors or disturbances ε i {\textstyle \varepsilon _{i}} have the same variance across all observation points. When this is not the case, the errors are said to be heteroskedastic, or to have heteroskedasticity, and this behaviour will be reflected in the residuals ε ^ i {\textstyle {\widehat {\varepsilon }}_{i}} estimated from a fitted model. Heteroskedasticity-consistent standard errors are used to allow the fitting of a model that does contain heteroskedastic residuals. The first such approach was proposed by Huber (1967), and further improved procedures have been produced since for cross-sectional data, time-series data and GARCH estimation. Heteroskedasticity-consistent standard errors that differ from classical standard errors may indicate model misspecification. Substituting heteroskedasticity-consistent standard errors does not resolve this misspecification, which may lead to bias in the coefficients. In most situations, the problem should be found and fixed. Other types of standard error adjustments, such as clustered standard errors or HAC standard errors, may be considered as extensions to HC standard errors.
History Heteroskedasticity-consistent standard errors are introduced by Friedhelm Eicker, and popularized in econometrics by Halbert White.
Problem Consider the linear regression model for the scalar y {\displaystyle y} .
y = x ⊤ β + ε , {\displaystyle y=\mathbf {x} ^{\top }{\boldsymbol {\beta }}+\varepsilon ,\,}
where x {\displaystyle \mathbf {x} } is a k × 1 column vector of explanatory variables (features), β {\displaystyle {\boldsymbol {\beta }}} is a k × 1 column vector of parameters to be estimated, and ε {\displaystyle \varepsilon } is the residual error. The ordinary least squares (OLS) estimator is
β ^ O L S = ( X ⊤ X ) − 1 X ⊤ y . {\displaystyle {\widehat {\boldsymbol {\beta }}}_{\mathrm {OLS} }=(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} .\,}
where y {\displaystyle \mathbf {y} } is a vector of observations y i {\displaystyle y_{i}} , and X {\displaystyle \mathbf {X} } denotes the matrix of stacked x i {\displaystyle \mathbf {x} _{i}} values observed in the data. If the sample errors have equal variance σ 2 {\displaystyle \sigma ^{2}} and are uncorrelated, then the least-squares estimate of β {\displaystyle {\boldsymbol {\beta }}} is BLUE (best linear unbiased estimator), and its variance is estimated with
V ^ [ β ^ O L S ] = s 2 ( X ⊤ X ) − 1 , s 2 = ∑ i = 0 n ε ^ i 2 n − k {\displaystyle {\hat {\mathbb {V} }}\left[{\widehat {\boldsymbol {\beta }}}_{\mathrm {OLS} }\right]=s^{2}(\mathbf {X} ^{\top }\mathbf {X} )^{-1},\quad s^{2}={\frac {\sum _{i=0}^{n}{\widehat {\varepsilon }}_{i}^{2}}{n-k}}}
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