Heteroscedasticity-consistent Standard Errors - Definition

Definition

Assume that we are regressing the linear regression model


y = X \beta + u, \,

where X is the design matrix and β is a k × 1 column vector of parameters to be estimated.

The ordinary least squares (OLS) estimator is


\widehat \beta_{OLS} = (X' X)^{-1} X' y. \,

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 are regression residuals.

When the assumptions of are violated, the OLS estimator loses its desirable properties. Indeed,

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 does not provide a consistent estimate of the variance of the OLS estimates.

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