HC0

HC0 is the original heteroskedasticity-robust covariance estimator used to obtain robust standard errors in linear regression when error variance is not constant. It is part of the class of heteroskedasticity-consistent covariance matrix estimators (HCCME) introduced to provide valid inference under heteroskedasticity.

Formally, let X be the n×k design matrix, β̂ the OLS estimator of the coefficients, and e the vector of residuals from the regression. The HC0 estimator of the covariance matrix of β̂ is V̂(HC0) = (X'X)^{-1} X' Diag(e_i^2) X (X'X)^{-1}, where Diag(e_i^2) is the diagonal matrix with the squared residuals on its diagonal. This estimator does not adjust for leverage or degrees of freedom.

HC0 is the simplest version and can be biased in small samples. Several variants exist to address such issues. HC1 multiplies HC0 by a degrees-of-freedom factor, typically n/(n−p), to reduce small-sample bias. HC2 and HC3 introduce leverage-based adjustments: they replace e_i^2 with e_i^2/(1−h_i) and e_i^2/(1−h_i)^2, respectively, where h_i is the i-th diagonal element of the hat matrix. These adjustments aim to stabilize the influence of high-leverage observations.

In practice, HC0 serves as a baseline robust method for inference under heteroskedasticity. HC1 is often preferred in small samples, while HC2 and HC3 provide more refined adjustments for high-leverage points. Implementations appear in various econometrics and statistics software, including packages that offer heteroskedasticity-robust covariance types.