Heteroskedastic linear regression: steps towards adaptivity, efficiency, and robustness

In linear regression with heteroscedastic errors, the Generalized Least Squares (GLS) estimator is optimal, i.e., it is the Best Linear Unbiased Estimator (BLUE). The Ordinary Least Squares (OLS) estimator is suboptimal but still valid, i.e., unbiased and consistent. Halbert White, in his seminal paper (Econometrica, 1980) used the OLS residuals in order to obtain an estimate of the standard error of the OLS estimator under an unknown structure of the underlying heteroscedasticity. The GLS estimator similarly depends on the unknown heteroscedasticity, and is thus intractable. In this paper, we introduce two different approximations to the optimal GLS estimator; the starting point for both approaches is in the spirit of White’s correction, i.e., using the OLS residuals to get a rough estimate of the underlying heteroscedasticity. We show how the new estimators can benefit from the Wild Bootstrap both in terms of optimising them, but also in terms of providing valid standard errors for them despite their complicated construction. The performance of the new estimators is compared via simulations to the OLS and to the exact (but intractable) GLS.