Improving weighted least squares inference

In the presence of conditional heteroskedasticity, inference about the coefficients in a linear regression model these days is typically based on the ordinary least squares estimator in conjunction with using heteroskedasticity consistent standard errors. Similarly, even when the true form of heteroskedasticity is unknown, heteroskedasticity consistent standard errors can be used to base valid inference on a weighted least squares estimator. Using a weighted least squares estimator can provide large gains in efficiency over the ordinary least squares estimator. However, intervals based on plug-in standard errors often have coverage that is below the nominal level, especially for small sample sizes. In this paper, it is shown that a bootstrap approximation to the sampling distribution of the weighted least squares estimate is valid, which allows for inference with improved finite-sample properties. Furthermore, when the model used to estimate the unknown form of the heteroskedasticity is misspecified, the weighted least squares estimator may be less efficient than the ordinary least squares estimator. To address this problem, a new estimator is proposed that is asymptotically at least as efficient as both the ordinary and the weighted least squares estimator. Simulation studies demonstrate the attractive finite-sample properties of this new estimator as well as the improvements in performance realized by bootstrap confidence intervals.