Subscedastic weighted least squares estimates

TL;DR

This paper investigates when feasible weighted least squares (FLS) estimates outperform ordinary least squares (OLS) in heteroscedastic linear models, providing conditions for their relative efficiency and insights into robust regression behavior.

Contribution

It identifies conditions under which FLS with fixed weights is more efficient than OLS, guiding practical regression analysis with unknown error variances.

Findings

FLS can outperform OLS under specific variance conditions.

Conditions for FLS efficiency depend on error variance structure.

Insights into robust regression behavior with heteroscedastic errors.

Abstract

In the heteroscedastic linear model, the weighted least squares (WLS) estimate of the model coefficients is more efficient than the ordinary least squares (OLS) esti- mate. However, the practical application of WLS is challenging because it requires knowledge of the error variances. Feasible weighted least squares (FLS) estimates, which use approximations of the variances when they are unknown, may either be more or less efficient than the OLS estimate depending on the quality of the approx- imation. A direct comparison between FLS and OLS has significant implications for the application of regression analysis in varied fields, yet such a comparison remains an unresolved challenge. In this study, we address this challenge by identifying the conditions under which FLS estimates using fixed weights demonstrate greater effi- ciency than the OLS estimate. These conditions provide guidance for the design of feasible estimates using random weights. They also shed light on how certain robust regression estimates behave with respect to the linear model with normal errors of unequal variance.