  When independent errors in a linear model have non-identity covariance, the
ordinary least squares estimate of the model coefficients is less efficient
than the weighted least squares estimate. However, the practical application of
weighted least squares is challenging due to its reliance on the unknown error
covariance matrix. Although feasible weighted least squares estimates, which
use an approximation of this matrix, often outperform the ordinary least
squares estimate in terms of efficiency, this is not always the case. In some
situations, feasible weighted least squares can be less efficient than ordinary
least squares. The comparison between these two estimates has significant
implications for the application of regression analysis in varied fields, yet
such a comparison remains an unresolved challenge despite its seemingly
straightforward nature. In this study, we directly address this challenge by
identifying the conditions under which feasible weighted least squares
estimates using fixed weights demonstrate greater efficiency than the ordinary
least squares 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.
