Optimal critical values of pre-tests when estimating the regression error variance: analytical findings under a general loss structure

This paper re-visits the problem of estimating the regression error variance in a linear multiple regression model after preliminary hypothesis tests for either linear restrictions on the coefficients or homogeneity of variances. There is an extensive literature that discusses these problems, particularly in terms of the sampling properties of the pre-test estimators using various loss functions as the basis for risk analysis. In this paper, a unified framework for analysing the risk properties of these estimators is developed under a general class of loss structures that incorporates virtually all first-order differentiable losses. Particular consideration is given to the choice of critical values for the pre-tests. Analytical results indicate that an α-level substantially higher than those normally used may be appropriate for optimal risk properties under a wide range of loss functions. The paper also generalizes some known analytical results in the pre-test literature and proves other results only previously shown numerically.