This paper enlarges on results of Wan and Zou [Journal of Econometrics 114 (2003), 165-96] on the choice of critical values for pre-test procedures based on the minimum risk criterion. We consider a modi. cation of the general theorem given in Wan and Zou (2003) to obtain the optimal critical value that minimizes the risks of various inequality pre-test estimators of the regression error variance under a general class of first-order differentiable loss functions. Theoretical proofs of earlier numerical results are provided. This paper also presents results on the optimal pre-test critical values for the simultaneous estimation of the error variance and coefficient vector.