When the choice of estimator for the coefficients in a linear regression model is determined by the outcome of a prior test of the validity of restrictions on the model, Brook (1976) has shown that a mini-max (risk) regret criterion leads to the simple rule that the optimal critical value for the preliminary test is approximately two in value, regardless of the degrees of freedom. We show that this result no longer holds in the (likely) event that relevant regressors are excluded from the model at the outset.