A note on comparing the unrestricted and restricted least-squares estimators

Abstract

The problem of comparing the ordinary least-squares estimator β̂ and the restricted least-squares estimator β∗ with respect to a weighted quadratic risk under the normal linear regression model in which restrictions are not known to be true is approached in the literature either (i) by choosing the weight matrix in a special form such that the corresponding necessary and sufficient conditions become operational or (ii) by replacing these necessary and sufficient conditions with weaker sufficient conditions, which may lead, however, to a region of uncertainty where neither dominance of β∗ over β̂ nor dominance of β̂ over β∗ is demonstrable. This note shows that the two approaches are partially reconciliable, in the sense that the class of weight matrices which are appropriate for approach (i) coincides with the class of weight matrices for which the region of uncertainty disappears in approach (ii). Moreover, a number of alternative characterizations of this class are given, and a useful invariance property is established.