The Geometry of the Wald Test

The issue of the non-invariance of the Wald test under nonlinear reparametrisations of the restrictions under test is studied from a differential geometric viewpoint. Quantities that can be defined in purely geometrical terms are by construction invariant under reparametrisation, and various attempts are made to construct a Wald test out of such invariant quantities only. Despite the existence of a wide variety of possibilities, no computationally convenient invariant test statistic emerges from the analysis, since all the statistics considered need calculations equivalent in difficulty to the estimation of the restricted model, contrary to the spirit of the Wald test. On the other hand a variant of a C(α) test is discussed which, while not completely invariant under reparametrisation, is very nearly so, at least in the context of the model discussed by Gregory and Veall (1985), for which the conventional Wald test is particularly badly behaved. This test is easily computed from estimates of the unrestricted model only, and Monte Carlo evidence supports the conclusion that it yields as reliable inference as any other classical test even in very small samples.