Nonlinear models, rescaling and test invariance

Abstract

Several asymptotic tests proposed in the literature are shown not to be invariant to changes in measurement units or, more generally, to various transformations that leave both the model and the null hypothesis invariant. The tests involved include the Wald test (and various generalizations of it), a variant of the Lagrange multiplier test, Neyman's C(α) test, Durbin's (1970) procedure, Hausman-type tests and a number of tests suggested by White (1982). For all these procedures, simply changing measurement units in a way that leaves both the form of the model and the null hypothesis invariant can lead to very different answers. We observe, in particular, that various consistent estimators of the information matrix lead to test procedures with different invariance properties. We then establish general sufficient conditions ensuring that Neyman's C(α) test is invariant to transformations that leave invariant the form of the model. In many practical cases where Wald-type tests lack invariance, we find that a modification of the C(α) test is invariant and hardly more costly to compute than Wald tests. This computational simplicity stands in contrast with other invariant tests such as the likelihood ratio test. The method suggested is applied to regression models with Box-Cox transformations on the variables.