Invariance and the Wald test

Many models and hypotheses of interest in econometrics are invariant to certain types of data transformations such as measurement unit changes. Dagenais and Dufour (Econometrica 59 (1991) 1601; Economics Letters 38 (1992) 251) and Dufour and Dagenais (J. Statist. Plann. Inference 32 (1992) 111) have shown that Wald tests are not invariant in general to such data transformations. In this paper, I provide a simple set of sufficient conditions to ensure that a Wald test for a null hypothesis is invariant to such a data transformation. I then use this set of conditions to help account for certain features of the Monte Carlo results from Gregory and Weall (Economics Letters 22 (1986) 203) on the properties of a variety of Wald tests for Sarganʹs COMmon FACtor restriction (Sargan (Econometrica 48 (1980) 879)) in the first-order autoregressive distributed-lag model.