On the Formulation of Wald Tests of Nonlinear Restrictions

This paper utilizes asymptotic expansions of the Edgeworth type to investigate alternative forms of the Wald test of nonlinear restrictions. Some formulae for the asymptotic expansion of the distribution of the Wald statistic are provided for a general case that should include most econometric applications. When specialized to the simple cases that have been studied recently in the literature, these formulae are found to explain rather well the discrepancies in sampling behavior that have been observed by other authors. It is further shown how the corrections delivered by Edgeworth expansions may be used to find transformations of the restrictions which accelerate convergence to the asymptotic distribution.