Wald tests when restrictions are locally singular

Wald-type tests are convenient because they allow one to test a wide array of linear and nonlinear restrictions from a single unrestricted estimator; we focus on the problem of implementing Wald-type tests for nonlinear restrictions. We provide examples showing that Wald statistics in non-regular cases can have several asymptotic distributions; the usual critical values based on a chi-square distribution can both lead to under-rejections and over-rejections; indeed, the Wald statistic may diverge under the null hypothesis. We study the asymptotic distribution of Wald-type statistics for the class of polynomial restrictions and show that the Wald statistic either has a non-degenerate asymptotic distribution, or diverges to infinity. We provide conditions for convergence and a general characterization of this distribution. We provide bounds on the asymptotic distribution (when it exists). In several cases of interest, this bound yields an easily available conservative critical value. We propose an adaptive consistent strategy for determining whether the asymptotic distribution exists and which form it takes.