  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.
