and over-rejections depending on the form of the function studied. Indeed, the Wald statistic may diverge under null hypothesis, so that arbitrary size distortions may occur. Second , we study the asymptotic distribution of Wald-type statistics in non-regular cases. Surprisingly, the asymptotic behavior of the Wald statistic has not been generally studied for full classes of restrictions; here we consider the class of polynomial restrictions. We show that the Wald statistic either has a non-degenerate asymptotic distribution even when the estimated co- variance converges to a singular matrix, or diverges to infinity. We provide conditions for convergence and a general characterization of this distribution. We find that the test can have several different asymptotic distributions un- der the null hypothesis – depending on the degree of singularity as well as various nuisance parameters – which may be non-chi-square distributions. Third , we provide bounds on the asymptotic distribution (when it exists), which turn out to be to be proportional to a chi-square distribution where the proportionality constant depends on the degree of singularity of the function considered. In several cases of interest, this bound yields an easily available conservative critical value. Even when the limit distribution is non-pivotal it is sometime possible to provide pivotal bounds that would yield conservative critical values. Fourth , we propose an adaptive consistent strategy for determining whether the asymptotic distribution exists and which form it takes; this approach also permits to determine what kind of bound is valid. 5