1 Introduction Tests based on asymptotic distributions typically require regularity assump- tions in order to be able to obtain critical values. This is the case, in par- ticular, for Wald-type statistics based on asymptotically normal estimators. Wald-type tests are especially convenient because they allow one to test a wide array of linear and nonlinear restrictions from a single unrestricted es- timator. We focus here on the problem of implementing Wald-type tests for nonlinear restrictions. The use of the Wald statistic has been criticized because of finite sam- ple non-invariance (Gregory and Veall (1985), Breusch and Shmidt (1988), Phillips-Park(1988), Dagenais-Dufour(1991)) and lack of robustness to iden- tification failure (Dufour(1997,2003)). We focus here on situations where the parameter tested is typically identified under the null hypothesis, but usual rank conditions on the Jacobian matrix may fail asymptotically. Under regularity conditions, the standard asymptotic distribution of the test statistic is chi-square with degrees-of-freedom equal to the number of restrictions. The regularity conditions involve the assumption that the re- strictions are differentiable with respect to the parameters considered, with a derivative matrix which has full column rank in an open neighborhood of the true value of the parameter vector. There are many problems, however, for which this regularity condition is violated. These include, among others: 1. hypothesis tests on bilinear and multilinear forms of model coefficients 2