problem , so that standard criticisms of Wald-type methods in the presence of identification problems (see Dufour (1997,2003)) do not apply in this case. If the covariance matrix estimator can be modified so that it remains con- sistent and its rank converges to the appropriate asymptotic rank, then the asymptotic distribution of the modified Wald-type statistic (based on a gen- eralized inverse of the covariance matrix) remains chi-square although with a reduced degrees-of-freedom number; see Andrews (1987). For example, Lutkepohl-Burda (1997) proposed such methods based on reducing the rank of the estimated covariance matrix by either using a form of randomization or setting “small eigenvalues” to zero. Such methods, however, effectively mod- ify the test statistic and involve arbitrary truncation parameter for which no practical guidelines are available: in finite samples, the test statistic can be- come as small as one wishes leading to largely arbitrary results and unlimited power reductions. Interestingly, except for a bound given by Sargan(1980) in a special case, the asymptotic distribution of Wald-type statistics in non-regular cases has not been studied. In this paper, we undertake this task and propose solu- tions to the problem that do not require modifying the test statistic. More specifically, the contributions of the paper can be summarized as follows. First , we provide examples showing that Wald statistics in such non- regular cases can have several asymptotic distributions. We also show that usual critical values based on a chi-square distribution (with degrees-of- freedom equal to the number of constraints) can both lead to under-rejections 4