This note presents (i) necessary and sufficient conditions for the consistency of estimators of Moore-Penrose inverted matrices, and (ii) sufficient conditions for convergence to a chi-square distribution of quadratic forms based on g-inverted weighting matrices. The latter results are needed to establish asymptotic significance levels and local power properties of generalized Wald tests (i.e., Wald tests with singular covariance matrices). Included in this class of tests are Hausman specification tests and various goodness of fit tests, among others. The results are relevant to procedures currently in the literature, since they illustrate that some results stated in the literature hold only under more restrictive assumptions than those given.