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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.
Andrews, Donald W.K., "Asymptotic Results for Generalized Wald Tests" (1985). Cowles Foundation Discussion Papers. 1001.