Asymptotic Results for Generalized Wald Tests
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This note presents conditions under which a quadratic form based on a g-inverted weighting matrix converges to a chi-square distribution as the sample size goes to inﬁnity. Subject to fairly weak underlying conditions, a necessary and suﬀicient condition is given for this result. The result is of interest, because it is needed to establish asymptotic signiﬁcance levels and local power properties of generalized Wald tests (i.e., Wald tests with singular limiting covariance matrices). Included in this class of tests are Hausman speciﬁcation tests and various goodness of ﬁt tests, among others. The necessary and suﬀicient condition is relevant to procedures currently in the econometrics literature, because it illustrates that some results stated in the literature only hold under more restrictive assumptions than those given.
Andrews, Donald W.K., "Asymptotic Results for Generalized Wald Tests" (1985). Cowles Foundation Discussion Papers. 1002.