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Some useful equivalence properties of Hausman's test
This paper describes in general terms situations in which apparently different expressions for Hausman's test statistic are actually either asymptotically equivalent or numerically equal. It also describes, in the most general possible way, cases where quadratic forms based on linear combinations of the constrained and unconstrained estimators of all the unknown parameters of a model are asymptotically equivalent to the classical test statistics. The theoretical results presented in this paper are illustrated with an example.
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