The Maximum Number of Parameters for the Hausman Test When the Estimators are from Different Sets of Equations

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Hausman (1978) developed a widely-used model specification test that has passed the test of time. The test is based on two estimators, one being consistent under the null hypothesis but inconsistent under the alternative, and the other being consistent under both the null and alternative hypotheses. In this paper, we show that the asymptotic variance of the difference of the two estimators can be a singular matrix. Moreover, in calculating the Hausman test there is a maximum number of parameters which is the number of different equations that are used to obtain the two estimators. Three illustrative examples are used, namely an exogeneity test for the linear regression model, a test for the Box-Cox transformation, and a test for sample selection bias.
JEL classifications: C2; C5; I18. This paper was supported by a Grant-in-Aid for Scientific Research “Analyses of the Large Scale Medical Survey Data and the Policy Evaluations in Japan (Grant Number: 24330067)” of the Japan Society of Science for the first author, and Australian Research Council and the National Science Council, Taiwan for the second author.
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