The Hausman test statistic can be negative even asymptotically

Published in the "Journal of Economics and Statistics" (Jahrbücher für Nationalökonomie und Statistik), 2008, vol. 228 no. 4, pp. 394-405.

Keywords: Hausman test, negative chi^2 statistic, nuisance parameter

Download: last working paper version August 2008.

Abstract: We show that under H1 the Hausman chi-square test statistic can be negative not only in small samples but even asymptotically. Therefore in large samples a negative test statistic is only compatible with H1 and should be interpreted accordingly. Applying a known insight from finite samples, this can only occur if the different estimation precisions (often the residual variance estimates) under H0 and under H1 both enter the test statistic. In finite samples, using the absolute value of the test statistic is a remedy that does not alter the test under the null hypothesis and is thus admissible.
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Even for positive test statistics the relevant covariance matrix difference should be routinely checked for positive semi-definiteness, because we also show that otherwise test results may be misleading. Of course the preferable solution still is to impose the same nuisance parameter (i.e., residual variance) estimate under the null and alternative hypotheses, if the model context permits that with relative ease. We complement the likelihood-based exposition by a formal proof in an omitted-variable context, we present simulation evidence for the test of panel random effects, and we illustrate the problems with a panel homogeneity test.

(Latest update: October 2017)