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.

[add the following paragraph for long summary:]

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)