  We study the distribution of Durbin-Wu-Hausman (DWH) and Revankar-Hartley
(RH) tests for exogeneity from a finite-sample viewpoint, under the null and
alternative hypotheses. We consider linear structural models with possibly
non-Gaussian errors, where structural parameters may not be identified and
where reduced forms can be incompletely specified (or nonparametric). On level
control, we characterize the null distributions of all the test statistics.
Through conditioning and invariance arguments, we show that these distributions
do not involve nuisance parameters. In particular, this applies to several test
statistics for which no finite-sample distributional theory is yet available,
such as the standard statistic proposed by Hausman (1978). The distributions of
the test statistics may be non-standard -- so corrections to usual asymptotic
critical values are needed -- but the characterizations are sufficiently
explicit to yield finite-sample (Monte-Carlo) tests of the exogeneity
hypothesis. The procedures so obtained are robust to weak identification,
missing instruments or misspecified reduced forms, and can easily be adapted to
allow for parametric non-Gaussian error distributions. We give a general
invariance result (block triangular invariance) for exogeneity test statistics.
This property yields a convenient exogeneity canonical form and a parsimonious
reduction of the parameters on which power depends. In the extreme case where
no structural parameter is identified, the distributions under the alternative
hypothesis and the null hypothesis are identical, so the power function is
flat, for all the exogeneity statistics. However, as soon as identification
does not fail completely, this phenomenon typically disappears.
