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

Exogeneity tests, incomplete models, weak identification and non-Gaussian distributions: invariance and finite-sample distributional theory