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

## TL;DR

This paper develops finite-sample distributional theory for exogeneity tests like DWH and RH, under non-Gaussian errors and incomplete models, enabling robust, exact inference even with weak identification or misspecification.

## Contribution

It provides explicit finite-sample null distributions for exogeneity tests under broad conditions, including non-Gaussian errors and incomplete models, and introduces invariance properties simplifying analysis.

## Key findings

- Null distributions do not involve nuisance parameters.
- Finite-sample tests are robust to weak identification and misspecification.
- Distributional characterizations enable Monte Carlo-based exact tests.

## Abstract

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.