Exogeneity Tests, Incomplete Models, Weak Identification and Non-Gaussian Distributions: Invariance and Finite-Sample Distributional Theory

55 Pages Posted: 4 Jan 2017

Date Written: December 30, 2016


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. We present simulation evidence which confirms the finite-sample theory. The theoretical results are illustrated with two empirical examples: the relation between trade and economic growth, and the widely studied problem of the return of education to earnings.

Keywords: Exogeneity; Durbin-Wu-Hausman Test; Weak Instrument; Incomplete Model; Non-Gaussian; Weak Identification; Identification Robust; Finite-Sample Theory; Pivotal; Invariance; Monte Carlo Test; Power

JEL Classification: C3; C12; C15; C52

Suggested Citation

Doko Tchatoka, Firmin and Dufour, Jean-Marie, Exogeneity Tests, Incomplete Models, Weak Identification and Non-Gaussian Distributions: Invariance and Finite-Sample Distributional Theory (December 30, 2016). Available at SSRN: https://ssrn.com/abstract=2891587 or http://dx.doi.org/10.2139/ssrn.2891587

Firmin Doko Tchatoka

University of Adelaide ( email )

No 233 North Terrace, School of Commerce
Adelaide, 5005

Jean-Marie Dufour (Contact Author)

McGill University ( email )

Department of Economics, McGill University
Leacock Building Room 443, 855 Sherbrooke West
Montreal, Quebec H3A 2T7
(1) 514 398 6071 (Phone)
(1) 514 398 4800 (Fax)

HOME PAGE: http://www.jeanmariedufour.com

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