On the validity of Durbin-Wu-Hausman tests for partial exogeneity with weak identification

This paper investigates the size and power properties of the subset (Wu, 1973,1974) T2-test and (Hausman, 1978) test for partial exogeniety when instrumental variables (IVs) are weakly associated with endogenous regressors (weak identification). We provide a characterization of the limiting distributions of the test statistics under both the null hypothesis of partial exogeneity (size), and the alternaitve hypothesis of endogeneity (power). The results show that when identification is weak, all statistics have non-standard limiting distributions. Therefore, the use of the usual chi-squared critical values within inference may be misleading. Furthermore, test consistency does no longer hold with weak instruments, and a Monte Carlo experiment indicates that both tests have low power over a wid range of cases when identification is weak. We provide an empirical application which supports our theoretical findings.