Specification Tests with Weak and Invalid Instruments

We focus on the classical linear simultaneous equations models and study the sensitivity to instrument endogeneity of six alternative versions of Durbin-Wu-Hausman (DWH) tests of exogeneity. To address this issue, we consider two setups for instrument endogeneity: (i) fixed instrument endogeneity, i.e., the parameter that controls instrument invalidity is fixed (does not depend on the sample size); (ii) local-to-zero instrument endogeneity, i.e., this parameter goes to zero at rate [n^1/2] as the sample size n increases [similar to Staiger and Stock (1997)]. In the first setup, we show that all tests have size converging to 1, as the sample size increases, no matter how weak the instruments are. In the second setup, we provide a characterization of the null limiting distribution of the statistics, which clearly shows that all statistics converge asymptotically to non-degenerated noncentral x^2 under the null hypothesis. This means that all tests have size greater than their nominal level asymptotically if the usual x^2 critical values are used, despite the fact that instrument endogeneity vanishes as the sample size increases. We propose size correction based on bootstrap techniques. Our analysis of the proposed bootstrap tests provides some new insights. More precisely, we show that even for moderate instrument endogeneity, the bootstrap provides a first-order approximation of the asymptotic size of the DWH tests, no matter how weak the instruments are. We present a Monte Carlo experiment that confirms our theoretical findings.