On Bootstrap Validity for Specification Tests with Weak Instruments

We investigate the asymptotic validity of the bootstrap for Durbin-Wu-Hausman (DWH) tests of exogeneity in linear IV regressions, with or without identification. Our analysis of the properties (size and power) of the proposed bootstrap tests provides some new insights and extensions of earlier studies. More precisely, we show that when identification is strong, the bootstrap: (1) provides an asymptotic refinement of the size of the DWH tests under exogeneity, and (2) is consistent under the alternative hypothesis if the endogeneity parameter is fixed. However, the bootstrap only provides a first-order approximation of the asymptotic distributions of these statistics when instruments are weak. Moreover, we characterize the necessary and sufficient condition under which the proposed bootstrap DWH tests exhibit power under fixed endogeneity and weak instruments. The latter condition may still hold over a wide range of cases, provided that at least one instrument is not irrelevant. But all bootstrap tests have low power when all instruments are irrelevant, a case of little interest in empirical work. We present a Monte Carlo experiment that confirms our theoretical findings.