On Bootstrap Validity for Specification Tests with Weak Instruments

This paper investigates the asymptotic validity of the bootstrap for Durbin-Wu-Hausman (DWH) specification tests when instrumental variables (IVs) may be arbitrary weak. It is shown that under strong identification, the bootstrap offers a better approximation than the usual asymptotic 2 distributions. However, the bootstrap provides only a first-order approximation when instruments are weak. These results show unlike theWald-statistic based on a k-class type estimator (Moreira et al., 2009), the bootstrap is valid even for the Wald-type of DWH statistics in the presence of weak instruments.