Pretesting for exogeneity has become routine in many empirical applications involving instrumental variables (IVs) to decide whether the ordinary least squares or IV-based method is appropriate. Guggenberger (2010a, Econometric Theory , 26, 369–382) shows that the second-stage test – based on the outcome of a Durbin-Wu-Hausman-type pretest in the first stage – exhibits extreme size distortion, with asymptotic size equal to 1 when the standard critical values are used. In this paper, we first show that both conditional and unconditional on the data, standard wild bootstrap procedures are invalid for two-stage testing. Second, we propose an identification-robust two-stage test statistic that switches between OLS-based and weak-IV-robust statistics. Third, we develop a size-adjusted wild bootstrap approach for our two-stage test that integrates specific wild bootstrap critical values with an appropriate size-adjustment method. We establish uniform validity of this procedure under conditional heteroskedasticity or clustering in the sense that the resulting tests achieve correct asymptotic size, regardless of whether the identification is strong or weak. Our procedure is especially valuable for empirical researchers facing potential weak identification. In such settings, its power advantage is notable: whereas weak-IV-robust methods maintain correct size but often suffer from relatively low power, our approach achieves better performance.
Tchatoka et al. (Wed,) studied this question.