Identification-Robust Two-Stage Bootstrap Tests with Pretesting for Exogeneity

Pretesting for exogeneity has become a routine in many empirical applications involving instrumental variables (IVs) to decide whether the ordinary least squares or IV-based method is appropriate. Guggenberger (2010a) shows that the second-stage test - based on the outcome of a Durbin-Wu-Hausman type pretest for exogeneity in the first stage - has extreme size distortion with asymptotic size equal to 1 when the standard asymptotic critical values are used, even under strong identification and conditional homoskedasticity. In this paper, we make the following contributions. First, we show that both conditional and unconditional on the data, standard wild bootstrap procedures are invalid for the two-stage testing and therefore are not viable solutions to such size-distortion problem. Second, we propose an identification-robust two-stage test statistic that switches between the OLS-based and the 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 no matter the identification is strong or weak.