Weak-Identification-Robust Bootstrap Tests after 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 (OLS) or the IV-based method is appropriate. Guggenberger (2010) shows that the second-stage t-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, even when the IVs are strong. In this paper, we propose a novel two-stage test procedure that switches between the OLS-based statistic and the weak-IV-robust statistic. Furthermore, we develop a size-corrected wild bootstrap approach, which combines certain wild bootstrap critical values along with an appropriate size-correction method. We establish uniform validity of this procedure under conditional heteroskedasticity in the sense that the resulting tests achieve correct asymptotic size no matter the identification is strong or weak. Monte Carlo simulations confirm our theoretical findings. In particular, our proposed method has remarkable power gains over the standard weak-identification-robust test.