Size-corrected Bootstrap Test after Pretesting for Exogeneity with Heteroskedastic or Clustered Data

Pretesting for exogeneity has become a routine in many empirical applications involving instrumental variables to decide whether the ordinary least squares or the two-stage least squares (2SLS) 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 when the standard asymptotic critical values are used. In this paper, we first show that both conditional and unconditional on the data, the standard wild bootstrap procedures are invalid for the two-stage testing and a closely related shrinkage method, and therefore are not viable solutions to such size-distortion problem. Then, we propose a novel 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 either conditional heteroskedasticity or clustering in the sense that the resulting tests achieve correct asymptotic size. Monte Carlo simulations confirm our theoretical findings. In particular, our proposed method has remarkable power gains over the standard 2SLS-based t-test in many settings, especially when the identification is not strong.