Wild Bootstrap for Instrumental Variables Regression with Weak Instruments and Few Clusters

Under a framework with a small number of clusters but large numbers of observations per cluster for instrumental variable (IV) regression, we show that an unstudentized wild bootstrap test based on IV estimators such as the two-stage least squares estimator is valid as long as the instruments are strong for at least one cluster. This is different from alternative methods proposed in the literature for inference with a small number of clusters, whose validity would require that the instruments be strong for all clusters. Moreover, for the leading case in empirical applications with a single instrument, the unstudentized wild bootstrap test generated by our procedure is fully robust to weak instrument in the sense that its limiting null rejection probability is no greater than the nominal level even if all clusters are ``weak". However, such robustness is not shared by its studentized version; the wild bootstrap test that is based on the t-test statistic can have serious size distortion in this case. Furthermore, in the general case with multiple instruments, we show that an unstudentized version of bootstrap Anderson-Rubin (AR) test is fully robust to weak instruments, and is superior with regard to both size and power properties to alternative asymptotic and bootstrap AR tests that employ cluster-robust variance estimators. By contrast, we �find that bootstrapping other weak-instrument-robust tests such as the Lagrange multiplier test and the conditional quasi-likelihood ratio test, no matter studentized or unstudentized, does not guarantee correct limiting null rejection probability when all clusters are ``weak".