Wild Bootstrap for Instrumental Variables Regressions with Weak and Few Clusters

We study the wild bootstrap inference for instrumental variable (quantile) regressions in the framework of a small number of large clusters, in which the number of clusters is viewed as fixed and the number of observations for each cluster diverges to infinity. For subvector inference, we show that the wild bootstrap Wald test with or without using the cluster-robust covariance matrix controls size asymptotically up to a small error as long as the parameters of endogenous variables are strongly identified in at least one of the clusters. We further develop a wild bootstrap Anderson-Rubin (AR) test for full-vector inference and show that it controls size asymptotically up to a small error even under weak or partial identification for all clusters. We illustrate the good finite-sample performance of the new inference methods using simulations and provide an empirical application to a well-known dataset about U.S. local labor markets.