Gradient Wild Bootstrap for Instrumental Variable Quantile Regressions with Weak and Few Clusters

TL;DR

This paper develops wild bootstrap inference methods for instrumental variable quantile regressions with a small number of large clusters, ensuring valid size control and improved power under various identification conditions.

Contribution

It introduces wild bootstrap Wald and Anderson-Rubin tests that are robust to weak and partial identification in clustered data settings.

Findings

Bootstrap Wald test controls size asymptotically.

Bootstrap AR test is robust under weak identification.

Simulation results show good finite-sample performance.

Abstract

We study the gradient wild bootstrap-based 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 the Wald inference, we show that our wild bootstrap Wald test, with or without studentization using the cluster-robust covariance estimator (CRVE), controls size asymptotically up to a small error as long as the parameter of endogenous variable is strongly identified in at least one of the clusters. We further show that the wild bootstrap Wald test with CRVE studentization is more powerful for distant local alternatives than that without. Last, we develop a wild bootstrap Anderson-Rubin (AR) test for the weak-identification-robust inference. We show 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 US local labor markets.