On bootstrap validity for specification testing with many weak instruments

Abstract

This paper studies the asymptotic validity of bootstrapping the J test of over-identifying restrictions and the Anderson–Rubin (AR) test under many/many weak instrument sequences. We show that the (residual-based) bootstrap consistently estimates the limiting distributions of interest under these asymptotic frameworks. Interestingly, such bootstrap validity holds even if the bootstrap cannot mimic well certain important properties in the model. In addition, the studied bootstrap procedures are easy to implement in practice because they do not require an a priori choice between the conventional asymptotics and the many/many weak instrument asymptotics. Monte Carlo simulation shows that the bootstrap techniques provide a more reliable method to approximate the null distribution of the J and AR test statistics under many/many weak instruments.