On Bootstrap Validity for Subset Anderson-Rubin Test in IV Regressions

This paper sheds new light on subset hypothesis testing in linear structural models in which instrumental variables (IVs) can be arbitrarily weak. For the first time, we investigate the validity of the bootstrap for Anderson-Rubin (AR) type tests of hypotheses specified on a subset of structural parameters, with or without identification. Our investigation focuses on two subset AR type statistics based on the plug-in principle. The first one uses the restricted limited information maximum likelihood (LIML) as the plug-in method, and the second exploits the restricted two-stage least squares (2SLS). We provide an analysis of the limiting distributions of both the standard and proposed bootstrap AR statistics under the subset null hypothesis of interest. Our results provide some new insights and extensions of earlier studies. In all cases, we show that when identification is strong and the number of instruments is fixed, the bootstrap provides a high-order approximation of the null limiting distributions of both plug-in subset statistics. However, the bootstrap is inconsistent when instruments are weak. This contrasts with the bootstrap of the AR statistic of the null hypothesis specified on the full vector of structural parameters, which remains valid even when identification is weak; see Moreira et al. (2009). We present a Monte Carlo experiment that confirms our theoretical findings.