Doubly Robust GMM Inference and Differentiated Products Demand Models

This paper develops robust inference methods for moment condition models implemented with a n1=2-consistent auxiliary estimator of the nuisance parameters. When applied to models subject to weak identification and boundary parameter problems; they simultaneously overcome both irregularities and are asymptotically pivotal with minimal assumptions on the parameter space. If these problems are not present in the data; they are asymptotically equivalent to standard statistics for nonlinear models. They also have similar computational requirements. We apply our tests to the differentiated products demand model; which may suffer from both problems: the variance of the random coefecients is often close to zero; causing the boundary parameter problem; and the strength of the available instruments is often put in doubt; which may cause weak identification. We evaluate the performance of the proposed tests by simulations.