Identification- and many instrument-robust inference via invariant moment conditions

Abstract: Identification-robust hypothesis tests are commonly based on the continuous updating GMM objective function. When the number of moment conditions grows proportionally with the sample size, the large-dimensional weighting matrix prohibits the use of conventional asymptotic approximations and the behavior of these tests remains unknown. We show that the structure of the weighting matrix opens up an alternative route to asymptotic results when, under the null hypothesis, the distribution of the moment conditions is reflection invariant. We provide several examples in which the invariance follows from standard assumptions. Our results show that existing tests will be asymptotically conservative, and we propose an adjustment to attain asymptotically nominal size. We illustrate our findings through simulations for various (non-)linear models, and an empirical application on the effect of the concentration of financial activities in banks on systemic risk.