Moment inequalities in the context of simulated and predicted variables

This paper explores the effects of simulated moments on the performance of inference methods based on moment inequalities. Commonly used confi dence sets for parameters are level sets of criterion functions whose boundary points may depend on sample moments in an irregular manner. Due to this feature, simulation errors can affect the performance of inference in non-standard ways. In particular, a (fi rst-order) bias due to the simulation errors may remain in the estimated boundary of the con fidence set. We demonstrate, through Monte Carlo experiments, that simulation errors can signi ficantly reduce the coverage probabilities of confi dence sets in small samples. The size distortion is particularly severe when the number of inequality restrictions is large. These results highlight the danger of ignoring the sampling variations due to the simulation errors in moment inequality models. Similar issues arise when using predicted variables in moment inequalities models. We propose a method for properly correcting for these variations based on regularizing the intersection of moments in parameter space, and we show that our proposed method performs well theoretically and in practice.