Moment Inequalities in the Context of Simulated and Predicted Variables

TL;DR

This paper investigates how simulation errors impact inference in moment inequality models, revealing potential biases and coverage issues, and proposes a correction method to improve reliability.

Contribution

It introduces a regularization-based correction method for simulation-induced variations in moment inequality inference, addressing a key gap in existing methodologies.

Findings

Simulation errors can significantly reduce confidence set coverage in small samples.

Bias from simulation errors can persist in boundary estimation of confidence sets.

The proposed correction method improves inference accuracy both theoretically and empirically.

Abstract

This paper explores the effects of simulated moments on the performance of inference methods based on moment inequalities. Commonly used confidence 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 (first-order) bias due to the simulation errors may remain in the estimated boundary of the confidence set. We demonstrate, through Monte Carlo experiments, that simulation errors can significantly reduce the coverage probabilities of confidence 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.