Weighted KS Statistics for Inference on Conditional Moment Inequalities

TL;DR

This paper introduces a new inverse variance weighted Kolmogorov-Smirnov statistic for constructing confidence regions in conditional moment inequality models, achieving faster convergence and requiring new asymptotic theory.

Contribution

It proposes a novel weighting scheme for KS statistics that improves convergence rates and develops a new asymptotic theory for critical value selection.

Findings

Confidence regions converge faster than existing methods.

New asymptotic theory for KS statistic behavior.

Monte Carlo study confirms finite sample performance.

Abstract

This paper proposes confidence regions for the identified set in conditional moment inequality models using Kolmogorov-Smirnov statistics with a truncated inverse variance weighting with increasing truncation points. The new weighting differs from those proposed in the literature in two important ways. First, confidence regions based on KS tests with the weighting function I propose converge to the identified set at a faster rate than existing procedures based on bounded weight functions in a broad class of models. This provides a theoretical justification for inverse variance weighting in this context, and contrasts with analogous results for conditional moment equalities in which optimal weighting only affects the asymptotic variance. Second, the new weighting changes the asymptotic behavior, including the rate of convergence, of the KS statistic itself, requiring a new asymptotic theory in choosing the critical value, which I provide. To make these comparisons, I derive rates of convergence for the confidence regions I propose along with new results for rates of convergence of existing estimators under a general set of conditions. A series of examples illustrates the broad applicability of the conditions. A monte carlo study examines the finite sample behavior of the confidence regions.