Weighted KS statistics for inference on conditional moment inequalities

This paper proposes set estimators and conservative 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, this paper shows that estimators based on KS statistics with the proposed weighting function 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. The results on rates of convergence of set estimators are the first such results even for the existing procedures, and involve developing the first general framework for determining consistency and rates of convergence for set estimators and confidence regions in this context. 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. A series of examples illustrate the broad applicability of the results.