Inference in Incomplete Models

TL;DR

This paper introduces a novel test for the correctness of structural models without requiring identification assumptions, using a Kolmogorov-Smirnov statistic for Choquet capacity functionals, applicable to various complex models.

Contribution

It develops a new specification test based on Choquet capacities, providing a method to verify model correctness without identification assumptions and enabling confidence region construction.

Findings

Test is consistent against certain alternatives

Limiting distribution derived under null hypothesis

Applicable to models with sample selection and multiple equilibria

Abstract

We provide a test for the specification of a structural model without identifying assumptions. We show the equivalence of several natural formulations of correct specification, which we take as our null hypothesis. From a natural empirical version of the latter, we derive a Kolmogorov-Smirnov statistic for Choquet capacity functionals, which we use to construct our test. We derive the limiting distribution of our test statistic under the null, and show that our test is consistent against certain classes of alternatives. When the model is given in parametric form, the test can be inverted to yield confidence regions for the identified parameter set. The approach can be applied to the estimation of models with sample selection, censored observables and to games with multiple equilibria.