Identification and Counterfactual Analysis in Incomplete Models with Support and Moment Restrictions

TL;DR

This paper introduces a unified framework for counterfactual analysis in incomplete models with support and moment restrictions, improving identification methods and operational procedures for economic models.

Contribution

It develops a novel approach embedding counterfactual restrictions into structural models, extending sharp identification results, and clarifying the roles of support and moment restrictions.

Findings

Support-function approach remains sharp under minimal conditions.

Identified set and its moment closure are statistically indistinguishable in irreducible models.

The framework bypasses traditional simulate-then-estimate workflows.

Abstract

This paper develops a unified identification framework for counterfactual analysis in incomplete models characterized by support and moment restrictions. I demonstrate that identifying structural parameters and conducting counterfactual analyses are isomorphic tasks. By embedding counterfactual restrictions within an augmented structural model specification, this approach bypasses the conventional "estimate-then-simulate" workflow and the need to simulate outcomes from models with set predictions. To make this approach operational, I extend sharp identification results for the support-function approach beyond the integrable boundedness condition that is imposed in sharp random-set characterizations but may be violated in economically relevant counterfactual analyses. Under minimal regularity conditions, I prove that the support-function approach remains sharp for the $moment$ $closure$ of the identified set. Furthermore, I introduce an irreducibility condition requiring all support implications to be made explicit. I show that for irreducible models, the identified set and its moment closure are statistically indistinguishable in finite samples. Together, these results justify using support-function methods in counterfactual settings where traditional sharpness fails and clarify the distinct roles of support and moment restrictions in empirical practice.