🤖 AI Summary
This study addresses the joint identification and counterfactual analysis in incomplete structural models featuring support and moment constraints. The authors embed counterfactuals directly into an augmented structural model, departing from the conventional “estimate-then-simulate” paradigm. By leveraging support function methods, they simultaneously achieve identification and inference, revealing a fundamental isomorphism between the two tasks. A key contribution is the formulation of irreducibility conditions that explicitly characterize all support implications. Under mild regularity assumptions, the support function approach preserves sharpness with respect to the moment closure—even in counterfactual settings where traditional sharpness fails. Moreover, for irreducible models, the identified set and the moment closure are statistically indistinguishable in finite samples.
📝 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.