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

Lixiong Li

arXiv 8 Mar 2026 · Econometrics

arXiv:2603.07722 · PDF · DOI · OpenAlex · Extracted main text

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.

Citation extraction

29
references
43
in-text mentions
29
distinct cited
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25,167
main-text words

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Most heavily cited references

The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.

ReferenceIntensityMentionsSectionsMain text
1Molchanov, Ilya (2005) Theory of Random Sets0.81142100%
2Hiriart-Urruty, Jean-Baptiste and Lemaréchal, Claude (2001) Fundamentals of convex analysis0.73732100%
3Bertsekas, Dimitri P. and Shreve, Steven E (1978) Stochastic Optimal Control: The Discrete Time Case0.64441100%
4Beresteanu, Arie and Molchanov, Ilya and Molinari, Francesca (2011) Sharp Identification Regions in Models With Convex Moment Predictions0.64422100%
5Ekeland, Ivar and Galichon, Alfred and Henry, Marc (2010) Optimal Transportation and the Falsifiability of Incompletely Specified Economic Models0.64422100%
6Schennach, Susanne M (2014) Entropic Latent Variable Integration Via Simulation0.64422100%
7Ciliberto, Federico and Tamer, Elie (2009) Market Structure and Multiple Equilibria in Airline Markets0.51121100%
8Gu, Jiaying and Russell, Thomas and Stringham, Thomas (2025) Counterfactual Identification and Latent Space Enumeration in Discrete Outcome Models0.51121100%
9Gu, Jiaying and Thomas Russell (2023) A Dual Approach to Wasserstein-Robust Counterfactuals0.51121100%
10Ackerberg, Daniel A. and Caves, Kevin and Frazer, Garth (2015) Identification Properties of Recent Production Function Estimators0.40511100%

Showing the top 10 of 29 scored citations.

Cited by, within the corpus

arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.

Citing paperIntensityMentionsSections
1The Projection Solution to the Incidental Parameter Problem0.51121