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

Lixiong Li

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.

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

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 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.
Paper Structure (25 sections, 16 theorems, 111 equations)

This paper contains 25 sections, 16 theorems, 111 equations.

Table of Contents

  1. Theoretical Framework
  2. Structural model
  3. Examples
  4. Counterfactual Analysis
  5. Counterfactual outcomes
  6. Counterfactual parameters
  7. Unified identification framework for structural and counterfactual parameters
  8. Identification Analysis
  9. Identified set and sharp identification results
  10. Moment closure of the identified set
  11. Finite-sample indistinguishability between the identified set and its moment closure
  12. Discussion
  13. Preliminaries on Random Set and Measurable Selection
  14. Basic results on random set
  15. Selection Theorem
  16. ...and 10 more sections

Key Result

theorem 1

Under Assumptions assu:reg and assu:compact, for any $F\in\mathcal{F}$ and any $\theta\in\Theta$, $\theta\in\Theta_I(F)$ if and only if $\theta$ satisfies eq:moment_ineq. Equivalently, for any $F\in\mathcal{F}$,

Theorems & Definitions (55)

  • Remark 1
  • example 1: Static entry game with complete information
  • example 2: Production-function estimation
  • example 3: Regression with an interval-censored dependent variable
  • example 4: continues=ex:entry_game
  • example 5: continues=ex:production_function
  • example 6: continues=ex:entry_game
  • example 7: continues=ex:production_function
  • Definition 1: Identified set
  • theorem 1
  • ...and 45 more