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

Imagine you are a detective trying to solve a crime, but you don't have a single, clear suspect. Instead, you have a list of possible suspects who could have done it, and you only know a few facts about them: they were in the neighborhood (a support restriction) and they bought a specific type of ticket at the train station (a moment restriction).

In economics, this is called an incomplete model. The "crime" is the real-world outcome (like a company's profit or a market's behavior), and the "suspects" are the hidden factors (like a CEO's secret motivation or a consumer's mood) that the model can't see directly. Because there are multiple possible suspects, the model can't predict exactly what will happen; it can only predict a range of possibilities.

This paper, written by Lixiong Li, tackles a very tricky problem: How do we predict what will happen if we change the rules of the game? (This is called counterfactual analysis). For example, "What if we doubled the tax?" or "What if two companies merged?"

Here is the breakdown of the paper's ideas using simple analogies:

1. The Old Way vs. The New Way

The Old Way (Estimate-Then-Simulate):
Imagine you try to guess the exact identity of the criminal first. You say, "I'm 90% sure it was Bob." Then, you try to simulate what Bob would do if the tax doubled.

The Problem: In incomplete models, you can't be sure it was Bob. It could be Bob, Alice, or Charlie. If you pick one and simulate, you might get it wrong. If you try to simulate all of them, the math gets incredibly messy and often breaks down because the "range" of possibilities becomes infinite (like trying to count every grain of sand on a beach that keeps growing).

The New Way (The Unified Framework):
Li suggests a smarter approach. Instead of guessing the criminal first, treat the "What if?" question as part of the original mystery.

The Analogy: Imagine you are building a single, giant puzzle. The original clues are on one side, and the "What if?" clues are on the other. You don't solve the first half to get to the second; you solve the whole picture at once.
The Result: You can figure out the range of possible outcomes for the new scenario (the counterfactual) without ever having to simulate a specific, impossible-to-determine scenario. It's like solving a maze by looking at the whole map at once, rather than trying to walk every single path one by one.

2. The "Boundedness" Problem (The Infinite Balloon)

In math, to make these calculations work, economists usually assume the "range of possibilities" is bounded. Think of it like a balloon that can get big, but not infinitely big. This is called "integrable boundedness."

The Issue: In real life, when we ask "What if profits doubled?" or "What if a company goes bankrupt?", the math often suggests the possibilities could be infinite. The balloon pops. The standard math tools say, "I can't solve this because the numbers are too wild."
Li's Insight: Li says, "Wait a minute. Just because the balloon is infinite doesn't mean we can't learn anything." He proves that even when the math gets wild and the numbers go to infinity, the method still gives us the best possible answer we can get from the data. It's like saying, "Even if we can't count every grain of sand, we can still accurately measure the volume of the beach."

3. The "Irreducibility" Rule (Don't Hide the Clues)

The paper introduces a concept called irreducibility. This is a rule about how you write down your clues.

The Analogy: Imagine you are writing a recipe.
- Bad Recipe: "Add a pinch of salt (which is actually 5 grams of salt, but I'm hiding that fact in the 'flavor' section)."
- Good Recipe (Irreducible): "Add 5 grams of salt."
The Point: Sometimes, economists hide a rule about the "neighborhood" (support) inside the "ticket purchase" (moment) section. Li argues that if you hide the rules, your math tools will fail. But if you lay all the rules out explicitly (make the model "irreducible"), the math works perfectly, even if the numbers are infinite.
The Takeaway: If you write your model clearly, the "best possible answer" (the moment closure) is statistically indistinguishable from the "perfect answer" in any real-world sample size. In plain English: If you set up the problem correctly, the method is as good as it gets.

4. Why This Matters

This paper is a game-changer for economists and policymakers because:

It saves time: You don't need to run thousands of simulations to guess the future.
It handles the messy stuff: It works even when the "what if" scenarios involve infinite possibilities (like huge profits or total losses), which happens often in real life.
It clarifies the rules: It tells researchers exactly how to write their models so they don't accidentally break the math.

In Summary:
Li has built a new, unified toolkit for predicting the future in uncertain worlds. He showed that you don't need to know the exact hidden details to make good predictions; you just need to combine the "what is" and the "what if" into one big, clear picture. Even when the numbers get crazy and infinite, this new method tells us exactly what we can and cannot know, preventing us from wasting time on impossible calculations.

Here is a detailed technical summary of the paper "Identification and Counterfactual Analysis in Incomplete Models with Support and Moment Restrictions" by Lixiong Li.

1. Problem Statement

The paper addresses the challenges of conducting counterfactual analysis in incomplete structural econometric models.

Incomplete Models: These are models where, conditional on observables ( $Z$ ), latent variables ( $U$ ), and parameters ( $\theta$ ), the outcome is not uniquely determined but rather a set of possible outcomes (e.g., games with multiple equilibria, models with limited attention).
The "Estimate-Then-Simulate" Bottleneck: Traditional counterfactual analysis involves estimating structural parameters and then simulating outcomes under new policies. In incomplete models, this is computationally difficult or infeasible because the model predicts a set of outcomes, requiring arbitrary equilibrium selection rules to generate point predictions.
Violation of Regularity Conditions: Existing sharp identification methods (based on random set theory and support functions) rely on the integrable boundedness condition. This condition often fails in economically relevant counterfactuals (e.g., welfare, profits, surplus) where latent variables may be unbounded from one side, causing the random sets associated with these counterfactual objects to be unbounded.

2. Methodological Framework

The paper proposes a unified framework based on Support and Moment Restrictions.

A. The Structural Model

The baseline model is defined by two conditions:

Support Restriction: $P((U, Z) \in \Gamma(\theta)) = 1$ . This encodes economic theory (e.g., equilibrium conditions) as a restriction on the joint support of latent and observed variables.
Moment Restrictions: $E[r(U, Z; \theta)] = 0$ . These restrict the distribution of latent variables without requiring full parametric specification (e.g., orthogonality, mean independence).

B. The Augmented Model for Counterfactuals

The core methodological innovation is treating counterfactual analysis as an identification problem within an augmented structural model.

Instead of simulating outcomes, the researcher defines counterfactual outcomes ( $\tilde{Y}$ ) and latent primitives ( $\tilde{U}$ ) via a new support restriction $\tilde{\Gamma}$ and potentially new moment restrictions $\tilde{r}$ .
The counterfactual parameter of interest ( $\tilde{\theta}$ ) is defined via a moment condition $\tilde{m}(\tilde{Y}, \tilde{U}, Z, U; \theta, \tilde{\theta}) = 0$ .
Isomorphism: These counterfactual restrictions are stacked with the baseline restrictions to form a single augmented model with parameters $\theta' = (\theta, \tilde{\theta})$ .
Result: Identifying the counterfactual parameter $\tilde{\theta}$ becomes isomorphic to identifying structural parameters in a partially identified model. Inference is conducted using standard subvector inference methods on the augmented model, bypassing the need for simulation.

3. Key Contributions and Theoretical Results

Contribution 1: Extension Beyond Integrable Boundedness

The paper extends the Support-Function Approach (Ekeland, Galichon, Henry; Beresteanu, Molchanov, Molinari) to settings where the integrable boundedness assumption (Assumption 2) fails.

The Problem: When support restrictions are one-sided (e.g., $U \geq c$ ) and moment functions involve higher-order terms (e.g., $U^2$ ), the random set of moment values is unbounded. Standard sharp identification results no longer apply.
The Solution: The author introduces the Moment Closure of the identified set, denoted $\Theta_I(F)$ $Θ_{I} (F)$ .
- Identified Set ( $\Theta_I$ ): Parameters where an exact joint distribution exists satisfying moments.
- Moment Closure ( $\Theta_I$ ): Parameters where the moment violation can be made arbitrarily small (approximated).
Theorem 2: Under minimal regularity conditions (Assumption 1 only), the support-function inequalities are sharp for the moment closure. That is, the set of parameters satisfying the support-function inequalities is exactly the moment closure of the identified set.

Contribution 2: Irreducibility and Finite-Sample Indistinguishability

The paper addresses the gap between the identified set and its moment closure.

Irreducibility Condition: A model is irreducible if its support restrictions are explicit and do not contain "hidden" support implications buried within moment conditions. If a model is reducible, one can reformulate it to move implicit support restrictions into the support set $\Gamma$ .
Theorem 3: If the model is formulated in an irreducible form, the identified set ( $\Theta_I$ $Θ_{I}$ ) and its moment closure ( $\Theta_I$ $Θ_{I}$ ) are statistically indistinguishable in finite samples.
- Meaning: No statistical test with controlled size can distinguish between a parameter belonging to the true identified set versus the moment closure.
- Implication: In practice, the support-function approach captures the "effective limit" of what can be learned from data, even when strict mathematical sharpness (exact equality of sets) fails due to unboundedness.

Contribution 3: Clarification of Support vs. Moment Restrictions

The paper demonstrates that treating support restrictions merely as moment conditions (e.g., $E[1\{(U,Z) \in \Gamma\}] = 0$ ) can lead to a loss of identifying power.

Example 3 (Interval Censoring): If an interval constraint is treated as a moment condition rather than a support restriction, the support-function approach yields an uninformative set (the entire parameter space), whereas the true identified set remains informative.
Takeaway: Explicitly separating support and moment restrictions is crucial for maintaining sharp identification.

4. Illustrative Examples

The paper validates the theory using three examples:

Static Entry Game:
- Scenario (i): Counterfactuals involving expected profits (unbounded latent shocks) lead to unbounded identified sets. The support-function approach correctly identifies this lack of upper bounds.
- Scenario (ii): Counterfactuals with variance constraints on latent shocks. Even though the random set is unbounded (violating integrable boundedness), the model still yields finite two-sided bounds for counterfactual profits. The support-function approach captures these bounds correctly.
Production Function Estimation: Uses cost-minimization inequalities to define support restrictions, avoiding invertibility assumptions.
Interval-Censored Regression: Demonstrates how mis-specifying support as moments leads to a failure of identification (Scenario iii), reinforcing the need for the irreducibility condition.

5. Significance and Implications

Practical Justification: The paper provides a rigorous justification for using support-function methods in complex counterfactual analyses (e.g., welfare, merger simulations) where traditional sharp identification fails due to unboundedness.
Unified Workflow: It eliminates the need for the "estimate-then-simulate" workflow, which is often infeasible in incomplete models. Counterfactuals are handled directly via partial identification techniques.
New Benchmark for Identification: The paper shifts the focus from "exact sharpness" (which requires strong regularity conditions) to finite-sample indistinguishability. This suggests that for irreducible models, the moment closure is the practically relevant object of study.
Policy Relevance: By clarifying when bounds are truly uninformative (due to model structure) versus when they are methodological artifacts, the paper helps researchers avoid drawing incorrect conclusions from counterfactual exercises.

Conclusion

Lixiong Li's paper establishes a robust theoretical foundation for counterfactual analysis in incomplete models. By embedding counterfactuals into an augmented support-and-moments framework and proving that the support-function approach characterizes the moment closure (which is indistinguishable from the identified set under irreducibility), the paper extends the applicability of partial identification methods to a broad class of empirically relevant problems previously considered intractable or requiring strong, often violated, regularity conditions.