Moment Inequalities, Partial Identification, and Sharp Bounds

Conceptual map

II-04.01Economic optimality
II-04.02Valid population moments
II-04.03Identified sets
II-04.04Sharp extrema
II-04.05Projection of uncertainty

1. Optimality can imply an inequality

A moment inequality restricts a population set. Bounds come from optimizing a declared target over that set, and endpoint witnesses establish sharpness.

Suppose an agent chooses $$D=1$$ over $$D=0$$ and the utility difference is $u_{1}-u_{0}=\theta-c$ . Revealed preference gives $D(c-\theta)\le0$ pointwise when the observed choice is optimal and ties are allowed. Hence every admissible population law satisfies $\mathbb{E}[D(c-\theta)]\le0$ whenever the product is integrable.

The validity of this inequality depends on the choice set, utility specification, observation of $$c$$ , and treatment of optimization error. Sampling variation affects its estimated analogue and leaves the population restriction conceptually separate.

2. Incomplete support gives a linear program

Let latent types have probabilities $p=(p_{1},p_{2},p_{3})$ with $p\ge0$ and $1^{\top}p=1$ . Suppose the observed event identifies $p_{1}+p_{2}=0.6$ . The target is $\tau=p_{2}+2p_{3}$ . Since $p_{3}=0.4$ and $0\le p_{2}\le0.6$ ,

\[0.8\le \tau \le 1.4.\](1)

The lower endpoint is attained by $p^{L}$ =(0.6,0,0.4); the upper endpoint is attained by $p^{U}$ =(0,0.6,0.4). These feasible distributions prove sharpness: every intermediate target is attained by their convex mixtures.

3. Primal witnesses close the sharpness argument

Proposition 1 · Linear identified interval

Let $\mathcal{P}=\{p\in\mathbb{R}^{K}:p\ge0,\ 1^{\top}p=1,\ Ap=b,\ Gp\le d\}$ be nonempty and compact. For linear target $c^{\top}p$ , the identified set is the closed interval [ $\operatorname{min}_{\mathcal{P}}c^{\top}p$ , $\operatorname{max}_{\mathcal{P}}c^{\top}p$ ]. Both endpoints are attained.

Proof. A continuous linear functional attains its extrema on nonempty compact $\mathcal{P}$ . Convexity of $\mathcal{P}$ makes its scalar image convex; a compact convex subset of $\mathbb{R}$ is a closed interval. ∎

A dual certificate can prove an endpoint value, while a primal optimizer supplies a compatible population witness. Numerical sharpness reporting should include primal feasibility, dual feasibility, and their objective gap.

4. Sampling uncertainty enlarges a different set

If the observed share 0.6 is replaced by a confidence set $q\in[0.55,0.65]$ , project the joint set over ( $$p,q$$ ) subject to $p_{1}+p_{2}=q$ . Then $p_{3}=1-q$ and $\tau=p_{2}+2(1-q)$ , yielding projected range [0.7,1.45]. This is a confidence projection built around the population model; it is not the population identified interval in (1).

Failure case · Three diagnoses merged into one wide interval

An empty population feasible set signals incompatible restrictions or data moments. A wide nonempty set signals limited identifying content. A wide confidence projection can also arise from imprecise estimated moments. Each diagnosis calls for a different response.

At probability boundaries zero and one, active nonnegativity constraints change the tangent geometry. Equality tolerances can falsely make an infeasible LP appear feasible, so solver residuals must be compared with the scale and uncertainty of the moments.

5. Implementation, exercises, and sources

Encode probability normalization explicitly, retain all zero-mass boundary types, and solve both target directions. Store primal and dual solutions, active constraints, feasibility residuals, complementarity, and the duality gap. To assess assumptions, remove one row or inequality at a time and recompute the set.

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Exercises

Verify the endpoints and every convex-mixture target in Section 2.
Add inequality $p_{2}\le0.2$ and recompute sharp bounds.
Set the observed share to 1.1 and distinguish mathematical infeasibility from a numerical tolerance issue.

Partial solutions

1. Write $p_{2}=t\in[0,0.6]$ , $p_{1}=0.6-t$ , and $p_{3}=0.4$ ; then $\tau=0.8+t$ . 2. The interval becomes [0.8,1.0]. 3. Nonnegative probabilities summing to one cannot have a subset share 1.1, so the exact set is empty.

Charles F. Manski, Partial Identification of Probability Distributions.Population identified sets and bounds.
Alfred Galichon and Marc Henry (2011), “Set Identification in Models with Multiple Equilibria,” Review of Economic Studies 78, 1264–1298.Moment inequalities and set identification.
Alexander Schrijver, Theory of Linear and Integer Programming, chapters on linear-program duality.Primal witnesses and dual certificates.

6. Audit checkpoint

State the behavioral inequality, integrability, latent support, probability simplex, equality and inequality directions, population feasibility, target, endpoint witnesses, dual residuals, active boundaries, sampling confidence set, and projection rule.

7. Scope boundary

The chapter covers finite-support linear moment models, sharp target bounds, and confidence projection. Optimal moment selection and general nonlinear set inference require further theory.