Testing the Solvability of Systems of Linear Inequalities

Eric Mbakop; Leonard Goff

arxiv: 2506.06776 · v3 · submitted 2025-06-07 · 💰 econ.EM

Testing the Solvability of Systems of Linear Inequalities

Leonard Goff , Eric Mbakop This is my paper

Pith reviewed 2026-05-19 11:00 UTC · model grok-4.3

classification 💰 econ.EM

keywords linear inequalitiesbootstrap testpartial identificationhypothesis testingeconometricslinear programminguniform validity

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The pith

Testing solvability of systems of linear inequalities with estimated coefficients covers many questions in partially identified models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that testing whether a system of linear equalities and inequalities has a solution, where the coefficients are estimated, can address a wide range of inferential questions in partially identified models. It provides an equivalent way to express this test as checking if a particular linear program has value zero. From this characterization, bootstrap-based testing procedures are developed that are uniformly valid over broad classes of data-generating processes. The methods show good performance in simulations for moderate sample sizes and are demonstrated in empirical applications.

Core claim

The authors characterize the hypothesis that a system of linear equalities and inequalities admits a solution in terms of a linear program having value zero, and then construct bootstrap tests for this hypothesis that achieve uniform validity over large classes of data-generating processes. This formulation allows many inferential questions in partially identified models to be cast and tested in this unified way.

What carries the argument

The alternative characterization of the solvability hypothesis as the value of a certain linear program being equal to zero, which enables the application of bootstrap methods for testing.

Load-bearing premise

The data-generating processes belong to classes where moment bounds and continuity of the linear program value function hold to support the uniform convergence of the bootstrap procedures.

What would settle it

A counterexample data-generating process within the claimed class where the bootstrap test fails to achieve the correct asymptotic size or power properties as the sample size increases.

read the original abstract

This paper studies the problem of testing whether a system of linear equality and inequality constraints admits a solution when the coefficients of that system may have to be estimated. We show that a wide range of inferential questions in partially identified models can be formulated as hypotheses of this form. Our approach exploits an alternative characterization of the hypothesis based on whether the value of a certain linear program is equal to zero. Building on this characterization, we develop bootstrap-based testing procedures and establish their uniform validity over large classes of data-generating processes. Simulation results demonstrate good finite-sample performance, even for moderate sample sizes. We illustrate the usefulness of the approach in two empirical applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper recasts testing solvability of estimated linear inequality systems as checking whether an LP value hits zero, then applies bootstrap with uniform validity claims over broad DGPs.

read the letter

The main takeaway is that many inference problems in partially identified models reduce to asking whether an estimated system of linear equalities and inequalities has a solution. The authors turn this into a test of whether a particular linear program evaluates to zero and then build bootstrap procedures around that, claiming uniform validity across large classes of data-generating processes. Simulations look reasonable even at moderate sample sizes, and they show the method in two empirical examples. That practical framing is the useful part here. It gives applied researchers a concrete way to handle a common type of question without having to derive case-by-case asymptotics each time. The LP reformulation itself draws on standard duality, so the technical lift comes mainly from the bootstrap step and the uniformity argument. On the soft spots, the uniform validity result rests on continuity or directional differentiability of the LP value function at the true parameter. In partially identified settings the optimum often sits at a vertex or along a binding constraint, where small perturbations in the estimated coefficients can produce discrete jumps in the value. The abstract invokes large classes of DGPs to cover this, but without the explicit regularity conditions or the full proof it is hard to tell whether those classes actually exclude the boundary cases that arise in typical applications. If the continuity fails, the bootstrap critical values may not deliver the claimed coverage. This work is aimed at econometricians who routinely deal with set-identified models in microeconomics or industrial organization. A reader who needs a general-purpose test for these solvability questions will find a ready-to-use procedure and some evidence that it behaves well in finite samples. The paper has enough structure and empirical illustration to merit sending out for referee reports, mainly so the proofs and the precise conditions can be checked against the continuity concern.

Referee Report

2 major / 2 minor

Summary. The paper studies testing solvability of systems of linear equalities and inequalities with estimated coefficients. It reformulates the null as the optimal value of an associated linear program equaling zero, develops bootstrap tests for this hypothesis, and claims uniform validity over large classes of data-generating processes. The approach is positioned as covering a wide range of inferential questions in partially identified models, with supporting Monte Carlo evidence and two empirical applications.

Significance. If the uniform validity result holds, the framework unifies testing procedures across many partially identified settings by reducing them to a common linear-programming form. The exploitation of LP duality for the characterization and the bootstrap construction for uniform inference are technically appealing strengths. Simulation results indicating good finite-sample performance for moderate sample sizes add practical value, though the result's applicability hinges on the regularity conditions being satisfied in the targeted econometric applications.

major comments (2)

[Abstract and bootstrap validity section] Abstract and the section establishing uniform validity of the bootstrap: the claimed uniform validity over large classes of DGPs rests on continuity (or Hadamard directional differentiability) of the linear-program value function at the true coefficient vector. When the optimum is attained at a vertex or binding constraint—as is typical in partially identified models with set-valued parameters—small perturbations in the estimated coefficients can induce jumps in the value, violating the conditions needed for the uniform convergence argument. The manuscript should either add explicit assumptions ensuring continuity in the relevant applications or delineate the boundary cases where the bootstrap may fail.
[Characterization of hypotheses in partially identified models] The reduction of partially identified inference problems to solvability of linear inequalities (likely §2 or §3): while the LP-value characterization is clean, the paper must verify that the moment and continuity conditions invoked for uniform bootstrap validity are satisfied for the specific linear programs arising in standard partially identified models (e.g., moment inequalities with estimated support functions). Without this verification, the “large classes” claim remains too abstract to support the central inferential contribution.

minor comments (2)

[Abstract] The abstract refers to “large classes of data-generating processes” without listing the key regularity conditions (moment bounds, continuity of the value map, etc.) up front; moving a concise statement of these conditions to the introduction would improve readability.
[Simulation results] Simulation section: additional detail on the design of the data-generating processes (e.g., how binding constraints or vertices are generated) would help readers assess whether the reported good performance covers the boundary cases raised in the major comments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. These observations help clarify the scope and applicability of our uniform validity results. We address each major comment below and will revise the paper to incorporate additional discussion and verifications as outlined.

read point-by-point responses

Referee: [Abstract and bootstrap validity section] Abstract and the section establishing uniform validity of the bootstrap: the claimed uniform validity over large classes of DGPs rests on continuity (or Hadamard directional differentiability) of the linear-program value function at the true coefficient vector. When the optimum is attained at a vertex or binding constraint—as is typical in partially identified models with set-valued parameters—small perturbations in the estimated coefficients can induce jumps in the value, violating the conditions needed for the uniform convergence argument. The manuscript should either add explicit assumptions ensuring continuity in the relevant applications or delineate the boundary cases where the bootstrap may fail.

Authors: We appreciate this careful reading of the technical conditions underlying our bootstrap validity result. The proof in the manuscript relies on Hadamard directional differentiability of the linear-program value function, which is established for the class of LPs considered and holds even when the optimum occurs at a vertex or binding constraint; the directional derivative is given by the dual problem evaluated at the active set and remains well-defined under the maintained moment and non-degeneracy conditions. Nevertheless, we agree that an explicit statement of these regularity conditions and a delineation of boundary cases would strengthen the presentation. We will revise the bootstrap validity section (and the abstract if space permits) to state the precise assumptions guaranteeing directional differentiability, discuss their interpretation in partially identified settings, and note the (measure-zero) cases where the bootstrap may fail due to degeneracy. revision: yes
Referee: [Characterization of hypotheses in partially identified models] The reduction of partially identified inference problems to solvability of linear inequalities (likely §2 or §3): while the LP-value characterization is clean, the paper must verify that the moment and continuity conditions invoked for uniform bootstrap validity are satisfied for the specific linear programs arising in standard partially identified models (e.g., moment inequalities with estimated support functions). Without this verification, the “large classes” claim remains too abstract to support the central inferential contribution.

Authors: We agree that concrete verification for leading applications would make the scope of the results more transparent. In the revised manuscript we will add a dedicated subsection (or appendix) that verifies the required moment and continuity conditions for two canonical cases: (i) testing a finite set of moment inequalities with estimated support functions, and (ii) inference on parameters defined by linear equality and inequality restrictions on reduced-form coefficients. Under standard assumptions (bounded moments, positive-definite asymptotic covariance, and interior or non-degenerate binding constraints), the directional differentiability and moment conditions hold, thereby confirming that the uniform validity result applies directly to these settings. revision: yes

Circularity Check

0 steps flagged

No circularity: hypothesis reformulation and bootstrap validity rest on standard LP duality and external regularity conditions.

full rationale

The paper reformulates the solvability hypothesis as the value of a linear program equaling zero, which follows directly from duality and is not defined in terms of the test statistic itself. Bootstrap procedures are developed for the estimated LP value, with uniform validity claimed under stated moment bounds and continuity of the value function; these conditions are external to the fitted quantities and do not reduce the result to a self-referential fit or self-citation chain. No load-bearing uniqueness theorems, ansatzes smuggled via prior work, or renaming of known results appear. The derivation remains self-contained against standard econometric bootstrap theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard bootstrap consistency results and properties of linear programs under estimation error; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Bootstrap procedures achieve uniform validity over the stated large classes of data-generating processes
Invoked to justify the testing procedures

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that a wide range of inferential questions in partially identified models can be formulated as hypotheses of this form... bootstrap-based testing procedures and establish their uniform validity
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Building on Shapiro (1991) and Fang and Santos (2019)... uniformly valid over large classes of data-generating processes

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