arxiv: 2604.24904 · v2 · submitted 2026-04-27 · 💰 econ.EM · math.ST· stat.TH

Recognition: unknown

Inference for Linear Systems with Unknown Coefficients

Yuehao Bai , Kirill Ponomarev , Andres Santos , Azeem M. Shaikh , Max Tabord-Meehan , Alexander Torgovitsky

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:02 UTC · model grok-4.3

classification 💰 econ.EM math.STstat.TH

keywords hypothesis testinglinear systemsunknown coefficientssample splittingpartial identificationtotal variation distancehigh-dimensional inferencenonparametric IV

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The pith

Sample-splitting tests remain valid for existence of non-negative solutions to linear systems with all coefficients unknown, even as dimension grows rapidly with sample size.

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

The paper establishes valid procedures to test whether a linear system of equations admits a solution obeying non-negativity constraints when every coefficient in the system, including slopes, must be estimated from data. This testing problem arises directly when forming confidence sets for partially identified parameters in nonparametric instrumental variables models, treatment effect models, and random coefficient models. The authors first characterize the closure of the null hypothesis under total variation distance to separate feasible from impossible testing problems. They then construct sample-splitting tests whose validity follows from weak, interpretable conditions on the linear system that explicitly allow the number of equations and unknowns to increase quickly with the sample size.

Core claim

The closure of the null hypothesis with respect to total variation distance admits a characterization that supports sample-splitting tests whose size and power properties hold under weak conditions on the linear system; these conditions permit the dimensionality to grow rapidly with the sample size and eliminate the need for simulation to obtain critical values.

What carries the argument

Sample-splitting tests built from the total-variation closure of the null hypothesis for non-negativity constrained linear systems with unknown coefficients.

If this is right

The tests control size under the null and deliver power under the stated weak conditions on the linear system.
The dimensionality of the system may increase rapidly with sample size while validity is preserved.
Critical values are obtained analytically without simulation.
The procedures directly support construction of confidence sets for partially identified parameters in nonparametric IV, treatment effect, and random coefficient models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sample-splitting approach could be examined for other inequality-constrained estimation problems that currently rely on simulation-based critical values.
Empirical researchers working with high-dimensional random coefficient models may obtain simpler inference by replacing existing methods with these tests when the weak conditions hold.
If the closure characterization extends to related distance metrics, the framework might apply to testing problems outside econometrics that involve high-dimensional linear inequalities.

Load-bearing premise

The characterization of the closure of the null hypothesis with respect to total variation distance, together with the weak conditions on the linear system that allow high-dimensional growth, must hold for the sample-splitting tests to be valid.

What would settle it

A data-generating process in which the linear system violates the stated weak conditions yet the sample-splitting test is applied, producing rejection probabilities under the null that exceed the nominal level by a non-negligible amount.

Figures

Figures reproduced from arXiv: 2604.24904 by Alexander Torgovitsky, Andres Santos, Azeem M. Shaikh, Kirill Ponomarev, Max Tabord-Meehan, Yuehao Bai.

**Figure 1.** Figure 1: Shaded region indicates the null set in the ( view at source ↗

**Figure 2.** Figure 2: Cox et al. (2025) simulation rejection curves, arranged by H (rows) and n (columns). In each plot, the hypothesized value of θ is on the horizontal axis. The shaded region is the identified set for θ. The dashed line represents the screening method and the solid line represents the direct method. 18 view at source ↗

**Figure 3.** Figure 3: Cox et al. (2025) simulation rejection curves, arranged by H (rows) and n (columns). In each plot, the hypothesized value of θ is on the horizontal axis. The shaded region is the identified set for θ. The dashed line represents the screening method and the solid line represents the direct method. 19 view at source ↗

**Figure 4.** Figure 4: Goff and Mbakop (2025) simulation rejection curves, arranged by n. In each plot, the hypothesized value of τ0 is on the horizontal axis. The shaded region is the identified set. The dashed line represents the screening method and the solid line represents the direct method. 21 view at source ↗

**Figure 5.** Figure 5: Freyberger and Horowitz (2015) simulation rejection curves, arranged by n. In each plot, the hypothesized value of L0 is on the horizontal axis. The shaded region is the identified set for L(g). The dashed line represents the screening method and the solid line represents the direct method. 23 view at source ↗

read the original abstract

This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system of equations, including the slope coefficients, to be unknown. For this reason, we describe the linear system as having unknown (as opposed to known) coefficients. This hypothesis testing problem arises naturally when constructing confidence sets for possibly partially identified parameters in the analysis of nonparametric instrumental variables models, treatment effect models, and random coefficient models, among other settings. To rule out certain instances in which the testing problem is impossible, in the sense that the power of any test will be bounded by its size, we begin our analysis by characterizing the closure of the null hypothesis with respect to the total variation distance. We then use this characterization to develop novel testing procedures based on sample-splitting. We establish the validity of our testing procedures under weak and interpretable conditions on the linear system. An important feature of these conditions is that they permit the dimensionality of the problem to grow rapidly with the sample size. A further attractive property of our tests is that they do not require simulation to compute suitable critical values. We illustrate the practical relevance of our theoretical results in a simulation study.

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

This paper gives sample-splitting tests for non-negative solutions to linear systems with all coefficients unknown, plus a total-variation closure that rules out impossible cases and supports high-dimensional growth.

read the letter

The main thing here is a set of tests for whether a linear system has a non-negative solution when none of the coefficients are treated as known. They start by characterizing the closure of that null in total variation distance, which excludes settings where any test would have power no better than size. From there they build sample-splitting procedures that control size under conditions allowing the dimension to grow quickly with sample size, and the tests avoid simulation-based critical values altogether. That combination looks like the actual advance over earlier work that kept some coefficients fixed. The conditions are presented as weak and interpretable, and the high-dimensional feature matches needs in modern applications like nonparametric IV, treatment-effect bounds, and random-coefficient models. The abstract and stress-test note give no sign of internal contradictions or hidden gaps in the argument. One soft spot is that the simulation study is described only as illustrative, so its ability to show finite-sample behavior under the stated conditions remains to be checked in the full proofs and tables. In practice, applied users will still have to confirm the linear-system conditions hold, which could be straightforward or not depending on the model. This is aimed at econometricians and statisticians who build confidence sets for partially identified parameters. A reader working on those problems would get usable procedures and a clearer sense of when the testing problem is feasible. It has enough formal grounding and practical relevance to deserve a serious referee.

Referee Report

0 major / 2 minor

Summary. The paper develops procedures for testing the existence of non-negative solutions to a linear system of equations in which all coefficients (including slopes) are unknown. It first characterizes the closure of the null hypothesis in total variation distance to exclude impossible testing problems, then constructs sample-splitting tests whose validity is established under weak, interpretable conditions on the linear system that explicitly allow the dimension to grow rapidly with sample size. The tests require no simulation for critical values and are illustrated via a simulation study. The setting arises in constructing confidence sets for partially identified parameters in nonparametric IV, treatment effect, and random coefficient models.

Significance. If the central results hold, the paper supplies a practical, simulation-free method for hypothesis testing in high-dimensional partially identified econometric models under conditions that are weaker and more interpretable than many existing approaches. The explicit allowance for rapid dimension growth and the total-variation closure characterization are notable strengths that could facilitate reliable inference in settings where conventional methods fail.

minor comments (2)

[Abstract] Abstract: the simulation study is mentioned but its design (e.g., dimension growth rates, specific linear systems, or performance metrics) is not described; a single additional sentence would help readers gauge the practical scope of the numerical evidence.
The manuscript would benefit from a short table or remark comparing the proposed tests' computational requirements and finite-sample size/power to existing simulation-based alternatives in the literature on partial identification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the paper, which correctly highlights the sample-splitting tests for existence of non-negative solutions to linear systems with unknown coefficients, the total variation closure characterization, and the allowance for dimension growth. The recommendation for minor revision is appreciated. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central derivation begins with a mathematical characterization of the closure of the null hypothesis (existence of non-negative solutions to the linear system) in total variation distance. This step is a direct analysis of the hypothesis set and does not reduce to any fitted parameter or self-referential definition. The subsequent sample-splitting tests are constructed from this characterization, and their validity is established by proving size control under explicit, weak conditions on the linear system that explicitly permit rapid growth in dimensionality with sample size. No step renames a known result, imports uniqueness via self-citation, or treats a fitted input as a prediction; the argument is self-contained against external benchmarks and does not rely on load-bearing self-citations or ansatzes smuggled from prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard econometric assumptions such as consistent estimation of system parameters and suitable convergence rates, but does not introduce new free parameters or invented entities; the central contribution is a new testing procedure rather than new primitives.

axioms (1)

domain assumption Parameters of the linear system are estimable from data at rates sufficient for the sample-splitting procedure to control size.
Implicit requirement for the validity claims in high-dimensional settings.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

max 1≤j≤d1+1 sup ∥y∥1≤1 1√n nX i=1 y′ξj(Zi, P) # =E P

+K 1,p∥ ˆA† 0,nˆbj,n −A † 0(P)b j(P)∥ 2 2 .(71) Further note that the arguments employed in (66) and (67) imply that uniformly inP∈Pand 1≤j≤d 1+1 ∥ ˆM0,nˆbj,n −M 0(P)b j(P)∥ 2 =O P r K1,p n +K 2,p( r (K0,p ∨K 1,p) log(1 +p) n + an n ) ∥ ˆA† 0,nˆbj,n −A † 0(P)b j(P)∥ 2 =O P r K1,p n +K 2,p( r (K0,p ∨K 1,p) log(1 +p) n + an n ) .(72) Therefore, combining re...

2012
[2]

Hence, we have ∥(A† 0)′∥2,2 = sup ∥x∥2≤1 (x′(A′ 0A0)−1A′ 0A0(A′ 0A0)−1x)1/2 = sup ∥x∥2≤1 (x′(A′ 0A0)−1x)1/2 = 1 s(A0) ,(106) where the first equality follows by definition of∥ · ∥ 2,2 and the final one from∥(A ′ 0A0)−1∥2,2 = 1/s(A0). 47