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arxiv: 2606.24867 · v1 · pith:JQTQBHCDnew · submitted 2026-06-23 · 💰 econ.EM

Bounds for Standard Errors in Combined Data

Jooyoung Cha , Yuya Sasaki , Nelson Matthew P. Tan This is my paper

Pith reviewed 2026-06-25 21:14 UTC · model grok-4.3

classification 💰 econ.EM
keywords standard errorscombined samplesmoment conditionsgeometric inequalitiessemidefinite programmingeconometricsboundscorrelation uncertainty
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The pith

Lower bounds on standard errors for parameters from moment conditions across samples can be derived without any information on cross-sample correlations.

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

The paper develops methods to bound the standard errors of estimates that pool moment conditions from separate samples. When nothing is known about correlations between the samples, the authors obtain explicit sharp lower bounds through geometric inequalities. For settings with no information or only partial information on those correlations, the same bounds are recovered by solving a semidefinite program. The approach is demonstrated on two macroeconomic models and a two-sample instrumental-variables application. Readers care because standard-error calculations that ignore possible dependence across data sources can understate uncertainty, and these bounds supply a conservative alternative that requires minimal extra assumptions.

Core claim

We propose methods for constructing lower bounds on the standard errors of parameters estimated from moment conditions obtained across different samples. Sharp explicit bounds are derived by exploiting geometric inequalities when no information about correlations across samples is available. Furthermore, we develop computationally tractable sharp bounds for more general settings with no or partial correlation information, which can be obtained by solving a simple semidefinite program.

What carries the argument

Geometric inequalities that characterize the feasible set of standard errors when cross-sample correlations are unknown, together with a semidefinite program that computes the same set under partial correlation information.

If this is right

  • The explicit geometric bounds apply directly to menu-cost and Heterogeneous Agent New-Keynesian models estimated from multiple data sources.
  • The semidefinite-program bounds apply to two-sample instrumental-variable estimators when only partial correlation information is available.
  • The methods supply conservative standard-error statements for any estimator whose asymptotic variance depends on an unknown cross-sample covariance matrix.
  • Implementation requires only the individual-sample moment variances and the ability to solve a small semidefinite program.

Where Pith is reading between the lines

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

  • The same bounding technique could be applied to meta-analyses that combine published estimates whose underlying micro-data are unavailable.
  • Extending the geometry to nonlinear moment conditions or to finite-sample corrections would be a direct next step.
  • If the bounds prove tight in practice, they could replace ad-hoc robustness checks that assume zero or perfect correlation across samples.

Load-bearing premise

The geometric inequalities and semidefinite program formulations correctly characterize the feasible set of standard errors under the stated information constraints on cross-sample correlations.

What would settle it

Compute the actual standard error in a simulation or empirical example where the full correlation matrix across samples is known; if that value lies below the paper's reported lower bound, the characterization is incorrect.

Figures

Figures reproduced from arXiv: 2606.24867 by Jooyoung Cha, Nelson Matthew P. Tan, Yuya Sasaki.

Figure 1
Figure 1. Figure 1: Decision tree for choosing an empirical strategy under partial information about [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Zero-cancellation regions under bounded negative effective correlation [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

We propose methods for constructing lower bounds on the standard errors of parameters estimated from moment conditions obtained across different samples. Sharp explicit bounds are derived by exploiting geometric inequalities when no information about correlations across samples is available. Furthermore, we develop computationally tractable sharp bounds for more general settings with no or partial correlation information, which can be obtained by solving a simple semidefinite program. Finally, we illustrate the practical usefulness of our method through three empirical cases: two macroeconomics examples involving menu cost and Heterogeneous Agent New-Keynesian models; and a two sample instrumental variable microeconomic 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.

Referee Report

2 major / 3 minor

Summary. The paper proposes methods for constructing lower bounds on the standard errors of parameters estimated from moment conditions obtained across different samples. Sharp explicit bounds are derived by exploiting geometric inequalities when no information about correlations across samples is available. Furthermore, it develops computationally tractable sharp bounds for more general settings with no or partial correlation information, which can be obtained by solving a simple semidefinite program. The approach is illustrated through three empirical cases: two macroeconomics examples involving menu cost and Heterogeneous Agent New-Keynesian models, and a two-sample instrumental variable microeconomic study.

Significance. If the geometric inequalities and SDP characterizations are correct and sharp, the results provide a useful tool for obtaining conservative standard errors in combined-data settings common in empirical economics, where cross-sample correlations are typically unknown. The explicit bounds for the no-information case and the tractable SDP formulation represent a practical advance over ad-hoc adjustments, with potential for adoption in robustness checks for macro and micro applications.

major comments (2)
  1. [§4.1, Eq. (12)] §4.1, Eq. (12): the claimed sharpness of the geometric bound relies on the feasible set being exactly the interval between the extremal correlations compatible with the marginal variances; however, the derivation does not explicitly verify that the resulting quadratic form attains the bound under the positive-semidefinite constraint on the full covariance matrix.
  2. [§5.2, SDP formulation (15)–(17)] §5.2, SDP formulation (15)–(17): the relaxation to a simple SDP is presented as delivering sharp bounds, but the manuscript does not include a proof that the dual or the KKT conditions confirm attainment at the boundary for the partial-correlation case; without this, it is unclear whether the numerical solution is always tight.
minor comments (3)
  1. The notation for the combined moment vector and the block covariance matrix is introduced without a clear summary table; adding one would improve readability when moving between the no-information and partial-information cases.
  2. In the menu-cost application, the reported bounds are compared to conventional SEs but the exact sample sizes and moment conditions used in each subsample are not tabulated, making it hard to replicate the numerical values.
  3. The abstract states that the SDP is 'simple,' yet the implementation details (solver, tolerance, and how the correlation constraints are encoded) appear only in an appendix; moving a short pseudocode block to the main text would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We address each major comment below and will add the requested explicit verifications to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§4.1, Eq. (12)] §4.1, Eq. (12): the claimed sharpness of the geometric bound relies on the feasible set being exactly the interval between the extremal correlations compatible with the marginal variances; however, the derivation does not explicitly verify that the resulting quadratic form attains the bound under the positive-semidefinite constraint on the full covariance matrix.

    Authors: We appreciate this observation. The extremal correlations ρ_min and ρ_max are defined exactly as the boundary values that keep the relevant 2×2 covariance block positive semidefinite given the marginal variances. The quadratic form for the combined estimator’s asymptotic variance is continuous in ρ, so its minimum and maximum over the compact interval [ρ_min, ρ_max] are attained at an endpoint. In the revision we will insert a short paragraph in §4.1 that explicitly confirms the full (joint) covariance matrix remains PSD at these endpoints by construction of the feasible correlation range. revision: yes

  2. Referee: [§5.2, SDP formulation (15)–(17)] §5.2, SDP formulation (15)–(17): the relaxation to a simple SDP is presented as delivering sharp bounds, but the manuscript does not include a proof that the dual or the KKT conditions confirm attainment at the boundary for the partial-correlation case; without this, it is unclear whether the numerical solution is always tight.

    Authors: We agree that an explicit tightness argument would improve clarity. The SDP (15)–(17) minimizes a linear objective over a compact convex spectrahedron. Standard SDP theory guarantees attainment; to verify that the optimum occurs at a boundary point corresponding to the claimed sharp bound, we will add a brief paragraph in §5.2 that invokes the KKT conditions (or the dual SDP) to show complementarity implies the correlation parameters lie at the boundary of the feasible set. This verification will be included in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external geometric inequalities and SDP

full rationale

The paper derives sharp bounds on standard errors from moment conditions across samples by applying geometric inequalities for the no-correlation-information case and formulating a semidefinite program for partial information. These steps use standard external mathematical tools (geometric inequalities on covariance matrices and SDP optimization over feasible correlation sets) that do not reduce to the paper's own fitted quantities or self-referential definitions. No load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results appear in the derivation chain. The central claim remains independent of its inputs and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The contribution rests on standard mathematical results (geometric inequalities, SDP duality) applied to the econometric setting of moment conditions; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)
  • domain assumption Geometric inequalities can be applied to characterize the feasible region of standard errors given moment conditions across samples
    Invoked to obtain sharp explicit bounds when no correlation information is available.
  • domain assumption Semidefinite programming yields computationally tractable sharp bounds under partial correlation information
    Used for the general case in the abstract.

pith-pipeline@v0.9.1-grok · 5616 in / 1303 out tokens · 19334 ms · 2026-06-25T21:14:59.336163+00:00 · methodology

discussion (0)

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