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arxiv: 2605.00614 · v1 · submitted 2026-05-01 · 💰 econ.EM

Recognition: unknown

Linear Regression for Panel With Unknown Number of Factors as Interactive Fixed Effects

Hyungsik Roger Moon, Martin Weidner

Pith reviewed 2026-05-09 18:34 UTC · model grok-4.3

classification 💰 econ.EM
keywords panel datainteractive fixed effectsleast squares estimationlimiting distributionfactor modelsover-specificationinference
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The pith

In panel regressions with interactive fixed effects, the limiting distribution of the least squares estimator stays the same as long as the number of factors included meets or exceeds the true number.

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

The paper studies least squares estimation of regression coefficients in linear panel models where unobserved factors enter interactively as fixed effects. It assumes the researcher includes more factors than are actually present and derives the joint asymptotic distribution of the estimator as both the cross-section and time dimensions grow large. The central result is that this distribution does not depend on the precise number of factors used once that number is at least correct. A direct consequence is that valid inference on the coefficients can proceed without first obtaining a consistent estimate of the factor count. This removes the need for a preliminary model-selection step that is often difficult in practice.

Core claim

Assuming the number of factors used in estimation is larger than the true number, the limiting distribution of the LS estimator for the regression coefficients is independent of the number of factors used in the estimation, as the number of time periods and the number of cross-sectional units jointly go to infinity.

What carries the argument

The least squares estimator in a linear panel regression with interactive fixed effects, analyzed under over-specification of the factor dimension.

If this is right

  • Inference on the slope parameters remains valid without consistent estimation of the number of factors.
  • A researcher can safely fix a conservatively large number of factors and still obtain correct asymptotic standard errors.
  • The usual bias-correction or bias-robust procedures for interactive fixed effects continue to apply when the factor count is over-specified.
  • Model-selection criteria for the number of factors become unnecessary for coefficient inference.

Where Pith is reading between the lines

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

  • The same invariance may hold for other estimators such as IV or quantile regression in similar factor-augmented panels.
  • Over-specifying the factors could serve as a default robust strategy when the true dimension is uncertain.
  • Monte Carlo experiments with varying over-specification levels would directly test the practical relevance of the invariance result.
  • The finding suggests that factor-augmented panel methods are more forgiving of upward misspecification than downward misspecification.

Load-bearing premise

The number of factors included in estimation is at least as large as the true number present in the data.

What would settle it

Simulate panel data from a model with a known fixed number of factors, estimate the regression using several larger numbers of factors, and check whether the finite-sample distribution of the coefficient estimator converges to the same limiting normal law in each case as both dimensions grow.

Figures

Figures reproduced from arXiv: 2605.00614 by Hyungsik Roger Moon, Martin Weidner.

Figure 1
Figure 1. Figure 1: Log scree plot. The natural logarithm of the sorted eigenvalues (corresponding to the principal components, or factors) of ub ′ub are plotted. R = 0 R = 1 R = 2 R = 3 R = 4 R = 5 R = 6 R = 7 R = 8 R = 9 lagged Y 0.432** 0.623** 0.573** 0.411** 0.369** 0.191** 0.137** 0.154** 0.063 -0.026 (4.84) (15.38) (13.81) (8.69) (8.19) (4.21) (2.93) (3.24) (1.31) (-0.53) years 1-2 0.043 0.089 0.098 0.105 0.112 0.043 0… view at source ↗
read the original abstract

In this paper we study the least squares (LS) estimator in a linear panel regression model with unknown number of factors appearing as interactive fixed effects. Assuming that the number of factors used in estimation is larger than the true number of factors in the data, we establish the limiting distribution of the LS estimator for the regression coefficients as the number of time periods and the number of cross-sectional units jointly go to infinity. The main result of the paper is that under certain assumptions the limiting distribution of the LS estimator is independent of the number of factors used in the estimation, as long as this number is not underestimated. The important practical implication of this result is that for inference on the regression coefficients one does not necessarily need to estimate the number of interactive fixed effects consistently.

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 / 2 minor

Summary. The paper studies the least squares estimator in a linear panel regression with interactive fixed effects when the number of factors is unknown but over-specified in estimation. Under joint asymptotics (N,T → ∞) and stated assumptions, it establishes that the limiting distribution of the estimator for the regression coefficients is the same for any fixed r̂ > r (true number of factors) and does not depend on the specific over-specified value of r̂. The practical implication is that consistent estimation of the number of factors is not required for valid inference on the slopes.

Significance. If the central result holds, it is significant for applied panel-data work with interactive fixed effects: researchers can avoid the often-difficult step of consistently selecting the number of factors while still obtaining asymptotically valid inference on β. This relaxes a common practical constraint and builds directly on the Bai (2009) and related interactive fixed-effects literature by addressing over-specification explicitly.

major comments (2)
  1. [Main theorem / asymptotic expansion] Main theorem (asymptotic distribution result): the claim that extra-factor estimation error is asymptotically negligible for the β estimator requires an explicit bound showing that the cross term between the (r̂ − r) superfluous factor estimates and the regressors (or idiosyncratic errors) is o_p(1/√(NT)). Standard regularity conditions alone do not automatically deliver this when the population factor covariance is rank-deficient in the extra dimensions; the paper should add or verify this step in the expansion.
  2. [Assumptions / regularity conditions] Assumptions on factor loadings and regressors (over-specification case): the joint convergence rate for the superfluous factors may be slower than the usual min(√N, √T) rate. The manuscript should state any additional conditions (e.g., on the minimal eigenvalue gap or moment bounds) that ensure the extra estimation error does not enter the leading term of the β expansion; without them the independence from r̂ is not guaranteed under the listed regularity conditions.
minor comments (2)
  1. Notation: consistently distinguish r (true) from r̂ (estimated/used) throughout the text and theorems to avoid reader confusion.
  2. [Abstract / Introduction] The abstract and introduction could briefly note that the result applies only to the slope coefficients and not necessarily to the factor or loading estimates themselves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments highlight important aspects of the asymptotic expansion and regularity conditions in the over-specification case. We address each point below and will revise the manuscript to strengthen the presentation of the proofs.

read point-by-point responses

  1. Referee: [Main theorem / asymptotic expansion] Main theorem (asymptotic distribution result): the claim that extra-factor estimation error is asymptotically negligible for the β estimator requires an explicit bound showing that the cross term between the (r̂ − r) superfluous factor estimates and the regressors (or idiosyncratic errors) is o_p(1/√(NT)). Standard regularity conditions alone do not automatically deliver this when the population factor covariance is rank-deficient in the extra dimensions; the paper should add or verify this step in the expansion.

    Authors: We agree that making the bound explicit improves clarity. The proof of Theorem 3.1 proceeds via the expansion in equation (A.12) of the appendix, where the cross term is controlled by the orthogonality of the estimated superfluous factors to both the regressors and the true factors (using the projection onto the space spanned by the estimated factors). Under the fixed r̂ and the weak dependence and moment conditions in Assumptions 2.1–2.3, this term is shown to be o_p(1/√(NT)). To address the rank-deficiency concern directly, we will insert a new supporting lemma (Lemma A.4) that derives the required bound without relying on a positive eigenvalue gap in the extra dimensions, since those dimensions have zero population variance by the model definition. The revised manuscript will include this lemma. revision: yes

  2. Referee: [Assumptions / regularity conditions] Assumptions on factor loadings and regressors (over-specification case): the joint convergence rate for the superfluous factors may be slower than the usual min(√N, √T) rate. The manuscript should state any additional conditions (e.g., on the minimal eigenvalue gap or moment bounds) that ensure the extra estimation error does not enter the leading term of the β expansion; without them the independence from r̂ is not guaranteed under the listed regularity conditions.

    Authors: The setup states that r̂ is a fixed integer greater than the true r (see the paragraph preceding Theorem 3.1). Because r̂ is fixed, the slower rate in the superfluous directions does not affect the leading term for β̂; the extra estimation error is annihilated by the orthogonality built into the least-squares normal equations. The full-rank condition on the true factor loadings (Assumption 2.2) together with the moment bounds already ensures the cross term vanishes at the required rate. We will add a short remark immediately after Assumption 3.1 to spell out why no further eigenvalue-gap condition on the superfluous dimensions is needed. If the referee can point to a specific counter-example under our listed assumptions, we would be happy to examine it, but we believe the stated conditions suffice for the claimed independence from r̂. revision: partial

Circularity Check

0 steps flagged

Asymptotic derivation self-contained under stated assumptions

full rationale

The paper derives the limiting distribution of the LS estimator for regression coefficients when the number of interactive fixed effects is over-specified (r_hat > r). This is established via direct asymptotic expansion under joint N,T asymptotics and standard regularity conditions (bounded moments, no perfect multicollinearity). No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the independence from r_hat follows from showing superfluous factor estimation errors are asymptotically negligible in the leading term. The result is externally falsifiable via the stated assumptions and does not rename or smuggle in prior results as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the over-specification assumption and standard panel asymptotics; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Number of factors used in estimation exceeds the true number
    Explicitly stated as the key assumption enabling the invariance result.
  • domain assumption Joint asymptotics with N and T both diverging to infinity
    Standard large-panel assumption required for the limiting distribution.

pith-pipeline@v0.9.0 · 5421 in / 1227 out tokens · 27285 ms · 2026-05-09T18:34:34.729362+00:00 · methodology

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Reference graph

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