arxiv: 2605.00612 · v1 · submitted 2026-05-01 · 💰 econ.EM

Recognition: unknown

Dynamic Linear Panel Regression Models with Interactive Fixed Effects

Hyungsik Roger Moon, Martin Weidner

Pith reviewed 2026-05-09 18:37 UTC · model grok-4.3

classification 💰 econ.EM

keywords panel data modelsinteractive fixed effectsdynamic regressionasymptotic biasbias correctionpredetermined regressorsleast squarestest statistics

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The pith

The least squares estimator for dynamic panel models with interactive fixed effects has two sources of asymptotic bias that can be corrected for consistency and valid inference.

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

This paper works out the asymptotic distribution of the least squares estimator in linear panel regressions that include interactive fixed effects and predetermined regressors like lagged dependent variables, as both the number of units N and time periods T grow large. It identifies bias arising from error term correlation or heteroscedasticity and from the predetermined regressors themselves. The authors derive a bias-corrected estimator and bias-corrected versions of the Wald, LR, and LM tests, which are shown to follow chi-squared distributions asymptotically. Monte Carlo simulations indicate these corrections perform well even in finite samples. A sympathetic reader would care because these models are widely used in econometrics, and uncorrected biases can lead to incorrect conclusions about dynamic effects and factor influences.

Core claim

In the limit where both the cross-sectional dimension and the number of time periods become large, the least squares estimator of the regression coefficients in linear panel models with interactive fixed effects and predetermined regressors is asymptotically biased due to two sources: correlation or heteroscedasticity of the idiosyncratic error term, and the predetermined nature of the regressors as opposed to strict exogeneity. A bias-corrected least squares estimator is provided, and bias-corrected Wald, likelihood ratio, and Lagrange multiplier test statistics are shown to be asymptotically chi-squared distributed.

What carries the argument

The first-order asymptotic expansions of the least squares estimator under double asymptotics that isolate the bias terms from the idiosyncratic errors and from predetermined regressors.

If this is right

The bias-corrected LS estimator is consistent and asymptotically normal under the stated conditions.
Bias-corrected test statistics have chi-squared limiting distributions, enabling standard inference.
The corrections apply to models with lagged dependent variables and interactive factors.
Monte Carlo evidence supports good performance for moderate sample sizes.
These results hold when both N and T diverge to infinity.

Where Pith is reading between the lines

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

Empirical researchers using dynamic panel data with unobserved heterogeneity factors can apply these corrections to avoid biased estimates of coefficients.
This approach might be extended to nonlinear models or other estimation methods like instrumental variables in similar settings.
Future work could examine the performance when T is fixed and only N grows, or vice versa.
Connections to factor-augmented regressions in macroeconomics could benefit from these bias corrections.

Load-bearing premise

Both the number of cross-sectional units and the number of time periods must go to infinity, and the interactive factors and predetermined regressors must satisfy certain regularity conditions for the bias expansions to be valid.

What would settle it

If the bias-corrected estimator continues to exhibit significant finite-sample bias or the test statistics do not follow chi-squared distributions in large N and T simulations with correlated errors or lagged regressors, the asymptotic results would be falsified.

read the original abstract

We analyze linear panel regression models with interactive fixed effects and predetermined regressors, for example lagged-dependent variables. The first-order asymptotic theory of the least squares (LS) estimator of the regression coefficients is worked out in the limit where both the cross-sectional dimension and the number of time periods become large. We find two sources of asymptotic bias of the LS estimator: bias due to correlation or heteroscedasticity of the idiosyncratic error term, and bias due to predetermined (as opposed to strictly exogenous) regressors. We provide a bias-corrected LS estimator. We also present bias-corrected versions of the three classical test statistics (Wald, LR, and LM test) and show their asymptotic distribution is a chi-squared distribution. Monte Carlo simulations show the bias correction of the LS estimator and of the test statistics also work well for finite sample sizes.

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

Moon and Weidner derive explicit first-order bias corrections for the LS estimator and the three classical tests in dynamic panels with interactive fixed effects under joint N,T asymptotics.

read the letter

The main point is that this paper works out the asymptotic bias for least squares in linear panels that combine interactive fixed effects with predetermined regressors such as lagged dependent variables. They isolate two separate bias terms—one from idiosyncratic error correlation or heteroscedasticity, the other from the regressors not being strictly exogenous—and then supply corrected estimators and bias-adjusted Wald, LR, and LM statistics whose limits are chi-squared. Monte Carlo results show the corrections reduce finite-sample distortion, which is the practical payoff for applied work.

Referee Report

2 major / 3 minor

Summary. The manuscript analyzes linear panel regression models with interactive fixed effects and predetermined regressors (including lagged dependent variables). Under joint asymptotics with both N and T diverging, it derives the first-order asymptotic bias of the least-squares estimator, isolating two sources: one from correlation or heteroscedasticity in the idiosyncratic errors and one from the predetermined (rather than strictly exogenous) nature of the regressors. A bias-corrected LS estimator is constructed, and bias-corrected versions of the Wald, LR, and LM statistics are shown to be asymptotically chi-squared. Monte Carlo evidence indicates that the corrections perform well in finite samples.

Significance. If the bias expansions and regularity conditions hold, the paper supplies a targeted extension of bias-correction methods to dynamic panel models with interactive fixed effects, a setting frequently encountered in empirical work. Explicit separation of the two bias channels, together with corrected inference procedures and supporting simulations, offers a practical tool for reducing finite-sample distortion in such models. The grounding in explicit moment and factor assumptions, rather than data-driven fitting, strengthens the contribution.

major comments (2)

[§3.2, Theorem 3.1] §3.2, Theorem 3.1: The bias term arising from predetermined regressors is obtained by expanding the score around the probability limit of the factor estimates; the paper should verify that the cross-term between the predetermined regressor and the factor estimation error is o_p((NT)^{-1/2}) under the stated conditions on the loadings and factors, as this term is load-bearing for the claimed separation of the two bias sources.
[§4.1, Assumption 4.2] §4.1, Assumption 4.2: The regularity conditions imposed on the predetermined regressors (e.g., uniform integrability and weak dependence across i) are used to control the bias expansion, but the paper does not provide a primitive example (such as an AR(1) process with factor-augmented errors) showing that these conditions are satisfied while still generating the claimed nonzero bias; this would strengthen the practical relevance of the correction.

minor comments (3)

[§3 and §5] The notation for the two bias components (B_1 and B_2) is introduced in §3 but reused with slight redefinition in the test-statistic corrections in §5; a single consistent definition across sections would improve readability.
[Table 1] Table 1 (Monte Carlo results) reports bias and RMSE but does not include the standard deviation of the bias-corrected estimator across replications; adding this column would allow readers to assess whether the correction preserves efficiency.
[Theorem 5.1] The abstract states that the corrected tests are asymptotically chi-squared, but the precise degrees of freedom (equal to the number of restrictions) should be restated explicitly in the theorem statement for the Wald statistic.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and have revised the manuscript to incorporate the requested clarifications and additions, which we believe strengthen the presentation of the bias separation and the practical relevance of the assumptions.

read point-by-point responses

Referee: [§3.2, Theorem 3.1] The bias term arising from predetermined regressors is obtained by expanding the score around the probability limit of the factor estimates; the paper should verify that the cross-term between the predetermined regressor and the factor estimation error is o_p((NT)^{-1/2}) under the stated conditions on the loadings and factors, as this term is load-bearing for the claimed separation of the two bias sources.

Authors: We agree that an explicit bound on this cross-term is useful for confirming the separation of the two bias channels. Under Assumptions 3.1–3.3, the factor estimation error is O_p(max(N^{-1/2}, T^{-1/2})) uniformly in the relevant norms, while the predetermined regressors satisfy uniform integrability and the moment conditions in Assumption 4.2. The product term is therefore o_p((NT)^{-1/2}) by a standard Cauchy–Schwarz argument combined with the weak dependence across i. We have added a new lemma (Lemma A.3 in the revised appendix) that states and proves this bound explicitly, together with a short remark in the proof of Theorem 3.1 directing the reader to it. revision: yes
Referee: [§4.1, Assumption 4.2] The regularity conditions imposed on the predetermined regressors (e.g., uniform integrability and weak dependence across i) are used to control the bias expansion, but the paper does not provide a primitive example (such as an AR(1) process with factor-augmented errors) showing that these conditions are satisfied while still generating the claimed nonzero bias; this would strengthen the practical relevance of the correction.

Authors: We concur that a concrete primitive example would improve interpretability. In the revised manuscript we have inserted a new Example 4.1 in Section 4.1 that constructs a dynamic panel AR(1) process in which the regressor is predetermined (via the lagged dependent variable), the idiosyncratic errors contain an interactive fixed-effects component, and the factor loadings and factors satisfy the paper’s assumptions. We verify that the uniform integrability and weak cross-sectional dependence conditions hold, yet the bias term arising from the predetermined regressor remains nonzero and of the order stated in Theorem 3.1. The example is kept simple enough to be analytically tractable while illustrating the two distinct bias sources. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the first-order asymptotic bias of the LS estimator analytically via expansions under joint (N,T) asymptotics in the interactive fixed effects model with predetermined regressors. The two bias sources (idiosyncratic error correlation/heteroscedasticity and predeterminedness) are obtained from the model's moment conditions and stated regularity assumptions on factors, loadings, and regressors rather than by parameter fitting or definitional equivalence. The bias-corrected estimator and adjusted Wald/LR/LM statistics are constructed directly from these expansions, with their chi-squared limits following from standard central limit theory; Monte Carlo results serve as separate finite-sample checks. No load-bearing step reduces to a self-citation chain, ansatz smuggling, or renaming of inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard econometric regularity conditions for panel asymptotics rather than on new free parameters or invented entities.

axioms (2)

domain assumption Both cross-sectional dimension N and time dimension T diverge to infinity
The first-order asymptotic theory and bias expansions are derived under joint large-N, large-T asymptotics.
domain assumption Regularity conditions on interactive factors, errors, and predetermined regressors
These conditions are invoked to obtain the bias expansions and the chi-squared limits for the corrected tests.

pith-pipeline@v0.9.0 · 5432 in / 1150 out tokens · 44338 ms · 2026-05-09T18:37:44.242644+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references

[1]

C., Lee, Y

Ahn, S. C., Lee, Y. H., and Schmidt, P. (2001). GMM estimation of linear panel data models with time-varying individual effects.Journal of Econometrics, 101(2):219–255. Andrews, D. W. K. (1999). Estimation when a parameter is on a boundary.Econometrica, 67(6):1341–1384. Andrews, D. W. K. (2001). Testing when a parameter is on the boundary of the maintaine...

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Golub, G. H. and Van Loan, C. F. (1996).Matrix Computations (Johns Hopkins Studies in Mathematical Sciences), Third Edition. The Johns Hopkins University Press. Hahn, J. and Kuersteiner, G. (2002). Asymptotically unbiased inference for a dynamic panel model with fixed effects when both ”n” and ”T” are large.Econometrica, 70(4):1639–1657. Hahn, J. and Kuer...

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[3]

Nickell, S. (1981). Biases in dynamic models with fixed effects.Econometrica, 49(6):1417–1426. Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenval- ues.The Review of Economics and Statistics, 92(4):1004–1016. Pesaran, M. H. (2006). Estimation and inference in large heterogeneous panels with a multifactor error str...

1981
[4]

std 0.1444 0.1480 0.0982 0.0723 0.1718 0.1301 rmse 0.1898 0.2050 0.1213 0.0750 0.4067 0.2669 T= 10 bias 0.1339 -0.0542 -0.0201 0.0218 -0.1019 -0.0623 (M=

2050
[5]

std 0.1454 0.1528 0.1299 0.0731 0.1216 0.1341 rmse 0.1910 0.5676 0.3942 0.0763 0.9792 0.7609 T= 10 bias 0.1343 -0.1874 -0.1001 0.0210 -0.4923 -0.3271 (M=

1910
[6]

std 0.0879 0.0469 0.0320 0.0354 0.0820 0.0528 rmse 0.1696 0.0648 0.0362 0.0436 0.1999 0.1207 T= 40 bias 0.1511 -0.0161 -0.0038 0.0300 -0.0227 -0.0128 (M=

1999
[7]

For different values of the AR(1) coefficientρ 0 and of the bandwidthM, we give the fraction of the LS estimator bias that is accounted for by the bias correction, i.e

std 0.0488 0.0123 0.0115 0.0182 0.0064 0.0057 rmse 0.1625 0.0143 0.0116 0.0372 0.0071 0.0058 Table S.3: Simulation results for the AR(1) model withN= 100,T= 20,ρ f = 0.5, and σf = 0.5. For different values of the AR(1) coefficientρ 0 and of the bandwidthM, we give the fraction of the LS estimator bias that is accounted for by the bias correction, i.e. the...

2006
[8]

std 0.1444 0.1480 0.0982 0.1681 0.0723 0.1718 0.1301 0.2203 rmse 0.1898 0.2050 0.1213 0.2430 0.0750 0.4067 0.2669 0.3966 T= 10 bias 0.1339 -0.0542 -0.0201 -0.0819 0.0218 -0.1019 -0.0623 -0.1436 (M=

2050
[9]

ρ0 = 0.3ρ 0 = 0.9 OLS FLS BC-FLS CCE OLS FLS BC-FLS CCE T= 5 bias 0.1239 -0.5467 -0.3721 -0.1767 0.0218 -0.9716 -0.7490 -0.3289 (M=

std 0.0487 0.0112 0.0109 0.0111 0.0179 0.0056 0.0053 0.0073 rmse 0.1627 0.0130 0.0109 0.0149 0.0372 0.0062 0.0053 0.0154 96 Dynamic Panel with Interactive Effects Table S.10: Same as Table S.2 in main paper, but also reporting pooled CCE estimator of Pesaran (2006). ρ0 = 0.3ρ 0 = 0.9 OLS FLS BC-FLS CCE OLS FLS BC-FLS CCE T= 5 bias 0.1239 -0.5467 -0.3721 -...

2006
[10]

std 0.1454 0.1528 0.1299 0.1678 0.0731 0.1216 0.1341 0.2203 rmse 0.1910 0.5676 0.3942 0.2437 0.0763 0.9792 0.7609 0.3958 T= 10 bias 0.1343 -0.1874 -0.1001 -0.0816 0.0210 -0.4923 -0.3271 -0.1414 (M=

1910
[11]

std 0.0879 0.0469 0.0320 0.0277 0.0354 0.0820 0.0528 0.0404 rmse 0.1696 0.0648 0.0362 0.0492 0.0436 0.1999 0.1207 0.0739 T= 40 bias 0.1511 -0.0161 -0.0038 -0.0199 0.0300 -0.0227 -0.0128 -0.0282 (M=

1999
[12]

ρ0 = 0.3ρ 0 = 0.9 OLS FLS BC-FLS CCE OLS FLS BC-FLS CCE T= 5 bias 0.1861 -0.4968 -0.3323 -0.1002 0.0309 -0.9305 -0.7057 -0.2750 (M=

std 0.0488 0.0123 0.0115 0.0111 0.0182 0.0064 0.0057 0.0074 rmse 0.1625 0.0143 0.0116 0.0149 0.0372 0.0071 0.0058 0.0155 97 Dynamic Panel with Interactive Effects Table S.11: Analogous to Table S.2 in main paper, but withR= 2 correctly specified, and also reporting pooled CCE estimator of Pesaran (2006). ρ0 = 0.3ρ 0 = 0.9 OLS FLS BC-FLS CCE OLS FLS BC-FLS...

2006
[13]

std 0.1562 0.1910 0.1580 0.2063 0.0801 0.1644 0.1754 0.2302 rmse 0.2429 0.5322 0.3680 0.2294 0.0859 0.9449 0.7272 0.3586 T= 10 bias 0.1989 -0.1569 -0.0758 0.0036 0.0326 -0.4209 -0.2732 -0.1040 (M=

1910
[14]

std 0.1185 0.1018 0.0700 0.1074 0.0543 0.1607 0.1235 0.1070 rmse 0.2315 0.1870 0.1031 0.1074 0.0633 0.4505 0.2998 0.1492 T= 20 bias 0.2096 -0.0592 -0.0185 0.0520 0.0366 -0.0741 -0.0406 -0.0310 (M=

2096