Dynamic Linear Panel Regression Models with Interactive Fixed Effects

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.

• The paper introduces a dynamic panel regression model with interactive fixed effects that extends to predetermined regressors.
• It establishes consistency and √(NT) asymptotic normality for the joint LS estimator while explicitly addressing identification challenges.
• The study provides bias correction techniques for feedback and Nickell-type biases, thereby improving inference and test statistic performance.

Dynamic Linear Panel Regression Models with Interactive Fixed Effects

Introduction and Context

This paper develops the asymptotic theory for linear panel regression models with interactive fixed effects and predetermined regressors, extending the literature to allow for feedback from past outcomes to future regressors (2605.00612). Interactive fixed effects represent an unobserved component structure where individual-specific loadings and time-varying factors jointly generate heterogeneity in outcomes. This specification subsumes traditional additive fixed effects and accommodates richer nonstationary and cross-sectional dependence structures.

The authors position their contributions relative to prior work in several ways. While Bai (2009) provided a $\sqrt{NT}$-consistent least squares (LS) estimator for interactive fixed effect models under strictly exogenous regressors, this paper’s primary innovation is to analyze scenarios with predetermined regressors, thereby incorporating dynamic models (e.g., those with lagged dependent variables). The theoretical results hold under “alternative” asymptotics in which both $N$ (cross-section) and $T$ (time) grow large at comparable rates.

Furthermore, the paper generalizes identification and estimation to accommodate panels with both low-rank regressors (e.g., time-invariant or common regressors, and their interactions) and high-rank regressors, which is critical for empirical settings where such regressors coexist. The identification challenges and solutions in the low-rank case, due to overlapping low-rank structures in regressors and unobserved effects, are rigorously addressed.

Model Specification and Identification

The main model is: $$Y_{it} = \beta^{\prime} X_{it} + \lambda_i^{\prime} f_t + e_{it}$$ with $K$-dimensional observable regressors $X_{it}$, $R$ unobserved factors $f_t$ and corresponding loadings $\lambda_i$, and idiosyncratic errors $e_{it}$. The panel has $N$0 individuals and $N$1 time periods.

Identification in this context is nontrivial due to the interactive structure: the unobserved $N$2 matrix $N$3 is itself low rank. The authors establish conditions for identification in the presence of both low- and high-rank regressors. Non-collinearity requirements are imposed, e.g., for linear combinations of low-rank regressors projected orthogonally to the space spanned by the factors, and for high-rank regressors to have sufficiently high rank relative to the dimension of interactive effects and low-rank regressors. These are formalized in Assumption ID and rigorously shown to guarantee identification of the regression coefficients $N$4.

Estimation and Consistency

The primary estimator is the joint least squares (LS) estimator of $N$5, minimizing the sum of squared residuals. The estimation of $N$6 involves profiling out the unobserved factors/loads using principal components, leading to a non-convex objective but with established minimization properties.

Consistency and rate results are derived under a set of regularity conditions on the regressors and error process, allowing for pre-determined regressors. The panel data generating process may include autoregressive dynamics such as AR(1) models with interactive fixed effects, or more generally, VAR(1) systems. The result is that $N$7 is consistent, with the “alternative” asymptotics rate $N$8.

Asymptotic Distribution and Bias Characterization

A core technical contribution is the expansion of the profile objective function for $N$9 in a neighborhood of the true value using perturbation theory, leveraging results from Moon and Weidner (2015). The expansion reveals two distinct sources of asymptotic bias in $T$0:

• Bias due to error dependence and heteroscedasticity: Present even with strictly exogenous regressors, as in Bai (2009), and related to the incidental parameter problem.
• Bias due to predetermined regressors: Analogous to the Nickell bias in dynamic panel models, this term arises when regressors are only weakly exogenous.

The asymptotic distribution is then

$T$1

where $T$2 is the limit Hessian, $T$3 aggregates explicit forms of both biases, and $T$4 is the limit covariance.

Bias Correction

Explicit estimators for both variance and bias are constructed: $T$5 addresses feedback bias (predeterminedness), while $T$6 and $T$7 address biases stemming from error structure. These are computed using sample residuals and projectors estimated via principal components. The paper presents a bias-corrected estimator

$T$8

which is shown to be asymptotically normal and unbiased to first order, allowing conventional inference.

Extensions to jackknife bias correction are discussed, and consequences for finite sample correction are illustrated. Empirically, the bias correction is substantial in panels of moderate $T$9.

Inference: Bias-Corrected Test Statistics

The authors derive the limiting distributions for three classical tests under the “incidental parameters” regime:

• Wald
• Likelihood Ratio (LR)
• Lagrange Multiplier (LM)

The uncorrected versions of these statistics are shown to be non-standard (non-central chi-square), even under the null, due to the non-negligible bias. The paper provides bias-corrected versions that restore asymptotic $$Y_{it} = \beta^{\prime} X_{it} + \lambda_i^{\prime} f_t + e_{it}$$0 null distributions. The construction and explicit forms of these corrections are an important practical output, as they improve both the size and power properties of standard inferential procedures in panel models with interactive effects and dynamics.

Endogeneity and Extensions

While the main asymptotic theory assumes no endogenous regressors, the paper briefly discusses how instrumental variables and minimum distance estimation can be integrated in the interactive effects setup, referencing the three-step LS-MD method. References are made to the work of Moon, Shum, and Weidner (2012) and further applications to panel demand models with product-market fixed effects.

Numerical Results and Monte Carlo Evidence

Extensive Monte Carlo studies highlight the finite-sample properties of the estimator and test statistics. The key findings include:

• Both sources of bias are substantial unless $$Y_{it} = \beta^{\prime} X_{it} + \lambda_i^{\prime} f_t + e_{it}$$1 is very large.
• The bias correction substantially reduces both the bias and size distortions in empirical sizes of classical tests.
• Incorrect specification of the number of factors mildly degrades efficiency and bias in moderate samples; the asymptotic theory remains robust when $$Y_{it} = \beta^{\prime} X_{it} + \lambda_i^{\prime} f_t + e_{it}$$2 at comparable rates.
• The performance of the OLS estimator and uncensored classical tests is inferior compared to both bias-corrected and properly specified interactive effect methods.

Implications and Future Directions

The theoretical and computational frameworks established here substantially improve the robustness of inference for linear panel models with interactive effects, especially those incorporating dynamics and a mix of regressor types. Practically, this opens up the use of LS estimation procedures in broad dynamic panel settings with unobserved heterogeneity, provided the factor structure is sufficiently strong.

On the theoretical side, the identification analysis for low-rank regressors relative to unobserved low-rank error components is essential for empirical macro, asset pricing, and labor economics panels, where such collinearities are frequently present. Methodologically, the work demonstrates that, under suitable conditions, the leading “incidental parameter” bias can be characterized, consistently estimated, and removed, allowing conventional inferential logic to be restored.

Future research directions include:

• Extension to models with an unknown number of factors and/or consistent information criteria for factor selection.
• Further exploration of robust inference when regressors are endogenous, using instruments or control functions within the interactive fixed effect context.
• Data-driven optimal choices of bandwidth and truncation parameters involved in the bias correction.
• Applications to large-scale panels with covariates of mixed rank, e.g., sectoral, cohort, or geographic dummies.

Conclusion

By delivering a comprehensive asymptotic analysis of the dynamic interactive fixed effects model, explicitly characterizing and correcting for both sources of “Nickell-type” and “incidental parameter” bias, this work provides practitioners with theoretically justified, finite-sample-improving estimation and inference methodologies. The results are broadly applicable to empirical contexts where both dynamics and complex forms of individual and time heterogeneity are key modeling features (2605.00612).