Fixed $T$ Estimation of Linear Panel Data Models with Interactive Fixed Effects

TL;DR

This paper introduces a novel transformation method for linear panel data models with fixed T, enabling consistent estimation of regression coefficients and reliable testing of the number of factors even with fixed T.

Contribution

The paper proposes a new transformation that reduces model dimension and allows for $\, ext{√n}$-consistent estimation with fixed T, extending the applicability of principal component estimators.

Findings

Transforming the model yields $\, ext{√n}$-consistent estimates with fixed T.

Estimates remain asymptotically unbiased under dependence and dynamics.

Eigenvalue ratio test remains consistent for factor number with transformed model.

Abstract

This paper studies the estimation of linear panel data models with interactive fixed effects, where one dimension of the panel, typically time, may be fixed. To this end, a novel transformation is introduced that reduces the model to a lower dimension, and, in doing so, relieves the model of incidental parameters in the cross-section. The central result of this paper demonstrates that transforming the model and then applying the principal component (PC) estimator of \cite{bai_panel_2009} delivers $\sqrt{n}$ consistent estimates of regression slope coefficients with $T$ fixed. Moreover, these estimates are shown to be asymptotically unbiased in the presence of cross-sectional dependence, serial dependence, and with the inclusion of dynamic regressors, in stark contrast to the usual case. The large $n$, large $T$ properties of this approach are also studied, where many of these results carry over to the case in which $n$ is growing sufficiently fast relative to $T$. Transforming the model also proves to be useful beyond estimation, a point illustrated by showing that with $T$ fixed, the eigenvalue ratio test of \cite{horenstein} provides a consistent test for the number of factors when applied to the transformed model.