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

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.