Fixed effect estimation of large T panel data models

Abstract

This article reviews recent advances in fixed effect estimation of panel data models for long panels, where the number of time periods is relatively large. We focus on semiparametric models with unobserved individual and time effects, where the distribution of the outcome variable conditional on covariates and unobserved effects is specified parametrically, while the distribution of the unobserved effects is left unrestricted. Compared to existing reviews on long panels (Arellano & Hahn, 2007; a section in Arellano & Bonhomme, 2011) we discuss models with both individual and time effects, split-panel Jackknife bias corrections, unbalanced panels, distribution and quantile effects, and other extensions. Understanding and correcting the incidental parameter bias caused by the estimation of many fixed effects is our main focus, and the unifying theme is that the order of this bias is given by the simple formula p/n for all models discussed, with p the number of estimated parameters and n the total sample size.

Two-Step Estimation and Inference with Possibly Many Included Covariates

“…Large-(N, T ) Panel Data. Our findings are also qualitatively connected to the literature on nonlinear panel models with fixed effects (Fernandez-Val and Weidner, 2018, and references therein) in at least two ways. First, in that context a first-order bias arises when the number of time periods (T )…”

Bias and consistency in three-way gravity models

“…To that end, let n be the total number of observations and let p be the total number of parameters being estimated, inclusive of any fixed effects. As described in Fernández-Val and Weidner (2018), what we need to be concerned with the number of observations that are available to estimate each fixed effect, i.e., n/p. More precisely, when appropriate regularity conditions are satisfied, the estimated β may generally be thought of as having the following bias and standard deviation:…”

“…The appropriate generalization of the operator D βα q i = D βγ q j in (18) to the case of vector-valued α i and γ j was described in Section 4.2 of Fernández-Val and Weidner (2018).…”

“…Specifically, if N is the number of countries, profiling out the large (on the order of N 2 ) number of pair fixed effects eliminates any "1/T " bias term that would normally be associated with a short time series. Using the heuristic suggested by Fernández-Val and Weidner (2018), we would then expect an asymptotic bias with an order given by the ratio between the order of the number of remaining parameters (N T ) and that of number of observations (N 2 T ), or 1/N . However, due to the special properties of FE-PPML, the asymptotic bias in our setting behaves more like a 1/(N T ) bias as N and T grow large at the same rate.…”

“…Aside from Fernández-Val and Weidner (2016)'s work on two-way nonlinear models, Pesaran (2006), Bai (2009), Hahn and Moon (2006), and Moon and Weidner (2017) have each conducted similar analyses for two-way linear models with interacted individual and time fixed effects. Turning to three-way models, Hinz, Stammann, and Wanner (2019) have recently developed bias corrections for dynamic three-way probit and logit models based on asymptotics suggested by Fernández-Val and Weidner (2018) where all three panel dimensions grow at the same rate. Though widely applicable, this approach is not appropriate for our setting because of the different role played by the time dimension when the estimator is FE-PPML.…”