(Debiased) Inference for Fixed Effects Estimators with Three-Dimensional Panel and Network Data

TL;DR

This paper develops new inference methods for fixed effects estimators in three-dimensional panel and network data, addressing biases and asymptotic issues not present in traditional two-dimensional panels.

Contribution

It introduces a comprehensive inferential theory for fixed effects estimators in 3D panels, including bias correction and handling of complex network structures.

Findings

Unbiased estimators when effects vary along a single dimension

Severe inference problems with effects varying along two dimensions

Explicit bias formulas and debiased estimators proposed

Abstract

Inference for fixed effects estimators is often unreliable due to Nickell- and incidental parameter biases. While these issues are well understood for classical two-dimensional panels, little is known about three-dimensional panel structures (e.g., sender x receiver x time). We develop inferential theory for a broad class of linear and nonlinear fixed effects M-estimators in this setting, covering bipartite, directed, and undirected network panel data, multiple specifications of additively separable unobserved effects, and both strictly exogenous and predetermined regressors. Our analysis reveals fundamentally different asymptotic properties compared to two-dimensional panels. In particular, we find a sharp dichotomy across specifications: (i) when unobserved effects vary along a single panel dimension, the estimator is asymptotically unbiased; (ii) when they vary along two panel dimensions, the estimator suffers from a severe inference problem characterized by a degenerate asymptotic distribution. We resolve the latter by deriving explicit bias formulas and proposing analytically debiased estimators with nondegenerate, correctly centered asymptotic distributions. An empirical application studies dynamic network formation in a directed panel of bilateral trade relationships.