Understanding Cross-sectional Dependence in Panel Data

TL;DR

This paper characterizes different types of cross-sectional dependence in panel data, examines their effects on estimator properties, and develops robust standard errors for valid inference under various dependence structures.

Contribution

It introduces norm-based definitions of cross-sectional dependence, analyzes their impact on estimators, and proposes robust standard errors for models with complex dependence.

Findings

Robust standard errors are effective for dependence structures.

Time dependence in errors does not affect asymptotic variance under strong dependence.

Standard inference tests remain valid with the proposed robust covariance estimators.

Abstract

We provide various norm-based definitions of different types of cross-sectional dependence and the relations between them. These definitions facilitate to comprehend and to characterize the various forms of cross-sectional dependence, such as strong, semi-strong, and weak dependence. Then we examine the asymptotic properties of parameter estimators both for fixed (within) effect estimator and random effect (pooled) estimator for linear panel data models incorporating various forms of cross-sectional dependence. The asymptotic properties are also derived when both cross-sectional and temporal dependence are present. Subsequently, we develop consistent and robust standard error of the parameter estimators both for fixed effect and random effect model separately. Robust standard errors are developed (i) for pure cross-sectional dependence; and (ii) also for cross-sectional and time series dependence. Under strong or semi-strong cross-sectional dependence, it is established that when the time dependence comes through the idiosyncratic errors, such time dependence does not have any influence in the asymptotic variance of $(\hat{\beta}_{FE/RE}). $ Hence, it is argued that in estimating $Var(\hat{\beta}_{FE/RE}),$ Newey-West kind of correction injects bias in the variance estimate. Furthermore, this article lay down conditions under which $t$, $F$ and the $Wald$ statistics based on the robust covariance matrix estimator give valid inference.