On John's test for sphericity in large panel data models

TL;DR

This paper analyzes John's sphericity test in large panel data models, establishing its asymptotic properties under various regimes and demonstrating its robustness and limitations across different asymptotic scenarios.

Contribution

The paper provides the first comprehensive asymptotic analysis of John's test for large panel data, revealing its dimension-proof property and consistency conditions.

Findings

John's test maintains the same null distribution across different asymptotic regimes.

The test is consistent except under specific covariance alternatives in large panels.

Asymptotic normality holds under general conditions, not limited to Gaussian populations.

Abstract

This paper studies John's test for sphericity of the error terms in large panel data models, where the number of cross-section units $n$ is large enough to be comparable to the number of times series observations $T$, or even larger. Based on recent random matrix theory results, John's test's asymptotic normality properties are established under both the null and the alternative hypotheses. These asymptotics are valid for general populations, i.e., not necessarily Gaussian provided certain finite moments. A fantastic phenomenon found in the paper is that John's test for panel data models possesses a powerful dimension-proof property. It keeps the same null distribution under different $(n,T)$-asymptotics, i.e., the small or medium panel regime $n/T\to 0$ as $T\to \infty$, the large panel regime $n/T\to c \in (0,\infty)$ as $ T\to \infty$, and the ultra-large panel regime $n/T\to \infty (T^\delta/n =O_p(1), 1<\delta<2)$ as $T\to \infty$. Moreover, John's test is always consistent except under the alternative of bounded-norm covariance with the large panel regime $n/T\to c \in (0,\infty)$.