  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 $n/T\to c \in (0,\infty)$.
