  Testing cross-sectional independence in panel data models is of fundamental
importance in econometric analysis with high-dimensional panels. Recently,
econometricians began to turn their attention to the problem in the presence of
serial dependence. The existing procedure for testing cross-sectional
independence with serial correlation is based on the sum of the sample
cross-sectional correlations, which generally performs well when the
alternative has dense cross-sectional correlations, but suffers from low power
against sparse alternatives. To deal with sparse alternatives, we propose a
test based on the maximum of the squared sample cross-sectional correlations.
Furthermore, we propose a combined test to combine the p-values of the max
based and sum based tests, which performs well under both dense and sparse
alternatives. The combined test relies on the asymptotic independence of the
max based and sum based test statistics, which we show rigorously. We show that
the proposed max based and combined tests have attractive theoretical
properties and demonstrate the superior performance via extensive simulation
results. We apply the two new tests to analyze the weekly returns on the
securities in the S\&P 500 index under the Fama-French three-factor model, and
confirm the usefulness of the proposed combined test in detecting
cross-sectional independence.
