  We consider a testing problem for cross-sectional dependence for
high-dimensional panel data, where the number of cross-sectional units is
potentially much larger than the number of observations. The cross-sectional
dependence is described through a linear regression model. We study three tests
named the sum test, the max test and the max-sum test, where the latter two are
new. The sum test is initially proposed by Breusch and Pagan (1980). We design
the max and sum tests for sparse and non-sparse residuals in the linear
regressions, respectively.And the max-sum test is devised to compromise both
situations on the residuals. Indeed, our simulation shows that the max-sum test
outperforms the previous two tests. This makes the max-sum test very useful in
practice where sparsity or not for a set of data is usually vague. Towards the
theoretical analysis of the three tests, we have settled two conjectures
regarding the sum of squares of sample correlation coefficients asked by
Pesaran (2004 and 2008). In addition, we establish the asymptotic theory for
maxima of sample correlations coefficients appeared in the linear regression
model for panel data, which is also the first successful attempt to our
knowledge. To study the max-sum test, we create a novel method to show
asymptotic independence between maxima and sums of dependent random variables.
We expect the method itself is useful for other problems of this nature.
Finally, an extensive simulation study as well as a case study are carried out.
They demonstrate advantages of our proposed methods in terms of both empirical
powers and robustness for residuals regardless of sparsity or not.
