Max-sum Tests for Cross-Sectional Independence of High-Dimensional Panel Data

We consider a testing problem for cross-sectional independence for high-dimensional panel data, where the number of cross-sectional units is potentially much larger than the number of observations. The cross-sectional independence is described through linear regression models. We study three tests named the sum, the max and the max-sum tests, 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 nonsparse correlation co-efficients of random errors between the linear regression models, respectively. And the max-sum test is devised to compromise both situations on the correlation coefficients. 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 numbers is usually vague. Toward 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 correlation 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 correlation coefficients of residuals regardless of sparsity or not.