## Abstract

We consider a testing problem for cross-sectional independence for highdimensional 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 coefficients 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.

Original language | English (US) |
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Pages (from-to) | 1124-1143 |

Number of pages | 20 |

Journal | Annals of Statistics |

Volume | 50 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2022 |

### Bibliographical note

Funding Information:Funding. The research of B. Liu was supported in part by NSFC grants 12171079, 11631003, 11690012, the China National Key R&D Program grant 2020YFA0714100 and the Special Fund for Key Laboratories of Jilin Province China grant 20190201285JC. T. Jiang’s research was supported in part by NSF Grant DMS-1916014. The research of W. Xiong was supported in part by NSFC Grants 12001101 and the Fundamental Research Funds for the Central Universities in UIBE CXTD10-09 and 20YQ18.

Publisher Copyright:

© 2022 Institute of Mathematical Statistics. All rights reserved.

## Keywords

- asymptotic independence
- asymptotic normality
- cross-sectional independence
- extreme-value distribution
- High-dimensional data
- hypothesis tests
- max-sum test
- panel data models