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One-way or two-way factor model for matrix sequences?
This paper investigates the issue of determining the dimensions of row and column factor spaces in matrix-valued data. Exploiting the eigen-gap in the spectrum of sample second moment matrices of the data, we propose a family of randomised tests to check whether a one-way or two-way factor structure exists or not. Our tests do not require any arbitrary thresholding on the eigenvalues, and can be applied with (virtually) no restrictions on the relative rate of divergence of the cross-sections to the sample sizes as they pass to infinity. Although tests are based on a randomisation which does not vanish asymptotically, we propose a de-randomised, “strong” (based on the Law of the Iterated Logarithm) decision rule to choose in favour or against the presence of common factors. We use the proposed tests and decision rule in two ways. We further cast our individual tests in a sequential procedure whose output is an estimate of the number of common factors. Our tests are built on two variants of the sample second moment matrix of the data: one based on a row (or column) “flattened” version of the matrix-valued sequence, and one based on a projection-based method. Our simulations show that both procedures work well in large samples and, in small samples, the one based on the projection method delivers a superior performance compared to existing methods in virtually all cases considered.
He’s work is supported by NSF China (12171282,11801316), National Statistical Scientific Research Key Project (2021LZ09), Project funded by China Postdoctoral Science Foundation (2021M701997) and the Fundamental Research Funds of Shandong University, Young Scholars Program of Shandong University, China. Kong’s work is partially supported by NSF China (71971118 and 11831008) and the WRJH-QNBJ Project and Qinglan Project of Jiangsu Province . The authors would like to thank the Editor Xiaohong Chen, an anonymous Associate Editor, and three anonymous Referees, whose helpful comments have greatly improved the quality and focus of the paper.
Author affiliationSchool of Business, University of Leicester
- AM (Accepted Manuscript)