Vector or Matrix Factor Model? A Strong Rule Helps!

This paper investigates the issue of determining the dimensions of row and column factor structures 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 an eigenvalue diverges as the sample size passes to infinity (corresponding to having a common factor) or not. Our tests do not require any arbitrary thresholding, and can be applied with no restrictions on the relative rate of divergence of the cross-sectional and time series 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. First, we propose a procedure to test whether a factor structure exists in the rows and/or in the columns. Second, we 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 "flattened" version of the matrix-valued series, 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.