Rank Determination in Tensor Factor Model

Factor model is an appealing and effective analytic tool for high-dimensional time series, with a wide range of applications in economics, finance and statistics. One of the fundamental issues in using factor model for time series in practice is the determination of the number of factors to use. This paper develops two criteria for such a task for tensor factor models where the signal part of an observed time series in tensor form assumes a Tucker decomposition with the core tensor as the factor tensor. The task is to determine the dimensions of the core tensor. One of the proposed criteria is similar to information based criteria of model selection, and the other is an extension of the approaches based on the ratios of consecutive eigenvalues often used in factor analysis for panel time series. The new criteria are designed to locate the gap between the true smallest non-zero eigenvalue and the zero eigenvalues of a functional of the population version of the auto-cross-covariances of the tensor time series using their sample versions. As sample size and tensor dimension increase, such a gap increases under regularity conditions, resulting in consistency of the rank estimator. The criteria are built upon the existing non-iterative and iterative estimation procedures of tensor factor model, yielding different performances. We provide sufficient conditions and convergence rate for the consistency of the criteria as the sample size $T$ and the dimensions of the observed tensor time series go to infinity. The results include the vector factor models as special cases, with an additional convergence rates. The results also include the cases when there exist factors with different signal strength. In addition, the convergence rates of the eigenvalue estimators are established. Simulation studies provide promising finite sample performance for the two criteria.