On testing Kronecker product structure in tensor factor models

We propose a test for testing the Kronecker product structure of a factor loading matrix implied by a tensor factor model with Tucker decomposition in the common component. Through defining a Kronecker product structure set, we define if a tensor time series has a Kronecker product structure, equivalent to the ability to decompose the series according to a tensor factor model. Our test is built on analysing and comparing the residuals from fitting a full tensor factor model, and the residuals from fitting a factor model on a reshaped version of the data. In the most extreme case, the reshaping is the vectorization of the tensor data, and the factor loading matrix in such a case can be general if there is no Kronecker product structure present. Our test is also generalized to testing the Khatri–Rao product structure in a tensor factor model with the CP (canonical polyadic) decomposition. Theoretical results are developed through asymptotic normality results on estimated residuals. Numerical experiments suggest that the size of the tests gets closer to the pre-set nominal value as the sample size or the order of the tensor gets larger, while the power increases with mode dimensions and the number of combined modes. We demonstrate our tests through extensive real data examples.