On testing Kronecker product structure in tensor factor models

We propose a test for the Kronecker product structure of a factor loading matrix implied by a tensor factor model with Tucker decomposition in the common component. By defining a Kronecker product structure set, we determine whether a tensor time series has a Kronecker product structure, equivalent to its 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 the Khatri–Rao product structure in a tensor factor model with canonical polyadic decomposition. Theoretical results are developed through asymptotic normality results on estimated residuals. Numerical experiments suggest that the size of the tests approaches the pre-set nominal value as the sample size or the order of the tensor increases, while the power increases with mode dimensions and the number of combined modes. We demonstrate our tests through extensive real data examples.