On Testing Kronecker Product Structure in Tensor Factor Models

TL;DR

This paper introduces a statistical test to determine if a tensor factor model's loading matrix has a Kronecker product structure, using residual analysis from different tensor decompositions, supported by theoretical and empirical validation.

Contribution

It develops a novel test for Kronecker product structure in tensor factor models, with asymptotic theory and practical demonstrations on real data.

Findings

Test size approaches nominal with larger samples and tensor order.

Test power increases with mode dimensions and combined modes.

Successful application to NYC taxi data and financial portfolio data.

Abstract

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 response $\{\mathcal{Y}_t\}$ has a Kronecker product structure, equivalent to the ability to decompose $\{\mathcal{Y}_t\}$ 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 (tensor) factor model on a reshaped version of the data. In the most extreme case, the reshaping is the vectorisation of the tensor data, and the factor loading matrix in such a case can be general if there is no Kronecker product structure present. 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 out tests through a NYC taxi traffic data and a Fama-French matrix portfolio of returns.