Testing matrix factor structure via generalized likelihood ratio

Abstract In this paper, we introduce a generalized likelihood ratio test for assessing whether the factor loading matrix in high-dimensional data exhibits a Kronecker structure, thereby offering a rigorous tool for validating two-way matrix factor models. A central challenge in likelihood-based inference for factor models arises from inherent rotational non-identifiability, which prevents the likelihood function from admitting a standard second-order Taylor expansion. To address this issue, we augment the quasi-likelihood with a quadratic identification penalty that removes rotational degeneracy locally. Building on this construction, we derive the asymptotic distribution of the proposed statistic under the null hypothesis and unveil a new Wilks-type phenomenon in a high-dimensional setting. Under the alternative hypothesis, we further establish the consistency of the test, demonstrating its ability to detect deviations from the Kronecker structure. Simulation studies demonstrate the finite-sample effectiveness of the proposed method. Furthermore, applications to real datasets illustrate the practical utility of the proposed procedure by assessing the presence of Kronecker structure in factor loading matrices.