Tensor-BEKK: Conditional Covariance Modeling and Inference for Tensor-Valued Time Series

Modern economic and financial data are increasingly organized as multiway arrays, with observations indexed simultaneously by geographic regions, industrial sectors, asset categories, and other economic characteristics. Representing such data as tensor-valued time series preserves their intrinsic multiway structure. Although substantial effort has been devoted to modeling the conditional mean of tensor-valued time series, comparatively less attention has been paid to their conditional covariance dynamics. The latter remains challenging because unrestricted multivariate covariance models involve many parameters and substantial computational cost. To address these challenges, we propose the Tensor-BEKK (T-BEKK) model, a tensor-structured BEKK specification that retains the positive definite covariance recursion for the vectorized process while imposing Kronecker structures on the intercept and the ARCH and GARCH coefficient matrices. The model reduces the parameter dimension and provides mode-specific interpretations of the covariance intercept, ARCH effects, and GARCH persistence. We establish stationarity, identification, and the asymptotic properties of the Gaussian quasi-maximum likelihood estimator. We further develop mode-specific restricted score tests tailored to the tensor structure, inference procedures for nonzero spillover intensities within each mode, and a portmanteau diagnostic test based on quadratic form residuals. For higher-dimensional settings, we also introduce the Tensor-Factor-BEKK (TF-BEKK) model. Under a first-step negligibility condition, its feasible second-step QMLE is asymptotically equivalent to the oracle QMLE based on the latent factors. Simulations and two empirical applications, covering currency futures and Chinese equity tensor portfolio allocation, illustrate the finite-sample behavior and empirical usefulness of the proposed methods.