arXiv 16 Sep 2026 · Econometrics
arXiv:2609.18157 · PDF · Extracted main text
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.
appendix boundary found by appendix_titled_section at “Supplementary Material” · 100% of the source is main text. Read the extracted text to check this.
The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Cheng Yu and Dong Li and Feiyu Jiang and Ke Zhu (2025) Matrix GARCH Model: Inference and Application self | 1.000 | 5 | 3 | 100% |
| 2 | Christian Francq and Jean-Michel Zakoian (2019) GARCH Models: Structure, Statistical Inference and Financial Applications | 0.928 | 4 | 3 | 100% |
| 3 | Pedersen, Rasmus S and Rahbek, Anders (2014) Multivariate variance targeting in the BEKK–GARCH model | 0.843 | 3 | 3 | 100% |
| 4 | Rong Chen and Dan Yang and Cun-Hui Zhang (2022) Factor Models for High-Dimensional Tensor Time Series | 0.811 | 4 | 2 | 100% |
| 5 | Robert F. Engle and Kenneth F. Kroner (1995) Multivariate Simultaneous Generalized ARCH | 0.737 | 3 | 2 | 100% |
| 6 | Robert F. Engle and Olivier Ledoit and Michael Wolf (2019) Large Dynamic Covariance Matrices | 0.737 | 3 | 2 | 100% |
| 7 | Yuefeng Han and Dan Yang and Cun-Hui Zhang and Rong Chen (2024) CP Factor Model for Dynamic Tensors | 0.737 | 3 | 2 | 100% |
| 8 | Cheng Yu and Zhoufan Zhu and Ke Zhu (2025) Tensor Dynamic Conditional Correlation Model: A New Way to Pursuit “Holy Grail of Investing” | 0.737 | 3 | 2 | 100% |
| 9 | Matteo Barigozzi and Haeran Cho and Hyeyoung Maeng (2026) Tail-Robust Factor Modelling of Vector and Tensor Time Series in High Dimensions | 0.644 | 2 | 2 | 100% |
| 10 | Farid Boussama and Florian Fuchs and Robert Stelzer (2011) Stationarity and Geometric Ergodicity of BEKK Multivariate GARCH Models | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 56 scored citations.