Tensor-BEKK：张量值时间序列的条件协方差建模与推断

中文摘要 AI 辅助

现代经济和金融数据日益以多维数组的形式组织，观测值同时按地理区域、工业部门、资产类别和其他经济特征进行索引。将此类数据表示为张量值时间序列可以保留其固有的多路结构。尽管已有大量工作致力于张量值时间序列的条件均值建模，但对其条件协方差动态的关注相对较少。后者仍然具有挑战性，因为无约束的多元协方差模型涉及大量参数和可观的计算成本。为解决这些挑战，我们提出了Tensor-BEKK（T-BEKK）模型，这是一种张量结构的BEKK规范，在保持向量化过程的正定协方差递归的同时，对截距项以及ARCH和GARCH系数矩阵施加Kronecker结构。该模型降低了参数维度，并为协方差截距、ARCH效应和GARCH持续性提供了特定模式解释。我们建立了平稳性、可识别性以及高斯拟极大似然估计量的渐近性质。我们进一步开发了针对张量结构定制的特定模式受限得分检验、每个模式内非零溢出强度的推断程序，以及基于二次型残差的portmanteau诊断检验。针对更高维度的设置，我们还引入了Tensor-Factor-BEKK（TF-BEKK）模型。在第一步可忽略性条件下，其可行的第二步QMLE在渐近上等价于基于潜在因子的oracle QMLE。模拟和两个实证应用（涵盖货币期货和中国股票张量投资组合配置）说明了所提出方法的有限样本行为和实证实用性。

英文摘要

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.

发表机构

• National University of Defense Technology（国防科技大学）
• National Key Laboratory of Digital Intelligent Modeling and Simulation（数字智能建模与仿真全国重点实验室）
• Fudan University（复旦大学）

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