Matrix GARCH model: Inference and application*

Matrix-variate time series data are largely available in applications. However, no attempt has been made to study their conditional heteroskedasticity that is often observed in economic and financial data. To address this gap, we propose a novel matrix generalized autoregressive conditional heteroskedasticity (GARCH) model to capture the dynamics of conditional row and column covariance matrices of matrix time series. The key innovation of the matrix GARCH model is the use of a univariate GARCH specification for the trace of conditional row or column covariance matrix, which allows for the model identification. Moreover, we introduce a quasi-maximum likelihood estimator (QMLE) for model estimation and develop a portmanteau test for model diagnostic checking. Simulation studies are conducted to assess the finite-sample performance of the QMLE and portmanteau test. To handle large dimensional matrix time series, we also propose a matrix factor GARCH model, and establish its theoretical properties. Finally, we demonstrate the superiority of the matrix GARCH and matrix factor GARCH models over existing multivariate GARCH-type models in volatility forecasting and portfolio allocations using three applications on credit default swap prices, global stock sector indices, and future prices.

矩阵值时间序列数据广泛存在于各类实际应用场景中。然而，针对经济与金融数据中普遍存在的条件异方差性，目前尚未有相关研究展开探讨。为填补这一研究空白，本文提出一种新颖的矩阵广义自回归条件异方差（GARCH）模型，用以刻画矩阵时间序列的条件行协方差矩阵与列协方差矩阵的动态演化规律。该矩阵GARCH模型的核心创新之处在于，针对条件行协方差矩阵或列协方差矩阵的迹采用单变量GARCH设定形式，以此实现模型识别。此外，本文提出用于模型估计的拟极大似然估计量（QMLE），并构建了用于模型诊断检验的混成检验。通过模拟实验评估了拟极大似然估计量与混成检验的有限样本表现。针对高维矩阵时间序列场景，本文进一步提出矩阵因子GARCH模型，并推导了其理论性质。最后，通过三个实证应用——信用违约互换（CDS）价格、全球股票行业指数与期货价格——验证了矩阵GARCH与矩阵因子GARCH模型在波动率预测与资产配置任务中，相较于现有多元GARCH类模型的性能优势。