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 identification of conditional row and column covariance matrices. 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. 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.

翻译：矩阵型时间序列数据在应用中大量存在。然而，目前尚无研究探讨经济与金融数据中常见的条件异方差性。为填补这一空白，我们提出一种新颖的矩阵广义自回归条件异方差（GARCH）模型，以捕捉矩阵时间序列条件行协方差矩阵与列协方差矩阵的动态变化。该矩阵GARCH模型的核心创新在于利用单变量GARCH规范来刻画条件行或列协方差矩阵的迹，从而实现对条件行协方差矩阵与列协方差矩阵的识别。此外，我们引入拟极大似然估计量（QMLE）进行模型估计，并开发了混成检验（portmanteau test）用于模型诊断。通过仿真研究评估QMLE与混成检验的有限样本性能。为处理高维矩阵时间序列，我们还提出了矩阵因子GARCH模型。最后，基于信用违约互换价格、全球股票行业指数及期货价格的三项应用，我们证明了矩阵GARCH模型与矩阵因子GARCH模型在波动率预测与投资组合配置中优于现有多元GARCH类模型。