  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.
