  We propose a new framework for modeling high-dimensional matrix-variate time
series by a two-way transformation, where the transformed data consist of a
matrix-variate factor process, which is dynamically dependent, and three other
blocks of white noises. Specifically, for a given $p_1\times p_2$
matrix-variate time series, we seek common nonsingular transformations to
project the rows and columns onto another $p_1$ and $p_2$ directions according
to the strength of the dynamic dependence of the series on the past values.
Consequently, we treat the data as nonsingular linear row and column
transformations of dynamically dependent common factors and white noise
idiosyncratic components. We propose a common orthonormal projection method to
estimate the front and back loading matrices of the matrix-variate factors.
Under the setting that the largest eigenvalues of the covariance of the
vectorized idiosyncratic term diverge for large $p_1$ and $p_2$, we introduce a
two-way projected Principal Component Analysis (PCA) to estimate the associated
loading matrices of the idiosyncratic terms to mitigate such diverging noise
effects. A diagonal-path white noise testing procedure is proposed to estimate
the order of the factor matrix. %under the assumption that the idiosyncratic
term is a matrix-variate white noise process. Asymptotic properties of the
proposed method are established for both fixed and diverging dimensions as the
sample size increases to infinity. We use simulated and real examples to assess
the performance of the proposed method. We also compare our method with some
existing ones in the literature and find that the proposed approach not only
provides interpretable results but also performs well in out-of-sample
forecasting.
