A Two-Way Transformed Factor Model for Matrix-Variate Time Series

Abstract

A new framework is proposed for modeling high-dimensional matrix-variate time series via 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. For a given $p_1\times p_2$ matrix-variate time series, nonsingular transformations are sought to project the rows and columns onto another $p_1$ and $p_2$ directions according to the strength of the dynamical dependence of the series on their past values. Consequently, the data are nonsingular linear row and column transformations of dynamically dependent common factors and white noise idiosyncratic components. A common orthonormal projection method is proposed 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 $p_2$, a two-way projected Principal Component Analysis is introduced to estimate the associated loading matrices of the idiosyncratic terms to mitigate such diverging noise effects. A new white-noise testing procedure is proposed to estimate the dimension of the factor matrix. Asymptotic properties of the proposed method are established for both fixed and diverging dimensions as the sample size increases to infinity. Simulated and real examples are used to assess the performance of the proposed method. Comparisons of the proposed method with some existing ones in the literature concerning the forecastability of the factors are studied and it is found that the proposed approach not only provides interpretable results, but also performs well in out-of-sample forecasting.