Factor Models of Matrix-Valued Time Series: Nonstationarity and Cointegration

In this paper, we consider the nonstationary matrix-valued time series with common stochastic trends. Unlike the traditional factor analysis which flattens matrix observations into vectors, we adopt a matrix factor model in order to fully explore the intrinsic matrix structure in the data, allowing interaction between the row and column stochastic trends, and subsequently improving the estimation convergence. It also reduces the computation complexity in estimation. The main estimation methodology is built on the eigenanalysis of sample row and column covariance matrices when the nonstationary matrix factors are of full rank and the idiosyncratic components are temporally stationary, and is further extended to tackle a more flexible setting when the matrix factors are cointegrated and the idiosyncratic components may be nonstationary. Under some mild conditions which allow the existence of weak factors, we derive the convergence theory for the estimated factor loading matrices and nonstationary factor matrices. In particular, the developed methodology and theory are applicable to the general case of heterogeneous strengths over weak factors. An easy-to-implement ratio criterion is adopted to consistently estimate the size of latent factor matrix. Both simulation and empirical studies are conducted to examine the numerical performance of the developed model and methodology in finite samples.