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

2025-08-15

AI Summary

This paper addresses the challenge of modeling common stochastic trends in nonstationary matrix-valued time series by proposing a cointegrated matrix factor model that preserves the intrinsic matrix structure. The model jointly captures dynamic dependencies along both rows and columns, accommodates weak factors and heterogeneous factor strengths, thereby extending the applicability beyond conventional stationary or scalar-factor settings. Estimation is achieved via eigenanalysis of row- and column-wise sample covariance matrices, coupled with a ratio-based criterion for automatic determination of the latent factor dimension—substantially reducing computational complexity. Theoretically, this work establishes, for the first time, consistency results for both the nonstationary factor matrices and their loadings. Extensive simulations and empirical analysis demonstrate the method’s high finite-sample estimation accuracy and robustness.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

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.

Problem

• Modeling nonstationary matrix-valued time series with trends
• Exploring matrix structure via row-column trend interactions
• Estimating factor matrices under cointegration and weak factors

Innovation

• Matrix factor model preserves intrinsic structure
• Eigenanalysis of covariance matrices for estimation
• Ratio criterion estimates latent factor size