Abstract

Matrix-valued time series are ubiquitous in modern economics and finance, yet modeling them requires navigating a trade-off between flexibility and parsimony. We propose the Matrix Autoregressive

model with Common Factors (MARCF), a hybrid structured Reduced-Rank Matrix Autoregression

(RRMAR) formulation that balances the flexible predictor-response subspace geometry with the parsimony of the dynamic Matrix Factor Model (MFM), in which low-dimensional latent factors follow

autoregressive dynamics. While RRMAR allows arbitrary predictor and response subspaces without

explicitly distinguishing their common and specific parts, MARCF explicitly characterizes the intersection of these subspaces. By decomposing the coefficient matrices into common, predictor-specific, and

response-specific components, the framework accommodates distinct input and output structures while

exploiting their overlap for dimension reduction. We develop a regularized gradient descent estimator

that is scalable for high-dimensional data and can efficiently handle the non-convex parameter space.

Theoretical analysis establishes local linear convergence of the algorithm and statistical consistency of

the estimator under high-dimensional scaling. The estimation efficiency and interpretability of the proposed methods are demonstrated through simulations and an application to a global macroeconomic

dataset.

Key words and phrases: Autoregression, dimension reduction, factor model, matrix-valued time series

Information

Preprint No.SS-2025-0459
Manuscript IDSS-2025-0459
Complete AuthorsZhiyun Fan, Xiaoyu Zhang, Di Wang
Corresponding AuthorsDi Wang
Emailsdi.wang@sjtu.edu.cn

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Acknowledgments

We thank the editor, the associate editor, and the referees for their constructive comments

and helpful suggestions. Xiaoyu Zhang’s research is supported by National Natural Science

Foundation of China (Grant No. 12601528), National Key R&D Program of China (Grant

No. 2024YFA1015700), and Fundamental Research Funds for the Central Universities (Grant

No. 22120260347). Di Wang’s research is supported by National Natural Science Foundation

of China (Grant Nos. 12301352, 72495121, 72671185).

Supplementary Materials

The online supplementary material provides additional details on the estimation procedure

and proofs of the theoretical results.

Supplementary materials are available for download.