A Hybrid Framework Combining Autoregression and Common Factors for Matrix Time Series

TL;DR

The paper introduces MARCF, a Matrix Autoregressive model with Common Factors, which unifies the flexible subspace structure of MAR with the parsimony of MFM by explicitly modeling intersections between predictor and response subspaces. It decomposes the bilinear coefficient matrices into common, predictor-specific, and response-specific components, enabling dimension reduction of roughly $p_1d_1+p_2d_2$ relative to standard reduced-rank MAR. A scalable regularized gradient descent algorithm with balanced identifiability penalties is developed, and its local convergence and statistical consistency are established, including rank-selection consistency. Through simulations and a multinational macroeconomic forecasting application, MARCF demonstrates improved estimation efficiency, interpretable dynamics (driver vs. responder countries and lead-lag indicators), and superior out-of-sample forecasting performance compared with existing MAR and MFM approaches. The framework thus provides a practical and theoretically sound tool for high-dimensional matrix-valued time series with complex cross-sectional interactions, with potential extensions to sparsity, tensors, and inferential procedures.

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 unified framework for high-dimensional matrix time series that bridges the structural gap between the Matrix Autoregression (MAR) and Matrix Factor Model (MFM). While MAR typically assumes distinct predictor and response subspaces and MFM enforces identical ones, 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 global macroeconomic forecasting.

Figures (7)

• Figure 1: Relative estimation error with various combinations of $(d_1,d_2)$ under two settings of $p_1$, $p_2$ and $T$: (a) $p_1=20$, $p_2=10$, and $T=500$; and (b) $p_1=30$, $p_2=20$, and $T=1000$. Results are based on 500 replications.
• Figure 2: Log estimation error of (a) $\mathcal{M}(\bm \Lambda_1^*)$; and (b) $\mathcal{M}(\bm \Lambda_2^*)$. In each subplot, the left (blue) and right (yellow) boxplot represent the estimation errors of the dynamic MFM and the MARCF model, respectively. Each pair of $(r_1,r_2)$ is evaluated over 500 replications.
• Figure 3: Estimates of common and specific projection matrices over countries.
• Figure 4: Estimates of full predictor and response projection matrices over countries.
• Figure 5: Estimates of common and specific projection matrices over economic indicators.

Theorems & Definitions (31)

• Remark 1: Interpretation vs. Estimation
• Remark 2
• Definition 1: RSC/RSS Condition
• Definition 2: Deviation Bound
• Theorem 1: Local Linear Convergence
• Proposition 1: Condition Verification
• Theorem 2: Statistical Error Rates
• Theorem 3: Rank Selection Consistency
• Lemma C.1
• proof : Proof of Lemma \ref{['lemma:H_upperBound']}