Estimation methods of Matrix-valued AR model

TL;DR

This work develops Yule-Walker and Burg-type estimation methods for Matrix-valued autoregressive (MAR) models, addressing causality and covariance preservation limitations of standard estimators. It defines MAR(1) and connects it to VAR(1) via vec representations, highlighting the significant parameter reduction relative to VAR. Through theoretical derivations and empirical benchmarking, the authors show that MAR estimators achieve comparable MAE and RMSE to VAR while using far fewer parameters, with Burg tending to offer fast, robust performance and Yule-Walker providing stability for long series. The approach offers a practical, interpretable framework for high-dimensional time series where preserving matrix structure and autocovariance is desirable.

Abstract

This article proposes novel estimation methods for the Matrix Autoregressive (MAR) model, specifically adaptations of the Yule-Walker equations and Burg's method, addressing limitations in existing techniques. The MAR model, by maintaining a matrix structure and requiring significantly fewer parameters than vector autoregressive (VAR) models, offers a parsimonious, yet effective, alternative for high-dimensional time series. Empirical results demonstrate that MAR models estimated via the proposed methods achieve a comparable fit to VAR models across metrics such as MAE and RMSE. These findings underscore the utility of Yule-Walker and Burg-type estimators in constructing efficient and interpretable models for complex temporal data.

Figures (7)

• Figure 1: Comparison of the Yule-Walker approach for the $\operatorname{MAR}(1)$ and $\operatorname{VAR}(1)$ models for a time series of length 300.
• Figure 2: Comparison of the Yule-Walker approach for the $\operatorname{MAR}(1)$ and $\operatorname{VAR}(1)$ models with a matrix size of 5.
• Figure 3: Comparison of the Burg approach for the $\operatorname{MAR}(1)$ and $\operatorname{VAR}(1)$ models for a time series of length 300. For metrics charts, the blue and orange lines overlap.
• Figure 4: Comparison of the Burg approach for the $\operatorname{MAR}(1)$ and $\operatorname{VAR}(1)$ models with a matrix size of 5. For metrics charts, the blue and orange lines overlap.
• Figure 5: Comparison of the YW, Burg and LSE approach for the $\operatorname{MAR}(1)$ model for a time series of length 300. For execution time chart, the red and orange lines overlap.

Theorems & Definitions (2)

• proof
• proof