[Paper Review] Autoregressive Models for Matrix-Valued Time Series

Rong Chen, Xiao Han+1 more|arXiv (Cornell University)|Dec 21, 2018

Complex Systems and Time Series Analysis28 references3 citations

TL;DR

This paper proposes a matrix autoregressive model (MAR) that preserves the matrix structure of time series data by modeling the conditional mean as a bilinear transformation of the previous matrix using two coefficient matrices. The approach achieves significant dimension reduction and improved interpretability compared to vector autoregressive models, with theoretical properties and estimation methods established for inference and prediction.

ABSTRACT

In finance, economics and many other fields, observations in a matrix form are often generated over time. For example, a set of key economic indicators are regularly reported in different countries every quarter. The observations at each quarter neatly form a matrix and are observed over many consecutive quarters. Dynamic transport networks with observations generated on the edges can be formed as a matrix observed over time. Although it is natural to turn the matrix observations into a long vector, and then use the standard vector time series models for analysis, it is often the case that the columns and rows of the matrix represent different types of structures that are closely interplayed. In this paper we follow the autoregressive structure for modeling time series and propose a novel matrix autoregressive model in a bilinear form that maintains and utilizes the matrix structure to achieve a greater dimensional reduction, as well as more interpretable results. Probabilistic properties of the models are investigated. Estimation procedures with their theoretical properties are presented and demonstrated with simulated and real examples.

Motivation & Objective

• To address the limitations of vector autoregressive models when applied to matrix-valued time series, which lose structural information by flattening matrices into vectors.
• To develop a parsimonious model that maintains the inherent row and column structures in matrix-structured time series data, such as economic indicators across countries or network edge observations.
• To provide a statistically interpretable model where the two coefficient matrices correspond to row and column dynamics, respectively, improving model interpretability.
• To establish theoretical properties of the model, including asymptotic distributions of estimators, under various error covariance structures.
• To demonstrate the model's performance through simulations and real data, including out-of-sample prediction and residual diagnostics.

Proposed method

• Proposes a first-order matrix autoregressive model of the form $\mathbf{X}_t = \mathbf{A}\mathbf{X}_{t-1}\mathbf{B}' + \mathbf{E}_t$, where $\mathbf{X}_t$ is an $m \times n$ matrix at time $t$, and $\mathbf{A}, \mathbf{B}$ are coefficient matrices.
• Extends the model to higher-order autoregressive structures (MAR(p)) by incorporating multiple lagged matrices.
• Employs three estimation methods: least squares (LSE), maximum likelihood (MLE), and nearest Kronecker product projection (PROJ), each suited to different error covariance assumptions.
• Derives asymptotic distributions of the estimators under regularity conditions, enabling statistical inference and hypothesis testing.
• Applies the Kronecker product structure to model the error covariance matrix, reducing the number of parameters compared to unstructured forms.
• Uses rolling forecast evaluation to compare MAR(1) performance against univariate AR(1), AR(2), and stacked VAR(1) models on real economic data.

Experimental results

Research questions

• RQ1 Can a bilinear matrix autoregressive model preserve the structural relationships between rows and columns in matrix-valued time series more effectively than vector autoregressive models?
• RQ2 How does the proposed MAR(1) model compare in predictive performance to univariate AR models and stacked vector autoregressive models on real economic data?
• RQ3 What are the theoretical properties of the estimators under different error covariance assumptions, particularly regarding asymptotic normality?
• RQ4 To what extent does the model achieve dimension reduction while maintaining interpretability and predictive accuracy?
• RQ5 Can the model be extended to higher-order autoregression or adapted for volatility modeling in financial or economic applications?

Key findings

• The MAR(1) model with least squares estimation achieved a sum of out-of-sample prediction error squares of 141.82, outperforming the stacked VAR(1) model (296.62) but slightly underperforming individual AR(1) models (136.00).
• The MAR(1) model with maximum likelihood estimation achieved the lowest prediction error (136.69), indicating strong performance among the three estimators tested.
• Residual diagnostics, including ACF plots and residual plots, showed no significant autocorrelation, supporting the model's adequacy in capturing the dynamic structure.
• The model demonstrated substantial dimension reduction: while a stacked VAR model requires $m^2n^2$ parameters, MAR(1) requires only $m^2 + n^2 - 1$ parameters.
• Theoretical asymptotic distributions of the estimators were derived, enabling valid statistical inference under regularity conditions.
• The MAR(1) model outperformed univariate AR(1) models in the later three years of the forecast period, indicating improved robustness over time despite initial underperformance.