# Autoregressive Models for Matrix-Valued Time Series

arXiv: https://arxiv.org/abs/1812.08916 · HTML: https://mlsanity.com/paper/1812.08916/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Dec 2018.

## Built on by (most cited first, 18 of 18)

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- [Bivariate Matrix-valued Linear Regression (BMLR): Finite-sample performance under Identifiability and Sparsity Assumptions](https://mlsanity.com/paper/2412.17749/) (2024)
- [Sparsity-Induced Global Matrix Autoregressive Model with Auxiliary Network Data](https://mlsanity.com/paper/2503.08579/) (2025)
- [Estimation methods of Matrix-valued AR model](https://mlsanity.com/paper/2505.15220/) (2025)
- [Learning Conditional Independence Differential Graphs From Time-Dependent Data](https://mlsanity.com/paper/2512.06960/) (2025)
- [Exact Recovery of Non-Random Missing Multidimensional Time Series via Temporal Isometric Delay-Embedding Transform](https://mlsanity.com/paper/2512.10191/) (2025)
- [Structured Adaptive Tensor Prediction for Streaming Data](https://mlsanity.com/paper/2606.10085/) (2026)

## Abstract (arXiv)

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.