Multi-linear Tensor Autoregressive Models

TL;DR

This paper introduces a tensor autoregressive model for tensor-valued time series, preserving tensor structure and providing new estimation methods with theoretical guarantees, demonstrated through simulations and real data.

Contribution

It proposes a novel tensor autoregressive model with three estimators, extending traditional vector autoregression to tensor data and analyzing both fixed and high-dimensional cases.

Findings

Establishes central limit theorems for estimators in fixed dimensions

Provides convergence rates and model selection criteria in high dimensions

Demonstrates effectiveness through simulations and real data examples

Abstract

Contemporary time series analysis has seen more and more tensor type data, from many fields. For example, stocks can be grouped according to Size, Book-to-Market ratio, and Operating Profitability, leading to a 3-way tensor observation at each month. We propose an autoregressive model for the tensor-valued time series, with autoregressive terms depending on multi-linear coefficient matrices. Comparing with the traditional approach of vectoring the tensor observations and then applying the vector autoregressive model, the tensor autoregressive model preserves the tensor structure and admits corresponding interpretations. We introduce three estimators based on projection, least squares, and maximum likelihood. Our analysis considers both fixed dimensional and high dimensional settings. For the former we establish the central limit theorems of the estimators, and for the latter we focus on the convergence rates and the model selection. The performance of the model is demonstrated by simulated and real examples.