  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.
