Vector autoregressions (VARs) are popular model for analyzing multivariate economic time series. However, VARs can be over-parameterized if the numbers of variables and lags are moderately large. Tensor VAR, a recent solution to over-parameterization, treats the coefficient matrix as a third-order tensor and estimates the corresponding tensor decomposition to achieve parsimony. In this paper, we employ the Tensor VAR structure with a CANDECOMP/PARAFAC (CP) decomposition and conduct Bayesian inference to estimate parameters. Firstly, we determine the rank by imposing the Multiplicative Gamma Prior to the tensor margins, i.e. elements in the decomposition, and accelerate the computation with an adaptive inferential scheme. Secondly, to obtain interpretable margins, we propose an interweaving algorithm to improve the mixing of margins and identify the margins using a post-processing procedure. In an application to the US macroeconomic data, our models outperform standard VARs in point and density forecasting and yield a summary of the dynamic of the US economy.

翻译：向量自回归（VAR）是分析多元经济时间序列的常用模型。然而，当变量数量和滞后阶数适中偏大时，VAR模型可能存在过度参数化问题。张量VAR作为近期解决过度参数化的一种方案，将系数矩阵视为三阶张量，并通过估计相应的张量分解来实现参数简约性。本文采用具有CANDECOMP/PARAFAC（CP）分解结构的张量VAR模型，并采用贝叶斯推断方法进行参数估计。首先，我们通过对张量边缘（即分解中的元素）施加乘法伽马先验来确定秩，并采用自适应推断方案加速计算过程。其次，为获得可解释的边缘分量，我们提出了一种交织算法以改进边缘分量的混合特性，并利用后处理程序对边缘分量进行识别。在美国宏观经济数据的实证应用中，我们的模型在点预测和密度预测方面均优于标准VAR模型，并提供了美国经济动态的概括性分析。