Bayesian Inference of Vector Autoregressions with Tensor Decompositions

Vector autoregression (VAR) is a 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 use 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.