Bayesian methods of vector autoregressions with tensor decompositions

Vector autoregressions (VARs) are popular in analyzing 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 overparameterization, treats the coefficient matrix as a third-order tensor and estimates the corresponding tensor decomposition to achieve parsimony. In this paper, the inference of Tensor VARs is inspired by the literature on factor models. Firstly, we determine the rank by imposing the Multiplicative Gamma Prior to 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 introduce a post-processing procedure to solve column permutations and sign-switching issues. In the application of the US macroeconomic data, our models outperform standard VARs in point and density forecasting and yield interpretable results consistent with the US economic history.