  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 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 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. In the application of the US
macroeconomic data, our models outperform standard VARs in point and density
forecasting and yield a summary of the US economic dynamic.
