  We propose a new flexible tensor model for multiple-equation regression that
accounts for latent regime changes. The model allows for dynamic coefficients
and multi-dimensional covariates that vary across equations. We assume the
coefficients are driven by a common hidden Markov process that addresses
structural breaks to enhance the model flexibility and preserve parsimony. We
introduce a new Soft PARAFAC hierarchical prior to achieve dimensionality
reduction while preserving the structural information of the covariate tensor.
The proposed prior includes a new multi-way shrinking effect to address
over-parametrization issues. We developed theoretical results to help
hyperparameter choice. An efficient MCMC algorithm based on random scan Gibbs
and back-fitting strategy is developed to achieve better computational
scalability of the posterior sampling. The validity of the MCMC algorithm is
demonstrated theoretically, and its computational efficiency is studied using
numerical experiments in different parameter settings. The effectiveness of the
model framework is illustrated using two original real data analyses. The
proposed model exhibits superior performance when compared to the current
benchmark, Lasso regression.
