Bayesian estimation of a multivariate TAR model when the noise process distribution belongs to the class of Gaussian variance mixtures

Threshold autoregressive (TAR) models are powerful tools for analyzing nonlinear and multivariate time series, which include Multivariate Self-Exciting Threshold Autoregressive (MSETAR) and Vector Autoregressive (VAR) models as special cases. In this paper, parameter estimation, inference, and forecasting are developed for multivariate TAR (MTAR) models using the Bayesian approach in a flexible setup, under which not only the Gaussian distribution but also the other distributions that belong to the class of Gaussian variance mixtures can be used to describe the noise process behavior. That class of distributions includes Student-t, Slash, symmetric hyperbolic, and contaminated normal, which are also symmetric but more flexible and have heavier tails than the Gaussian one. Inferences from MTAR models based on that kind of distribution may be less affected by extreme or outlying observations than those based on the Gaussian distribution. All parameters in the MTAR model are included in the proposed MCMC-type algorithm, except the number of regimes and the autoregressive orders, which can be chosen using Deviance Information Criterion (DIC) and/or Watanabe-Akaike Information Criterion (WAIC) and/or Root Mean Square Error (RMSE) and/or log-score. A library for the language and environment for statistical computing R was developed to assess the effectiveness of the proposed methodology using simulation studies and analysis of a real multivariate time series.