Bayesian analysis of predictive Non-Homogeneous hidden Markov models using Polya-Gamma data augmentation.

We consider Non-Homogeneous Hidden Markov Models (NHHMMs) for forecastingunivariate time series. We introduce two state NHHMMs where the time series aremodeled via different predictive regression models for each state. Also, thetime-varying transition probabilities depend on exogenous variables through alogistic function. In a hidden Markov setting, inference for logisticregression coefficients becomes complicated and in some cases impossible due toconvergence issues. To address this problem, we use a new latent variablescheme, that utilizes the P\'{o}lya-Gamma class of distributions, introduced by\citet{Po13}. Given an available set of predictors, we allow for modeluncertainty regarding the predictors that affect the series both linearly -- inthe mean -- and non-linearly -- in the transition matrix. Predictor selectionand inference on the model parameters are based on a MCMC scheme withreversible jump steps. Single-step and multiple-steps-ahead predictions areobtained based on the most probable model, median probability model or aBayesian Model Averaging (BMA) approach. Simulation experiments, as well as anempirical study on real financial data, illustrate the performance of ouralgorithm in various setups, in terms of mixing properties, model selection andpredictive ability.