Bayesian analysis of predictive Non-Homogeneous hidden Markov models using Polya-Gamma data augmentation
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We consider Non-Homogeneous Hidden Markov Models (NHHMMs) for forecasting univariate time series. We introduce two state NHHMMs where the time series are modeled via different predictive regression models for each state. Also, the time-varying transition probabilities depend on exogenous variables through a logistic function. In a hidden Markov setting, inference for logistic regression coefficients becomes complicated and in some cases impossible due to convergence issues. To address this problem, we use a new latent variable scheme, that utilizes the Pólya-Gamma class of distributions, introduced by Po13. Given an available set of predictors, we allow for model uncertainty regarding the predictors that affect the series both linearly -- in the mean -- and non-linearly -- in the transition matrix. Predictor selection and inference on the model parameters are based on a MCMC scheme with reversible jump steps. Single-step and multiple-steps-ahead predictions are obtained based on the most probable model, median probability model or a Bayesian Model Averaging (BMA) approach. Simulation experiments, as well as an empirical study on real financial data, illustrate the performance of our algorithm in various setups, in terms of mixing properties, model selection and predictive ability.
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