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We consider two-state Non-Homogeneous Hidden Markov Models (NHHMMs) forforecasting univariate time series. Given a set of predictors, the time seriesare modeled via predictive regressions with state dependent coefficients andtime-varying transition probabilities that depend on the predictors via alogistic function. In a hidden Markov setting, inference for logisticregression coefficients becomes complicated and in some cases impossible due toconvergence issues. In this paper, we aim to address this problem using a newlatent variable scheme that utilizes the P\'{o}lya-Gamma class ofdistributions. We allow for model uncertainty regarding the predictors thataffect the series both linearly -- in the mean -- and non-linearly -- in thetransition matrix. Predictor selection and inference on the model parametersare based on a MCMC scheme with reversible jump steps. Single-step andmultiple-steps-ahead predictions are obtained by the most probable model,median probability model or a Bayesian Model Averaging approach. Usingsimulation experiments, we illustrate the performance of our algorithm invarious setups, in terms of mixing properties, model selection and predictiveability. An empirical study on realized volatility data shows that ourmethodology gives improved forecasts compared to benchmark models.

Forecasting under model uncertainty:Non-homogeneous hidden Markov models with Polya-Gamma data augmentation