  We consider two-state Non-Homogeneous Hidden Markov Models (NHHMMs) for
forecasting univariate time series. Given a set of predictors, the time series
are modeled via predictive regressions with state dependent coefficients and
time-varying transition probabilities that depend on the predictors via a
logistic function. In a hidden Markov setting, inference for logistic
regression coefficients becomes complicated and in some cases impossible due to
convergence issues. In this paper, we aim to address this problem using a new
latent variable scheme that utilizes the P\'{o}lya-Gamma class of
distributions. 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 by the most probable model,
median probability model or a Bayesian Model Averaging approach. Using
simulation experiments, we illustrate the performance of our algorithm in
various setups, in terms of mixing properties, model selection and predictive
ability. An empirical study on realized volatility data shows that our
methodology gives improved forecasts compared to benchmark models.
