Semi-parametric Bayesian Forecasting with an Application to Stochastic Volatility

We propose a new and highly exible Bayesian sampling algorithm for nonlinear state space models under nonparametric distributions. The estimation framework combines a particle filtering and smoothing algorithm for the latent process with a Dirichlet process mixture model for the error term of the observable variables. In particular, we overcome the problem of constraining the models by transformations or the need for conjugate distributions. We use the Chinese restaurant representation of the Dirichlet process mixture, which allows for a parsimonious and generally applicable sampling algorithm. Thus, our estimation algorithm combines a pseudo marginal Metropolis Hastings scheme with a marginalized hierarchical semiparametric model. We test our approach for several nested model specifications using simulated data and provide density forecasts. Furthermore, we carry out a real data example using S&P 500 returns.