  In this paper, we explore the class of the Hidden Semi-Markov Model (HSMM), a
flexible extension of the popular Hidden Markov Model (HMM) that allows the
underlying stochastic process to be a semi-Markov chain. HSMMs are typically
used less frequently than their basic HMM counterpart due to the increased
computational challenges when evaluating the likelihood function. Moreover,
while both models are sequential in nature, parameter estimation is mainly
conducted via batch estimation methods. Thus, a major motivation of this paper
is to provide methods to estimate HSMMs (1) in a computationally feasible time,
(2) in an exact manner, i.e. only subject to Monte Carlo error, and (3) in a
sequential setting. We provide and verify an efficient computational scheme for
Bayesian parameter estimation on HSMMs. Additionally, we explore the
performance of HSMMs on the VIX time series using Autoregressive (AR) models
with hidden semi-Markov states and demonstrate how this algorithm can be used
for regime switching, model selection and clustering purposes.
