A Hidden Markov Renewal Model

Hidden semi-Markov models (HSMMs) are a powerful class of statistical model that have been applied to a wide range of areas such as speech recognition, protein structure prediction, Internet-traffic modeling, financial time-series modeling, and classification of music. Three basic problems of hidden Markov model inference are: Computation of the likelihood, computation of the maximum likelihood-estimate of the model parameters, and computation of the maximum a posteriori estimate of the hidden state sequence. We address these inference problems for a set of models closely related to HSMMs.

Our contributions are: (i) We extend the HSMM to allow observations to depend not only on the current underlying hidden state, but on the next underlying hidden state also. This extension can be used to model behavior whereby the observed data gradually transitions between states, rather than abruptly. (ii) We formulate the hidden portion of the model as a Markov renewal process. This allows us to naturally perform inference on models with hidden events other than state changes, e.g., jumps. (iii) We show that by augmenting the state space of our hidden Markov renewal model (HMRM), we can perform inference on an even larger class of phenomena, including models with stochastic volatility. Hence our HMRM can address three key areas of modern financial time series: regime-switching, jumps, and stochastic volatility.

We develop algorithms to solve the three basic problems of inference for the HMRM. We validate the algorithms by performing inference on simulated data.

We apply our model to two real-world datasets appearing in previously published analyses. The first dataset contains the log-returns of four European sector indices. Specifications of the HMRM improve the modeling of the auto-correlation function of squared returns compared to the HSMMs used in this first analysis. The second dataset consists of weekly returns from a weighted portfolio of NYSE stocks. Another specification of the HMRM gives improved volatility forecasts compared to the regime-switching GARCH models published in the second analysis.