Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models

We introduce a number of nonstandard stochastic volatility (SV) models and examine their performance when applied to the series of daily returns on several stocks listed on the New York Stock Exchange. The nonstandard models under investigation extend both the observation process and the volatility-generating process of basic SV models. In particular, we consider dependent as well as independent mixtures of autoregressive components as the log-volatility process, and include in the observation equation a lower bound on the volatility. We also consider an experimental SV model that is based on conditionally gamma-distributed volatilities.Our estimation method is based on the fact that an SV model can be approximated arbitrarily accurately by a hidden Markov model (HMM), whose likelihood is easy to compute and to maximize. The method is close, but not identical, to those of Fridman and Harris (1998), Bartolucci and De Luca (2001, 2003) and Clements et al. (2006), and makes explicit the useful link between HMMs and the methods of those authors. Likelihood-based estimation of the parameters of SV models is usually regarded as challenging because the likelihood is a high-dimensional multiple integral. The HMM approximation is easy to implement and particularly convenient for fitting experimental extensions and variants of SV models such as those we introduce here. In addition, and in contrast to the case of SV models themselves, simple formulae are available for the forecast distributions of HMMs, for computing appropriately defined residuals, and for decoding, i.e. estimating the volatility of the process.