Maximum likelihood estimation for stochastic volatility in mean models with heavy-tailed distributions

In this article, we introduce a likelihood-based estimation method for the stochastic volatility in mean (SVM) model with scale mixtures of normal (SMN) distributions (Abanto-Valle et al., 2012). Our estimation method is based on the fact that the powerful hidden Markov model (HMM) machinery can be applied in order to evaluate an arbitrarily accurate approximation of the likelihood of an SVM model with SMN distributions. The method is based on the proposal of Langrock et al. (2012) and makes explicit the useful link between HMMs and SVM models with SMN distributions. Likelihood-based estimation of the parameters of stochastic volatility models in general, and SVM models with SMN distributions in particular, is usually regarded as challenging as the likelihood is a high-dimensional multiple integral. However, the HMM approximation, which is very easy to implement, makes numerical maximum of the likelihood feasible and leads to simple formulae for forecast distributions, for computing appropriately defined residuals, and for decoding, i.e., estimating the volatility of the process.