A Bayesian GED-Gamma stochastic volatility model for return data: a marginal likelihood approach

Several studies explore inferences based on stochastic volatility (SV) models, taking into account the stylized facts of return data. The common problem is that the latent parameters of many volatility models are high-dimensional and analytically intractable, which means inferences require approximations using, for example, the Markov Chain Monte Carlo or Laplace methods. Some SV models are expressed as a linear Gaussian state-space model that leads to a marginal likelihood, reducing the dimensionality of the problem. Others are not linearized, and the latent parameters are integrated out. However, these present a quite restrictive evolution equation. Thus, we propose a Bayesian GED-Gamma SV model with a direct marginal likelihood that is a product of the generalized Student's t-distributions in which the latent states are related across time through a stationary Gaussian evolution equation. Then, an approximation is made for the prior distribution of log-precision/volatility, without the need for model linearization. This also allows for the computation of the marginal likelihood function, where the high-dimensional latent states are integrated out and easily sampled in blocks using a smoothing procedure. In addition, extensions of our GED-Gamma model are easily made to incorporate skew heavy-tailed distributions. We use the Bayesian estimator for the inference of static parameters, and perform a simulation study on several properties of the estimator. Our results show that the proposed model can be reasonably estimated. Furthermore, we provide case studies of a Brazilian asset and the pound/dollar exchange rate to show the performance of our approach in terms of fit and prediction. Keywords: SV model, New sequential and smoothing procedures, Generalized Student's t-distribution, Non-Gaussian errors, Heavy tails, Skewness