GARCH-type Models with Generalized Secant Hyperbolic Innovations

GARCH-type models have been analyzed assuming various nongaussian distributions of errors. In general, the asymmetric generalized Student-t random variable seems to be the distribution which better captures the nonnormality features of financial data. However, a drawback of this distribution is represented by the technical dificulties due to the evaluation of moments, especially in the case of fractional degrees of freedom. In the paper we propose to model high frequency time series returns using GARCH-type models with a generalized secant hyperbolic (GSH) distribution. The main advantage of the GSH distribution over the Student-t distribution is that all the moments are finite for each value of the shape parameter. The distribution is symmetric with respect to the mean, but we show that it is still possible to obtain the density in a closed form introducing a skewness parameter according to the method proposed by Fernandez and Steel. We use a Monte Carlo experiment to validate this distribution in the context of GARCH models with maximum likelihood estimates of parameters. Finally, we show an application to log returns of a stock index.