Practical likelihood-based inference for the univariate generalized hyperbolic distribution

Abstract

Maximum likelihood estimation is a powerful estimation tool that is widely used to fit models to data. In this study, the behaviour of the log-likelihood function, and the ensuing impact on the maximum likelihood estimation process is explored. This exploration is conducted using the univariate generalized hyperbolic distribution, a highly flexible distribution with tail properties making it desirable as a model for financial returns data. The study aims to explore potential issues that may present when estimating the parameters of such flexible distributions, especially those stemming from the behaviour of the log-likelihood function. Different numerical methods are applied to showcase the effect of not only the shape and behaviour of the log-likelihood function, but the structure of the parameters themselves on the outcome of the estimation process. Application to real-world financial data shows that the behaviour of the log-likelihood function has a significant impact on the estimation outcome, and that an understanding of these components is fundamental to the success of the process of estimation.