On Gegenbauer long memory stochastic volatility models: A Bayesian Markov chain Monte Carlo approach with applications

Abstract

This thesis begins by developing a time series model which has generalised (Gegenbauer) long memory in the mean process with stochastic volatility errors where each process is assumed to have Gaussian errors. We subsequently develop and derive a new Bayesian posterior simulator that couples advanced posterior maximisation techniques, as well as traditional latent stochastic volatility estimation procedures. This model provides a distinct advantage by measuring the long memory attributes in the mean process, whilst also estimating the daily time varying volatility of the error process. These findings are then extended to a Gegenbauer long memory stochastic volatility model with leverage and a bivariate Student's t-error distribution to describe the innovations of the observation and latent volatility jointly, with applications to Cryptocurrency time series. The main advantage of pursuing such a model is incorporating the robustness of the Student's t-distribution, and leverage effects to address the deep rooted characteristics found in Cryptocurrencies. The models are applied to Value-at-Risk (VaR) forecasts and several measures are used to assess the forecast performance. Finally, it is found that Cryptocurrencies do indeed show highly distinct behaviours that are not present in fiat currencies, and thus require specialized analysis. Cryptocurrencies as of late have commanded global attention on a number of fronts. Most notably, their variance properties are known for being notoriously wild, unlike their fiat counterparts. The third part of this thesis highlights some stylized facts about the variance measures of Cryptocurrencies and relates these results to their respective cryptographic designs such as intended transaction speed. The overarching implication of these result is the volatility of Cryptocurrencies can be better understood and measured via the use of fast moving autocorrelation functions, as opposed to smoothly decaying functions for fiat currencies.