This dissertation aims to extend on the idea of Bollerslev (1987), estimating ARCH models with Student-t distributed errors, to estimating Stochastic Volatility (SV) models with Student-t distributed errors. It is unclear whether Gaussian distributed errors sufficiently account for the observed leptokurtosis in financial time series and hence the extension to examine Student-t distributed errors for these models. The quasi-maximum likelihood estimation approach introduced by Harvey (1989) and the conventional Kalman filter technique are described so that the SV model with Gaussian distributed errors and SV model with Student-t distributed errors can be estimated. Estimation of GARCH (1,1) models is also described using the method maximum likelihood. The empirical study estimated four models using data on four different share return series and one index return, namely: Anglo American, BHP, FirstRand, Standard Bank Group and JSE Top 40 index. The GARCH and SV model with Student-t distributed errors both perform best on the series examined in this dissertation. The metric used to determine the best performing model was the Akaike information criterion (AIC).

Pathogens, such as HIV and influenza, evolve in response to the selective pressures of their host environments accumulating changes in their genomes that offer fitness benefits. This selective pressure is characterised by three properties: (1.) it is episodic, tracking changes in the adaptive immune response and drug therapy, (2.) it is directional in that only particular amino acid substitutions are favoured and (3.) it varies between genomic loci. Most previous models have ignored or inadequately addressed some of these phenomena. This work extends recent approaches to modelling episodic directional selection acting on protein-coding sequences. We use inference techniques within the topic model framework to identify loci evolving under natural selection. A notable example of such techniques are the variational Bayesian methods. We show that our approach performs well in terms of specificity and power, and demonstrate its utility by applying it to some real datasets of HIV sequences.