A Novel Non-Linear GARCH framework for Modelling the Volatility of Heteroskedastic Time Series

Conventional methods of modelling financial data using Generalised AutoRegressive Conditional Heteroskedastic (GARCH) models make many assumptions involving the distribution of the data and the structures of the underlying mean and variance models. This research intends to develop a unified framework for estimating GARCH models using Non-Linear AutoRegressive Moving Average with eXogenous inputs (NARMAX) methodology without making many assumptions about the structures of the mean and variance models. This thesis starts with a review of financial volatility and the different models that have been used to model financial volatility. These models are collectively termed as GARCH-class models. All GARCH-class models attempt to model financial volatility of a given return series by fitting a mean model, obtaining the residuals, and then fitting a variance model using the obtained residuals. Whilst a great deal of research has been done to develop different types of linear and non-linear variance model, researchers have ignored the possibility of the mean model being non-linear in nature, and most GARCH models use a very simple constant mean model to describe the means process. In 2010, Zhao developed a NARMAX based Weighted Orthogonal Forward Regression method to identify non-linear mean models whilst assuming that the structure of the variance model is known. In this thesis, this method is extended to accommodate the case where the variance is unobservable. The working of this method is demonstrated with a simulated example. The method is also used to select and estimate the mean models of two real financial data sets and to demonstrate that a constant mean model is often inadequate and can result in inaccurate variance estimates. A new Weighted Least Squares approach for the estimation of the variance model is also developed. Since the true variance is unobservable and unknown, a linear ARCH estimate of the variance is used as a proxy for the true variance. The results of simulations to demonstrate the working of the new method are also shown. Identification of a non-linear variance model is not possible using this method, since a linear estimate of the variance is used. The thesis goes on to generate a non-linear estimate of the variance without making strong assumptions about the structure of the variance model making use of Radial Basis Function models. These have been used to create a generalised representation of linear and non-linear multivariate functions in other fields. In order to create a generalised non-linear variance estimate, a generalised RBF representation of a linear ARCH variance estimate is first created. The parameters of the obtained RBF model are optimised using Maximum Likelihood to generate a variance estimate that is much more accurate. The proposed method is tested and demonstrated in three simulations and succeeds in creating a non-linear variance estimate that is more accurate than a linear ARCH variance estimate in all the demonstrated simulations. The methods introduced in this thesis build upon the newly developed application of NARMAX methodology to modelling financial volatility using GARCH models and provides a fresh new perspective to estimating financial volatility using GARCH-class models without making many assumptions about the structures of the underlying mean and variance processes.