Memory and persistence in models of volatility in financial time series

Abstract

This thesis first investigates the moment and memory properties of exponential-type conditional heteroscedasticity models. This primarily includes exponential generalised autoregressive conditional heteroscedastic (EGARCH) models, the fractionally integrated EGARCH model of Bollerslev and Mikkelsen (1996) (FIEGARCH(BM)), the hyperbolic EGARCH (HYEGARCH) model and the FIEGARCH(DL) model, as presented in Chapter 2. The moment conditions of these models are derived from previous literature, and the memory properties are measured by using the near-epoch dependence (NED) functions of an independent process approach. The existence of moments supports the limited memory properties of these models. This study shows that exponential autoregressive conditional heteroscedastic (EARCH)(∞) processes may exhibit geometric memory, hyperbolic memory or long memory. The EGARCH is a case of a geometric memory process. The FIEGARCH(BM) and HY/FIEGARCH(DL) processes can exhibit hyperbolic memory or long memory, depending on the sign of the memory parameter. The study also derives the functional central limit theorem (FCLT) or fractional FCLT for the relevant processes in these exponential-type conditional heteroscedasticity models. Finally, the results of the simulation show that the HYEGARCH model has a hyperbolic memory and that the FIEGARCH(DL) model can capture long memory in absolute return series. Next, the study investigates the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) in autoregressive moving average (ARMA) models with EGARCH or HY/FIEGARCH(DL) errors in Chapter 3. This part of the study aims to investigate the asymptotic theory of the ARMA(1,1)-EGARCH(1,1) models and that of the pure HY/FIEGARCH(DL) models. First, the literature on the asymptotic properties of the ARMA-GARCH and EGARCH processes is reviewed. The conditions for the consistency and asymptotic normality of the QMLE of the ARMA-EGARCH models are then demonstrated. This analysis also provides an investigation of that of the QMLE in the HY/FIEGARCH(DL) processes. A Monte Carlo simulation is used to study the properties of the QMLE in the pure HY/FIEGARCH(DL) processes. Lastly, in a study co-authored with Professor James Davidson, we derive a simple sufficient condition for strict stationarity in the ARCH(∞) class of processes with conditional heteroscedasticity. The concept of persistence in these processes is explored, and is the subject of a set of simulations showing how persistence depends on both the pattern of the lag coefficients of the ARCH model and the distribution of the driving shocks. The results are used to argue that an alternative to the usual method of ARCH/GARCH volatility forecasting should be considered.