Generalised linear Gegenbauer long memory models for time series of counts with financial and insurance applications

Abstract

The contribution of this thesis is on developing new models and applying them to analyse data in finance and insurance. The proposed generalised linear Gegenbauer autoregressive moving average models (GLGARMA) incorporate GARMA into the mean functions of the four different count distributions within a GLM framework under the parameter-driven and observation-driven approaches. Model properties in the time domain and spectral density function in the frequency domain are studied and compared to seasonal long memory model. The approximated spectral density function of the PD GLGARMA model is derived to facilitate Whittle likelihood estimation. To estimate and forecast these models, we adopt a Bayesian approach implemented using the R package Rstan. Various model selection criteria including deviance information criterion are evaluated to select some best-fitting models to undertake forecasting of future events. We test 136 open interest series for the types of long memory structures. The GLGARMA models outperform these models in both in-sample fittings and out-of-sample forecasts for each type of long memory structures. The prevalence of long memory in death count series is demonstrated. We extend this Lee-Carter type GLGARMA resulting in three modelling components, namely, period effect, graduation effect and long memory. Results show that the long memory structures enhance the accuracy of in-sample fitting, out-of-sample forecast and life expectancy. A stationary long memory mortality model with long memory cohort effect (LMLM) model is proposed. The in-sample fitting, out-of-sample forecast performances and life expectancies are calculated to exhibit the enhancement by adopting LMLM model. Assuming both constant interest rate and stochastic interest rate model with four dependency models, the annuity pricing and guaranteed annuity options demonstrate the enhancement of adopting LMLM model.