This talk presents an approximation of a Gegenbauer autoregressive moving average (GARMA) process driven by Gaussian white noise. The process model characterised by long memory using a finite order moving average is considered. Using a derived state space form the parameters are estimated by pseudo maximum likelihood via the Kalman filter. It is comparatively assessed initially with a finite order autoregressive approximation for choice and feasibility in establishing the order of the model. An extensive Monte Carlo experiment is executed to show that the optimal order is not very large (around 35) and rather insensitive to the series length. A rolling forecasting experiment is performed to validate the choice of the order of approximation in terms of predictive accuracy. Proposed state space methodology is applied to two different yearly sunspot series, and compared with other conventional and hybrid time series methods in the literature. The effect of a seasonal filter on GARMA processes is also examined.
This methodology is extended to the class of GARMA models with Generalized Autoregressive Conditionally Heteroskedastic (GARCH) errors. Finally, a unit root test of the index of a Gegenbauer polynomial in terms of the psedo maximum likelihood estimator is also considered.