State space modeling of Gegenbauer processes with long memory

An approximation of a Gegenbauer autoregressive moving average (GARMA) process with long memory using a finite order moving average (MA) representation is considered. The state space form of the MA approximation is developed and the corresponding estimates are obtained by pseudo maximum likelihood using the Kalman filter. For comparative purposes the same exercise is executed with an autoregressive (AR) approximation. Using an extensive Monte Carlo experiment, optimal order of the chosen MA approximation is established, and found it was not very large (around 35) and rather insensitive to the sample size. Further evidence suggests the approximation is reliable for forecasting and signal extraction with periodic long memory components. A rolling forecasting experiment was performed to validate the choice of optimal order of both AR and MA approximations in terms of predictive accuracy. Finally, the proposed methodology was applied to two yearly sunspots time series, and compared with corresponding results proposed in the literature.