State Space Modeling of GARMA Processes with Generalized Long Memory

This chapter extends state space methodology to the Gegenbauer ARMA (GARMA) process, which generalizes long memory by allowing spectral peaks at frequencies away from the origin. The GARMA(p,d,q) model is defined using the Gegenbauer polynomial (1-2uB+B 2 )^d, where u ∈ (-1,1) determines the frequency location of the spectral peak and d controls the memory intensity. This formulation captures cyclical long memory behavior observed in phenomena such as sunspots and El Niño oscillations. The chapter develops MA and AR approximations for GARMA processes using Gegenbauer coefficients computed via recursive formulas. The Kalman filter recursions and QMLE methodology are adapted for this generalized setting. An extensive Monte Carlo study across different combinations of d, u, and sample sizes establishes optimal truncation lag orders for both approximations. The MA approximation is found superior in terms of implementation ease, lower estimation variance, and computational efficiency. Theoretical results including consistency, asymptotic normality, and efficiency of the QMLE estimators are presented. Empirical applications to Wolfer’s and Tong’s sunspot series demonstrate forecasting performance superior to traditional ARIMA models. The chapter also extends the methodology to seasonal GARSMA models, incorporating both Gegenbauer long memory and seasonal components.