A k‐Factor GARMA Long‐memory Model

Long-memory models have been used by several authors to model data with persistent autocorrelations. The fractional and fractional autoregressive moving-average (FARMA) models describe long-memory behavior associated with an infinite peak in the spectrum at f = 0. The Gegenbauer and Gegenbauer ARMA (GARMA) processes of Gray, Zhang and Woodward (On generalized fractional processes. J. Time Ser. Anal. 10 (1989), 233–57) can model long-term periodic behavior for any frequency 0 ≤f≤ 0.5. In this paper we introduce a k-factor extension of the Gegenbauer and GARMA models that allows for long-memory behavior to be associated with each of k frequencies in [0, 0.5]. We prove stationarity conditions for the k-factor model and discuss issues such as parameter estimation, model iden- tification, realization generation and forecasting. A two-factor GARMA model is then applied to the Mauna Loa atmospheric CO2 data. It is shown that this model provides a reasonable fit to the CO2 data and produces excellent forecasts.