A <i>k</i>‐Factor GARMA Long‐memory Model

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].

Abstract

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 CO 2 data. It is shown that this model provides a reasonable fit to the CO 2 data and produces excellent forecasts.