225 ON THE AUTOREGRESSIVE FRACTIONAL UNIT INTEGRATED MOVING AVERAGE (ARFUIMA) PROCESS Olanrewaju I. Shittu and OlaOluwa S. Yaya Department of Statistics, University of Ibadan, Nigeria ABSTRACT This paper presents a nonstationary fractional unit integrated moving average process that can model time series data that are not stationary after the first difference. This is autoregressive fractionally unit integrated moving average, ( ) , , ARFUIMA p d q process with 0.5 2.5 d < < . The classical unit differencing of Box-Jenkins is combined with the semi-parametric approach to estimate the fractional difference parameter. The model when applied on quarterly Nigerian gross domestic products (GDP) series indicates that ARFUIMA model is better than the corresponding autoregressive integrated moving average (ARIMA) model when compared based on diagnostic tests and forecast. Keywords: Fractional unit integration, ARFIMA, semi-parametric estimation. INTRODUCTION It has been observed that several economic and financial data have been modelled on the assumption that differencing parameter is usually an integer. Although models obtained from integer differencing have been found to be fairly adequate and reliable, however, better models with higher forecast performance can be obtained if appropriate fractional difference parameter is used. When fractional differencing parameter is non-zero, non-stationarity is suspected. This implies there is strong dependence between distant observations. A number of studies have been carried out on this subject and these include studies on real national product (Diebold et al., 1989); consumer and wholesale price (Geweke and Porter-Hudak, 1982) and stock market prices (Lo, 1991). The motivation for this paper and presentation is derived from Fatoki, et al., (2010) who investigated the annual GDP from 1980 to 2007. The series was found to be integrated of order 2 ( ) 2 I ⎡ ⎤ ⎣ ⎦ and autoregressive integrated moving average, ( ) 1, 2,1 ARIMA was fitted using the usual model identification and order determination tools. We are of the opinion that further study could reveal a possibility of the series being integrated of a fractional order and hence could lead to better forecast performance. It is observed that classical ADF unit root test may not give conclusive remark on fractional difference but a test designed for this purpose such as Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test could. Detail about KPSS test can be found in Kwiatkowski, Phillips, Schmidt and Shin (1992). Following Shittu and Yaya (2009), an improved model in terms of parameter estimates and forecasts can be obtained by modelling the remaining long memory in a series after the first or second difference. This paper therefore proposes a class of Journal of Sustainable Development in Africa (Volume 13, No.5, 2011) ISSN: 1520-5509 Clarion University of Pennsylvania, Clarion, Pennsylvania