Modeling persistence in hydrological time series using fractional differencing

The class of autoregressive integrated moving average (ARIMA) time series models may be generalized by permitting the degree of differencing d to take fractional values. Models including fractional differencing are capable of representing persistent series ( d > 0) or short‐memory series ( d = 0). The class of fractionally differenced ARIMA processes provides a more flexible way than has hitherto been available of simultaneously modeling the long‐term and short‐term behavior of a time series. In this paper some fundamental properties of fractionally differenced ARIMA processes are presented. Methods of simulating these processes are described. Estimation of the parameters of fractionally differenced ARIMA models is discussed, and an approximate maximum likelihood method is proposed. The methodology is illustrated by fitting fractionally differenced models to time series of streamflows and annual temperatures.