Fractional Differencing Modeling in Hydrology¹

Fractional differencing is a tool for modeling time series which have long‐term dependence; i.e., series in which the correlation between distant observations, though small, is not negligible. Fractionally differenced ARIMA models are formed by permitting the differencing parameter d in the familiar Box‐Jenkins ARIMA(p, d, q) models to take nonintegral values; they permit the simultaneous modeling of the long‐term and short‐term behavior of an observed time series. This paper discusses the usefulness of fractional differencing to time‐series modeling, with emphasis on hydrologic applications. A methodology for fitting fractionally differenced ARIMA models is described, and examples are presented.