Fractional Differencing and ARFIMA Process with White Noise

This chapter marks a pivotal transition from integer differencing to fractional differencing, introducing the autoregressive fractionally integrated moving average (ARFIMA) model that characterizes long memory time series. The chapter traces the historical development of long memory processes from the foundational work of Granger and Joyeux (1980) and Hosking (1981), who extended traditional ARIMA models by allowing the differencing parameter d to take fractional values. The fractional difference operator (1-B)^d is defined through its binomial expansion for values of d in the interval (0, 0.5), where the series exhibits both stationarity and long memory. The chapter contrasts the hyperbolic decay of autocorrelations in ARFIMA models with the exponential decay characteristic of short memory ARMA processes, and discusses the unbounded spectral density peak at the origin as a frequency-domain signature of long memory. Practical estimation using R’s arfima package is demonstrated through the analysis of varve data (glacial sediment measurements). The generalized fractional difference operator is introduced, extending to second-order models and laying the groundwork for Gegenbauer processes. The chapter establishes the theoretical and practical foundations for understanding how fractional differencing captures persistent dependencies that integer differencing destroys.