A study of fractionally integrated time series using descriptive methods

We demonstrate the use of some descriptive methods for nonstationary time series to better understand the sample path behaviours of fractionally integrated processes for a range of different fractional orders of integration. We are particularly interested in better understanding the behaviours of I(d) series when d is an element of [1/2, 1). In fact, we will point out that there is considerable disagreement in the literature when it comes to describing such processes, and we show that descriptive methods can be useful tools for better understanding their sample path properties. We also present an empirical example to compare conclusions from some of the descriptive methods and inference from two state-of-the-art estimators for fractional orders of integration.