Fractional Integration

Abstract

This chapter is the first of two that consider the case of fractional values of the integration parameter. That is suppose a process generates a time series that is integrated of order d, I(d), then the techniques in UR, Vol. 1, were wholly concerned with the case where d is an integer, the minimum number of differences necessary, when applied to the original series, to produce a series that is stationary. What happens if we relax the assumption that d is an integer? There has been much recent research on this topic, so that the approach in these two chapters must necessarily be selective. Like so many developments in the area of time series analysis one again finds influential original contributions from Granger; two of note in this case are Granger (1980) on the aggregation of ‘micro’ time series into an aggregate time series with fractional I(d) properties, and Granger and Joyeux (1980) on long-memory time series.