Fractionally cointegrated systems
As discussed earlier in this chapter, one of the main characteristics of the existence of unit roots in the Wold representation of a time series is that they have "long memory," in the sense that shocks have permanent effects on the levels of the series so that the variance of the levels of the series explodes. In general, it is known that if the differencing filter (1 – L)d, d being now a real number, is needed to achieve stationarity, then the coefficient of e— in the Wold representation of the I(d) process has a leading term jd-1 (e. g. the coefficient in an I(1) process is unity, since d = 1) and the process is said to be fractionally integrated of order d. In this case, the variance of the series in levels will explode at the rate T2d-1 (e. g. at the rate T when d = 1) and then all that is needed to have this kind of long memory is a degree of differencing d > 1/2.
Consequently, it is clear that a wide range of dynamic behavior is ruled out a priori if d is restricted to integer values and that a much broader range of cointegration possibilities are entailed when fractional cases are considered. For example, we could have a pair of series which are I(d1), d1 > 1/2, which cointegrate to obtain an I(d0) linear combination such that 0 < d0 < 1/2. A further complication arises in this case if the various integration orders are not assumed to be known and need to be estimated for which frequency domain regression methods are normally used. Extensions of least squares and maximum likelihood methods of estimation and testing for cointegration within this more general framework can be found in Jeganathan (1996), Marmol (1998) and Robinson and Marinucci (1998).