9 - Fractional unit roots and fractional cointegration

Summary

In the preceding chapters we assumed that the time series is I(d) with d being 0 or 1 or some greater integer. The model with d = 1 corresponds to a model with persistence of shocks. However, persistence or long memory can also be modeled with d > 0 but < 1. The introduction of autoregressive fractionally integrated moving average (ARFIMA) model by Granger and Joyeux (1980) and Hosking (1981) allows us to replace the discrete choice of unit root versus no unit root with a continuous parameter estimate of the long memory component of time series. Likewise, if two series y_t and x_t are I(d), they are said to cointegrated if y_t + βx_t is I(b) with b < d, and we allow integer values for b and d. Typically d = 1 and b = 0. In the case of fractional cointegration, suppose that y_t and x_t are both I(d) and y_t + βx_t is I(b) with b < d. If b = 0, then the long memory components in y_t and x_t are common and we say that y_t and x_t are fractionally cointegrated. If b > 0, then y_t + βx_t has a long memory component left in it.

In this chapter we shall discuss the issues related to these problems. The paper by Granger (1980) gives a motivation behind the I(d) process. Granger shows that the sum of a large number of stationary AR(1) processes with random parameters can result in a process with long memory. Thus it makes sense to consider I(d) processes while analyzing aggregate date.