Cointegration, Fractional Cointegration, and Exchange Rate Dynamics

Multivariate tests due to Johansen (1988, 1991) as implemented by Baillie and Bollerslev (1989a) and Diebold, Gardeazabal, and Yilmaz (1994) reveal mixed evidence on whether a group of exchange rates are cointegrated. Further analysis of the deviations from the cointegrating relationship suggests that it possesses long memory and may possibly be well described as a fractionally integrated process. Hence, the influence of shocks to the equilibrium exchange rates may only vanish at very long horizons. IN THE ARTICLE BY Baillie and Bollerslev (1989a), it is argued that seven different nominal spot and forward exchange rates all contain unit roots in their univariate time series representations. At the same time, however, the spot exchange rates appear to be tied together in the long run through a cointegration-type relationship. The latter finding of Baillie and Bollerslev has attracted particular interest and several studies such as those by Hakkio and Rush (1991) and Sephton and Larsen (1991) have already addressed this issue. Using the same data as the Baillie and Bollerslev article, Sephton and Larsen (1991) describe the evidence for the presence of cointegration as being "fragile" and note that mixed conclusions are reached by truncating the Baillie and Bollerslev sample at different points in time. Diebold, Gardeazabal, and Yilmaz (1994) henceforth Diebold et al., provide interesting evidence that application of the Johansen (1988, 1991) tests with and without an intercept will result in different inferences on the Baillie and Bollerslev data set. Furthermore, Diebold et al. carry out an ex ante forecasting experiment and find that the addition of an error correction term to the martingale model, as implied by the standard cointegration paradigm, fails to reduce the prediction mean square error when compared to a simple martingale model. This therefore leads Diebold et al. to conclude that "there exists substantial uncertainty regarding the existence of cointegration relationships among nominal dollar exchange rates." This article provides some additional evidence on the existence of such a long-run relationship among the same seven nominal spot exchange rates. After further analysis it appears that a form of cointegration does exist between the exchange rates, so that they do not drift apart in the long run. We argue that this form of cointegration is *Ballie is from Michigan State University and Bollerslev is from Northwestern University and NBER. The authors would like to thank an anonymous referee, Frank Diebold and Rene Stulz (the editor) for helpful comments.