452 Fractional Integration and Cointegration
version of PPP being characterized by a slope equal to 1, while evidence of frac-
tional cointegration implies a weaker (relative) version of PPP. Using the GPH test,
Choudhry provided evidence of relative PPP for Russia and Slovenia (with unit root
observables and estimates of the integration order of the cointegrating error close
to 0 and 0.5, respectively), but failed to find evidence for absolute PPP.
In a different setting, Baillie and Bollerslev (1994a) argued whether seven spot
exchange rates appear to be tied together in the long run or not, taking into account
that there does not seem to be much discussion in the literature about the unit root
character of these series, which makes much more fragile the idea that they are
cointegrated (see, for example, Sephton and Larsen, 1991; Diebold, Gardeazabal
and Yilmaz, 1994). Baillie and Bollerslev’s (1994a) explanation was that unit root
tests, which serve traditionally to detect the presence of unit roots, have very low
power against fractional alternatives, so that a situation of fractional cointegration
with long memory cointegrating error could be hidden. In fact, their estimate of the
memory of the cointegrating error was 0.89, over five standard errors away from 1,
thus providing evidence of fractional cointegration withδ−γ<0.5. Similarly, Pan
and Liu (1999), using the same nominal exchange rate data as Baillie and Bollerslev
(1994a), analyzed the presence of cointegration in different sub-samples by means
of the GPH test. Interestingly, they only found evidence of fractional cointegration
(withδ−γ<0.5)for the 1980–84 sample, whereas standard cointegration was
found for the most recent period 1985–92, supporting the conjecture that the
fractional cointegration feature could vary across different time spans.^12
Another important topic in the literature of exchange rates is the analysis of
the forward premium,ft−st, wherestandftare logs of the spot exchange rate
and of the forward rate respectively. Here, noting the “overwhelming” evidence of
unit roots in spot exchange rates, the differenceft−stcould be considered as a
cointegrating error with cointegrating vector(1,− 1 )′. Baillie and Bollerslev (1994b)
claimed that unit root tests generally reject that the forward premium isI( 0 ), which
is paradoxical as, given that the forward premium is associated with risk, it seems
hard to see any theoretical reason for anI( 1 )risk premium. The purpose of their
paper was to show that the forward premium is indeed mean-reverting, the esti-
mates of the memory of the forward premium for Canada, Germany and UK (with
respect to US) being 0.45, 0.77 and 0.55 respectively, suggesting stronger evidence
in favor of weak cointegration relations.
Choudhry (1999b) analyzed, by means of the GPH test, nine forward premi-
ums (with respect to US), showing evidence of (weak) cointegration for three of
them (Canada, Hong Kong and Italy). He also tested the unbiased forward rate
hypothesis (in short, that the forward exchange rate is an unbiased predictor of
the corresponding future spot rate), which was examined by analyzing the exis-
tence of fractional cointegration in a regression ofst+konft(although two other
alternative specifications were also considered), and testing for a unit slope, which
ensures that the forward rate is an unbiased predictor of the future spot rate. Evi-
dence of cointegration was found but not of the unbiasedness hypothesis (with the
exception of South Africa).