Luis A. Gil-Alana and Javier Hualde 45110.3.3.1 Applications to exchange rates
The initial applications of the concept of fractional cointegration were devoted to
analyzing PPP. This refers to the tendency for nominal exchange rates and prices
to adjust in such a way that the real exchange rate reverts (perhaps slowly) to its
parity value. Thus the (log) real exchange rate could be viewed as the cointegrat-
ing error in a linear combination of (log) nominal exchange rates and (log) prices
with cointegrating vector(1,− 1 )′. Here the literature mainly provided evidence of
weak fractional cointegration(δ−γ<0.5), with (approximately) unit root observ-
ables with non-stationary but mean-reverting cointegrating errors, as presented by
Diebold, Husted and Rush (1991). Although the authors approximated the log of
the real exchange rate in a particular way, and not as the difference of the logs of
the nominal exchange rate and prices, their empirical analysis, taking into account
that the (log) nominal exchange rate isI(1) (see, for example, Baillie and Boller-
slev, 1994a, 1994b), reported some cases where the estimated memory of the real
exchange rates (for example, France–Germany, Germany–UK) were non-stationary
mean-reverting, while for others there was evidence of stationary long memory. In
a similar framework, Cheung and Lai (1993) proposed checking the PPP hypoth-
esis via a regression of a foreign price index, converted to domestic (US) currency
units, on a domestic price index, with the errors of this relation capturing devi-
ations from PPP. While they provided evidence of the unit root character of the
observables, they stated that PPP will be characterized by stationarity, or at least
mean reversion, in the cointegrating error. They estimated the degree of memory
of the cointegrating error for different countries and bandwidths and, in 11 out of
15 cases, these estimates were suggestive ofδ−γ<0.5 instead ofδ−γ>0.5.
In a similar setting to Diebold, Husted and Rush (1991), Crato and Rothman
(1994b) provided estimates of the (log) real bilateral sterling exchange rates for
different countries. Of the nine countries analyzed, for only two of them (Nether-
lands and Italy) was there very clear evidence thatδ−γ<0.5, whereas for another
two (Canada and Sweden) there were conflicting results.
Chou and Shih (1997) investigated long-run PPP in the relationship of four Asian
countries (Hong Kong, Singapore, South Korea and Taiwan) with the US dollar in an
unrestricted trivariate model (involving the logs of the nominal exchange rate and
domestic and foreign price levels), allowing for a linear trend and using quarterly
data from 1965 to 1992. Using the Geweke and Porter-Hudak (1983) (GPH) test
for fractional cointegration (that is, estimating the memory of the cointegrating
error by the GPH estimate applied to the cointegrating regression residuals and
testing whether this memory is significantly smaller than the integration order of
the observables, assumed to be one in this case), they found some evidence of
fractional cointegration, withδ−γ>0.5 for the South Korea–US relationship.
Choudhry (1999a) investigated PPP between the US and four high-inflation East-
ern European countries (Poland, Romania, Russia and Slovenia), using monthly
data from 1991 to 1997. The author estimated a regression model of the log of the
nominal exchange rate on the log ratio of domestic to foreign prices, the absolute