442 Fractional Integration and Cointegration
that a fractional model could be more appropriate than standard ARIMAs. Advan-
tages of seasonally fractionally integrated models for forecasting are illustrated in
Ray (1993) and Sutcliffe (1994), and other empirical applications using quarterly
seasonal models can be found in Gil-Alana and Robinson (2001) and Gil-Alana
(2005b). Monthly data in the context of seasonal fractional integration have been
examined in Gil-Alana (1999) and Ooms and Franses (2001).
Applications using the cyclical model based on the Gegenbauer process described
by equation (10.3) can be found, for example, in Arteche and Robinson (2000),
Bierens (2001) and Gil-Alana (2001), and empirical work based on multiple cyclical
structures (k-factor Gegenbauer processes) can be found in Ferrara and Guegan
(2001), Sadek and Khotanzad (2004) and Gil-Alana (2007).
10.3 Fractional cointegration
The concept of fractional integration leads naturally to an extension of the standard
notion of cointegration (which involves series with integer orders of integration) to
the fractional case, where equilibrium relations among fractional processes could
be captured. In the present section, we introduce this concept, give an overview
of the different estimation methods proposed so far in the literature, and give
evidence of the empirical relevance of this idea.
10.3.1 The concept and modelization of fractional cointegration
Engle and Granger (1987) suggested that, if two processesxtandytare bothI(d),
then it is generally true that, for a certain scalara =0, a linear combination
wt=yt−axtwill also beI(d), although it is possible thatwtbeI(d−b)with
b>0. This idea characterizes the concept of cointegration, which they adapted
from Granger (1981) and Granger and Weiss (1983). They provided the following
definition for multivariate series. Given two real numbersd,b, the components of
the vectorztare said to be cointegrated of orderd,b, denotedzt∼CI(d,b), if:
(i) all the components ofztareI(d),
(ii) there exists a vectorα =0 such thatwt=α′zt∼I(d−b),b>0.
Here,αandwtare called the cointegrating vector and error respectively. Engle
and Granger (1987) offered some intuition behind this crucial concept in modern
time series econometrics, suggesting the existence of forces which tend to keep
series not too far apart. Given a vector of economic variableszt, and a certain
vectorα =0, economic theory would say that the variables are in equilibrium
ifα′zt =0. This is a very tight notion of equilibrium, and it is a very narrow
view that this equality could hold for every time periodt. Alternatively, we might
think of an equilibrium error aswt=α′zt, which accommodates deviations from
equilibrium. If, for example, in Engle and Granger’s (1987) definitiond=b=1,
what characterizes cointegration as a “long-run equilibrium” relationship is that a
linear combination ofI( 1 )processes isI( 0 ), so that the series inztcannot drift too
far apart.