Luis A. Gil-Alana and Javier Hualde 443To be fair, the idea of equilibrium betweenI( 1 )processes was hinted at long
before in the statistical literature. In the AR model:
yt=ρyt− 1 +εt, t>0; yt=0, t≤0,εtbeing a sequence of independent normally distributed random variables with
mean 0 and finite variance, Dickey and Fuller (1979) studied the properties of the
regression estimate ofρ,ρˆ, under the assumption thatρ=1. In fact, this represents
a situation of cointegration between theI( 1 )processesytandyt− 1 , as the linear
combinationyt−yt− 1 isI( 0 ). This is a particular case of what Park (1992) called
“singular cointegration,” which is characterized by cointegrating errors being linear
combinations of innovations which generate the regressors.
Engle and Granger (1987) introduced another important concept. If the multi-
variateI(d)processzthasp>2 components, there may be several linearly inde-
pendent cointegrating vectors, representing the case where several equilibrium
relations drive the joint movement of the variables inzt. It is easy to see that the
maximum number of linearly independent cointegrating vectors isr≤p−1, the
valuerdefining the “cointegrating rank” ofzt.
Even considering only integer orders of integration, a more general definition
of cointegration than the one given by Engle and Granger (1987) is possible, one
that allows for a multivariate process with components having different orders
of integration. Denotingd 1 anddpto be the largest and smallest of these orders,
respectively, Johansen (1996) proposed that any vectorztsuch thatα′zt∼I(dw),
withdw < d 1 , is a cointegrating vector. Flôres and Szafarz (1996) narrowed
Johansen’s definition, proposing instead that the vector series is cointegrated if
there is a non-trivial linear combination of its components (with at least a non-zero
scalar multiplyingd 1 ) which is integrated of orderdw<d 1. Alternatively, Robinson
and Marinucci (2003) defineztto be cointegrated if there exists a vectorα =0 such
thatα′zt∼I(dw), withdw<dp, which is a much stronger requirement. Robinson
and Yajima (2002) offered an alternative (although rather more involved) definition
and a comparison of the different definitions that have appeared in the literature.
Once fractional integration is defined, the concept of fractional cointegration
appears as a natural extension of traditional cointegration. In fact, the standard
definition of cointegration by Engle and Granger (1987) does not necessarily refer
to integer orders of integration. In the simple bivariate case, two seriesyt,xt, shar-
ing the same order of integration, sayδ, are cointegrated if there exists a vector
α =0 such thatα′zt∼I(γ ), withγ<δ, withzt=(yt,xt)′. This prompts considera-
tion of an extension of Phillips’ (1991a) triangular system, which, for a very simple
bivariate case, is:
yt=νxt+u 1 t(−γ), (10.5)
xt=u 2 t(−δ), (10.6)fort=0,±1,..., where, for any vector or scalar sequencewtand anyc, we intro-
duce the notationwt(c)=cw#t.ut=(u 1 t,u 2 t)′is a bivariate zero mean covariance
stationaryI( 0 )unobservable process andν =0,γ<δ. The truncation in (10.6)