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Ch. 23 Cointegration
1 Introduction
An important property of I(1) variables is that there can be binations
of theses variables that are I(0). If this is so then these variables are said to be
cointegrated. Suppose that we consider two variables Yt and Xt that are I(1).
(For example, Yt = Yt−1 + ζt and Xt = Xt−1 + ηt.) Then, Yt and Xt are said to
be cointegrated if there exists a β such that Yt −βXt is I(0). What this mean is
that the regression equation
Yt = βXt + ut
make sense because Yt and Xt do not drift too far apart from each other over
time. Thus, there is na long-run equilibrium relationship between them. If Yt
and Xt are not cointegrated, that is, Yt −βXt = ut is also I(1), then Yt and Xt
would drift apart from each other over time. In this case the relationship between
Yt and Xt that we obtain by regressing Yt and Xt would be spurious.
Let us continue the cointegration with the spurious regression setup in which
Xt and Yt are independent random walks, consider what happens if we take a
nontrivial bination of Xt and Yt:
a1Yt + a2Xt = a1Yt−1 + a2Xt−1 + a1ζt + a2ηt,
where a1 and a2 are not both zero. We can write this as
Zt = Zt−1 + vt,
where Zt = a1Yt + a2Xt and vt = a1ζt + a2ηt. Thus, Zt is again a random walk
process, as vt is . with mean zero and finite variance, given that ζt and ηt
each are . with mean zero and finite variance. No matter what coefficients a1
and a2 we choose, the resulting bination is again a random walk, hence
an integrated or unit root process.
Now consider what happens when Xt is a random walk as before, but Yt is
instead generated according to Yt = βXt + ut, with ut again .. By itself, Yt is
an integrated process, because
Yt − Yt−1 = (Xt − Xt−1)β+ ut − ut−1,
1
so that
Yt = Yt−1 + βηt + ut − ut−1
= Yt−1 + εt,
where εt = βηt + ut − ut−1 is readily verified to be I(0) process.
Despite the fact that both Xt and Yt are integrated processes, the situation is
very different from that co