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Lecture 26 – Cointegration:I
Introduction
Let yt = [y1t y2t]’ where y1t any y2t are I(1)
process. (yt is a 2-dimensional I(1) proc.)
In general, α1y1t + α2y2t will be I(1) for all
[α1 α2]’ ε R2 (s.t. α1 and/or α2 is nonzero).
However, there may be a nonzero α ε R2 s.t.
α1y1t + α2y2t ~ I(0) in which case we say that
yt is a cointegrated process and we call α a
cointegrating vector for yt.
A more general definition – A qdimensional I(1) process yt is CI if there
exists a nonzero α ε Rq s.t. α’yt ~ I(0). α is
called a CI vector for yt.
Examples:
1. It may be that Ct , real consumption per
capita, and It, real disposable income per capita
are each I(1) but for some α and some β ε (0,1),
Ct = α + βIt + ut,
where ut is a zero-mean I(0) process. In this
case, Ct and It are cointegrated with
cointegrating vector [1 –β].
2. Let et denote the (log) nominal exchange rate
in period t (foreign currency units per unit of
home currency), let pt denote the (log) home
country price level and let pt*denote the (log)
foreign price level. Unit root tests applied to
each of these series suggest that they are I(1)
series. Economic theory suggests that the (log)
real exchange rate
et + pt* - pt
should be I(0). So, we might expect et, pt*, and pt
to be cointegrated with cointegrating vector [1 1
-1].
Note that both of these examples assume longrun equilibrium conditions that imply
cointegration if the component variables are I(1).
Although each series individually can wander
around like an I(1), economic theory implies that
in this case there is a linear combination of these
series that cannot wander around.
In other words, if economic time series are
typically I(1) series, then the equilibrium
conditions provided by economic theory suggest
that cointegrating relationships among these
series will be common.
Other examples:
3. Money Demand: (M/P)t, it, Yt
4. P. Ireland (JME, 1979) : Barro-Gordon time
consistency model of central bank behavior
implies that if the unemployment rate is I(1)
then the inflation rate is also I(1) and the
two are cointegrated.
Some Econometric Issues:
Suppose yt and xt are I(1) and not cointegrated.
Then there is not a stable long-run relationship
between these two series (though there may be a
stable relationship between their first
differences).
If, on the other hand, yt and xt are cointegrated
then there exists a unique β0 and β1 such that
yt – β0 – β1xt = εt ~ I(0)
The OLS estimator of β from the regression of y
on 1, x is a super-consistent estimator (even if y
and x are jointly determined, i.e., even if E(xtεt)
is nonzero.
1. However, standard inference procedures
cannot be applied to the cointegrating regression
(and the proper inference procedures will
depend on whether the x’s are exogenous,…)
2. If yt and xt are I(1), the proper specification of
a VAR form for (yt , xt) will depend on whether
y and x are cointegrated. (More later)
Suppose that yt is a q-dimensional CI process.
By definition, there must be at least one nonzero
α in Rq s.t. α’yt ~ I(0). Can there be more than
one CI vector?
Note that if α is a CI vector then so is cα for any
nonzero real number c. (α1y1t + α2y2t ~ I(0)
implies that 2α1y1t + 2α2y2t ~ I(0).)
The more interesting question, how many
linearly independent CI vectors can there be for
yt?
We know that there cannot be more than q
linearly independent q-dimensional vectors so
there cannot be more than q linearly independent
CI vectors.
There cannot be q linearly independent CI
vectors. Suppose α(1),…,α(q) are linearly
independent q-dim vectors s.t. α(i)’yt ~ I(0) for
i = 1,…,q. Let A’ = [α(i)’] be the qxq matrix
whose i-th row is α(i)’. Then
A’yt = ut
where ut is a q-dimensional I(0) process. (Why?)
So,
yt = (A’)-1ut ~ I(0),
which is a contradiction.
If the q-dimensional process yt is cointegrated
then it can have as few as one linearly
independent CI vector and as many as
q-1. The number of linearly independent CI
vectors is called the CI rank of yt.
In the money demand example suppose that the
money demand function is one cointegrating
relationship among M/P, i, and Y. Suppose that
the monetary authority uses a money supply rule
in which the real money supply depends upon
real GDP. The money supply rule provides a
second CI relationship among M/P, i, and Y.
(Can there be a 3rd?)
If r is the CI rank of yt the CI vectors for an rdim subspace of Rq called the CI space.
If yt is an n-dimensional I(1) process –
1. Are the elements of y cointegrated?
2. What is the cointegrating rank of y?
3. What is the CI space?