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x
t
y
t
ε
t
ε
t
c
x
t
−
1
y
t
−
1
−
c
Δ
=
−
,
(18.33)
−
dd
then
c
−
c
Π
=
,
−
dd
and so, rank
1. This shows that the cointegrating relationship between the
drunk lady and her puppy has the cointegration rank 1.
Note that rank
(
Π
)=
0. In such case, (18.31) becomes
simply a system of Equations (18.29) and (18.30), which models
two
independent
random walks driven by independent white noise process
(
Π
)=
0, if and only if
c
=
d
=
. On the other hand, when
at least one of the coefficients
c
and
d
is nonzero, then by multiplying system (18.33)
by a vector
ε
]
,wehave
[
d
,
c
d
Δ
x
t
+
c
Δ
y
t
=
d
ε
t
+
c
ε
t
,
t
=
1
,
2
,...,
(18.34)
which means that the model is driven by a
single
common stochastic trend
d
ε
t
.
Although the example described by Murray is clearly a bivariate cointegrated
VAR(1), it can be extended to an illustration of the multivariate cointegrated pro-
cess. Consider, for example, a herd of sheep guarded by two dogs, where the sheep
wonder aimlessly in the field, while the dogs run around and bring the sheep that
have strayed too far back into the flock. Say, for example, a faster dog guards sheep
from the east, south, and west, whereas a slower dog - from the north, then the coin-
tegrated process appears to have the cointegration rank of 2. Clearly, two dogs are
able to keep a flock of sheep closer together, than a single dog can. In other words,
the higher cointegration rank the more restrictive it is.
In fact, let us consider a
K
-dimensional cointegrated vector autoregressive pro-
cess, and let
r
denote the cointegration rank of the process. Similarly to the bivariate
example above, we can see that when the rank is zero (
r
ε
t
+
c
=
0), the univariate com-
ponents of the process are independent, and the model is driven by
K
independent
white noise processes (i.e., there is no cointegration). In the case of
r
=
1, we can
decompose the multivariate process onto
K
2 independent components, and two
dependent components that form a common stochastic trend. Hence, in the case
r
−
1 independent stochas-
tic processes. By induction, we can show that for a cointegrated VAR process with
the cointegration rank
r
,0
=
1, the cointegrated model is driven by
(
K
−
2
)+
1
=
K
−
<
r
<
K
−
1, the VAR model is generated by
K
−
r
inde-
pendent stochastic trends.
Therefore, the smaller is the cointegration rank
r
, the larger is the number
K
r
of
the underlying independent stochastic trends, and so (the larger) is the vector space
in which our cointegrated model can travel. And the other way around, increasing
the cointegration rank of the model shrinks the underlying domain of the process,
i.e., makes it bounded to a smaller hyperplane. For
r
−
K
,theVAR(
p
) is a stable
process, which clearly has the most constrained domain. For
r
=
=
0, the VAR process
is not cointegrated and unrestricted.
