Information Technology Reference
In-Depth Information
this is the case for systems with cointegrated variables. It turns out that fitting
VA R (
p
) model after differencing the original
cointegrated
process produces inade-
quate results.
Suppose that sampled values
y
it
of
K
different variables of interest
Y
i
(
t
)
are com-
y
Kt
)
. In addition, suppose that
bined into the
K
-dimensional vectors
y
t
=(
y
1
t
,...,
the variables are in a
long-run
equilibrium relation:
cY
(
t
)
:
=
c
1
·
Y
1
(
t
)+
...
+
c
K
·
Y
K
(
t
)=
0
,
(18.2)
c
K
)
is a
K
-dimensional real vector. During any given time inter-
val, the relation (18.2) may not necessarily be satisfied precisely by the sample
y
t
,
instead we may have
where
c
=(
c
1
,...,
cy
t
:
=
c
1
·
y
1
t
+
...
+
c
K
·
y
Kt
=
ε
t
,
(18.3)
where
ε
t
is a stochastic process that denotes the deviation from the equilibrium
relation at time
t
. If our assumption about the long-run equilibrium among individual
variables
Y
i
(
)
=
,...,
t
,
i
1
K
is valid then it is reasonable to expect that the variables
(
)
Y
i
t
is stable. On the other hand, this
does not contradict the possibility that the variables deviate substantially as a group.
Therefore, it is possible that although each individual component
Y
i
t
move together, i.e., the stochastic process
ε
(
t
)
is integrated,
there is a linear combination of
Y
i
(
K
, which is stationary. Integrated
processes with such property are called
cointegrated
.
Without loss of generality, we assume that all individual 1D components
Y
i
(
t
)
,
i
=
1
,...,
t
)
(
i
=
1
,...,
K
) are either
I
(
1
)
or
I
(
0
)
processes. Then the combined
K
-dimensional
VA R (
p
) process
Y
(
t
)=
ν
+
A
1
Y
(
t
−
1
)+
...
+
A
p
Y
(
t
−
p
)+
ε
t
(18.4)
is said to be
cointegrated of rank r
, when the correspondent matrix
Π
=
I
K
−
A
1
−...−
A
p
(18.5)
has rank
r
.
Since some 1D components of the cointegrated VAR(
p
) process are integrated
processes, one may be interested in testing the presence of a unit root in the uni-
variate series. In the following section, we present a commonly used unit root test,
which was derived by Dickey and Fuller [7].
18.2.1 Augmented Dickey-Fuller Test for Testing the Null
Hypothesis of a Unit Root
The augmented Dickey-Fuller (or ADF) test is a widely used statistical test for
detecting the existence of a unit root of the reverse characteristic polynomial in a
