Information Technology Reference
In-Depth Information
where
∞
i
=
1
d
i
X
i
Γ
=
,
(18.10)
i
=
1
√
2
d
i
X
i
,
∞
W
=
(18.11)
i
+
1
2
(
−
1
)
d
i
=
)
,
(18.12)
π
(
2
i
−
1
random variables
X
i
,
i
, are independent and identically distributed ac-
cording to the normal distribution with zero mean and variance
=
1
,
2
,...
2
, and
σ
⇒
denotes
convergence in distribution.
In [7], Dickey and Fuller considered the following “Studentized” statistic based
on the likelihood ratio test of the hypothesis
H
0
:
c
=
1:
T
t
=
2
y
t
−
1
2
τ
=
c
−
1
,
(18.13)
S
where
T
t
=
2
(
y
t
−
cy
t
−
1
)
2
1
S
2
=
,
(18.14)
T
−
2
and
c
is computed from (18.7).
Tables of the critical values for the asymptotic distributions of the ADF test statis-
tic
T
(
c
−
1
)
and the statistic
τ
can be found in Fuller [10].
18.2.2 Estimation of Cointegrated VAR(
p
) Processes
Several methods can be employed to estimate the parameters of a cointegrated
VA R (
p
) model, including modifications of the approaches used for estimation of
the standard VAR(
p
) processes.
In this section we present the maximum likelihood approach to estimating a
Gaussian
cointegrated
VAR(
p
)
process.
Suppose
y
t
is
a
realization
of
a
K
-dimensional VAR(
p
) process with cointegration rank
r
, such that 0
<
r
<
K
.
Without loss of generality, we assume that
Y
(
t
)
has zero mean, i.e., the intercept
ν
=
0 in (18.4).
Given a realization
y
t
,
t
=
1
,
2
,...
,of
Y
(
t
)
, one seeks to determine the coefficients
of the following model:
y
t
=
A
1
y
t
−
1
+
...
+
A
p
y
t
+
p
+
ε
t
,
t
=
1
,
2
,...,
(18.15)
