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Then the log-likelihood function in
(18.19)
is maximized for
v
r
]
G
C
:
=[
v
1
,...,
,
YMY
−
p
C
CY
−
p
MY
−
p
C
−
1
H
:
=
−
Δ
S
01
C
CS
11
C
−
1
=
−
,
X
Δ
X
−
1
D
:
=(
Δ
Y
+
HCY
−
p
)
Δ
X
Δ
,
1
T
(
Δ
HCY
−
p
)
.
Σ
:
=
Y
−
D
Δ
X
+
HCY
−
p
)(
Δ
Y
−
D
Δ
X
+
The maximum is
ln
r
i
=
1
ln
(
1
−
λ
i
)
KT
2
T
2
KT
2
.
max
[
ln
l
]=
−
ln
[
2
π
]
−
[
det
S
00
]+
−
(18.20)
18.2.3 Testing for the Rank of Cointegration
Based on Theorem 18.1, one can easily derive the likelihood ratio statistic for testing
a candidate value
r
0
of the cointegration rank
r
of a VAR(
p
) process against a larger
cointegration rank
r
1
.
Given a VAR(
p
) process
y
defined by (18.4), suppose we wish to test a hy-
pothesis
H
0
against an alternative
H
1
, where
(
t
)
H
0
:
r
=
r
0
against
H
1
:
r
0
<
r
≤
r
1
.
(18.21)
Under assumption that the noise
t
is a Gaussian process, the maximum of the
likelihood function for a cointegrated VAR(
p
) model with cointegration rank
r
is
computed in Theorem 18.1. From that result, the value of the LR statistic for test-
ing (18.21) can be determined in the following manner:
ε
λ
LR
(
r
0
,
r
1
)=
2
[
ln
L
max
(
r
1
)
−
ln
L
max
(
r
0
)]
(18.22)
T
r
1
i
=
1
ln
(
1
−
λ
i
)+
r
0
i
=
1
ln
(
1
−
λ
i
)
=
−
r
1
∑
i
=
r
0
+
1
=
−
T
ln
(
1
−
λ
i
)
,
where
L
max
(
1, denotes the maximum of the Gaussian likelihood function
for cointegration rank
r
i
. The advantage of this test is in the simplicity with which
the LR statistic can be computed. On the other hand, the asymptotic distribution of
the LR statistic (18.22) is nonstandard. Specifically, the LR statistic is not asymp-
totically distributed according to
r
i
)
,
i
=
0
,
2
-distribution. Nevertheless, the asymptotic dis-
tribution of the cointegration rank test statistic
χ
λ
LR
depends only on two factors:
