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subject to the constraint
rank
(
Π
)=
rank
(
I
K
−
A
1
−...−
A
p
)=
r
.
(18.16)
Note that
ε
t
is assumed to be a Gaussian white noise with a nonsingular covariance
matrix
y
0
are supposed to be fixed.
In order to impose the cointegration constraint, the model (18.15) is reparame-
terized in the following fashion [17]:
Σ
ε
. Furthermore, the initial conditions
y
−
p
+
1
,...,
Δ
y
t
=
D
1
Δ
y
t
−
1
+
...
+
D
p
−
1
Δ
y
t
−
p
+
1
+
Π
y
t
−
p
+
ε
t
,
t
=
1
,
2
,...,
(18.17)
where
Δ
y
t
=
y
t
−
y
t
−
1
, and matrix
Π
can be represented as a product
Π
=
HC
of
matrices of rank
r
, i.e.,
H
is
(
K
×
r
)
and
C
is
(
r
×
K
)
.
Consider
Δ
Y
:
=[
Δ
y
1
,...,
Δ
y
T
]
,
⎡
⎣
⎤
⎦
,
Δ
y
t
.
Δ
X
t
:
=
(18.18)
Δ
y
t
−
p
+
2
Δ
X
:
=[
Δ
X
0
,...,
Δ
X
T
−
1
]
,
D
:
=[
D
1
,...,
D
p
−
1
]
,
Y
−
p
:
=[
y
1
−
p
,...,
y
T
−
p
]
.
Then the log-likelihood function for a sample of size
T
can be written as
KT
2
T
2
ln
l
=
−
ln
[
2
π
]
−
ln
[
det
Σ
ε
]
2
trace
(
Δ
HCY
−
p
)
.
1
HCY
−
p
)
Σ
−
1
−
Y
−
D
Δ
X
+
ε
(
Δ
Y
−
D
Δ
X
+
(18.19)
The proof of the following theorem on the maximum likelihood estimators of a
cointegrated VAR process can be found in [17] (Proposition 11.1).
Theorem 18.1.
(reproduced from [17])
Define
X
(
Δ
X
)
−
1
M
:
=
I
−
Δ
X
Δ
Δ
X
,
R
0
:
=
Δ
YM
,
R
1
:
=
Y
−
p
M
,
1
T
R
i
R
j
S
ij
:
=
,
i
=
0
,
1
.
Let G be the lower triangular matrix with positive diagonal such that GS
11
G
=
I
K
.
λ
1
≥ ...≥
λ
K
to be the eigenvalues of GS
10
S
−
1
00
S
01
G
,
Denote
and
v
1
,...,
v
2
be the corresponding orthonormal eigenvectors.