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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.
 
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