K . Neusser Economics Letters 67 2000 273 –281
277
converges to zero. Therefore d
d d
d 2
d
D X 5 Z 1 F 2 I
Z 1
F 2 F 1
I Z
1 ? ? ? 1 F 2 I Z
S S D D S S D S D D
s d
t t
n t 21
n t 22
n t 2d
1 1
2
` j
d
1
O
F 2 I F Z s
d
n t 2d 2j
j 51 d
is a stationary representation of hD X j. h
t
2. Representations and basic properties of the group inverse
In most of the following I restrict myself to the case where hX j | I0, thus assuming:
t
Assumption 2. Integration of order one
2
rank P 5 rankP or IndP 5 1
This implies r 5 p and N 50 . In the literature Assumption 2 is not always made explicit and its
n2r
nature often not discussed. The best treatment is given in Johansen 1995. The main implications of Assumptions 1 and 2 are summarized in the following two lemmas and one corollary. The proof of
these assertions can be found in Campbell and Meyer 1979.
Lemma 1. Assumptions 1 and 2 imply
[
1. P
exists.
[ 22
2. The r 3 r matrix b9a is nonsingular and P 5 a b9a
b9. 3.
F has a full set of eigenvectors corresponding to the eigenvalue 1. 4. The canonical form representations specialize to
:
n 2r 21
P 5 P P
,
S D
I 2 J
r
I
n 2r n 2r
21 [
21
F 5 P P
and P 5 P
P .
S D S
D
21
J I 2 J
s d
r
5. R P and NP are complementary subspaces.
n 2r [
[ 21
6. E 5 P P 5 PP 5 P
P is a projection on R
P along NP .
S D
I
r
I
t 21
[ n 2r
21
Lemma 2.
F is semiconvergent: lim F 5 C 5 P
P 5 I 2
P P 5 I 2 a b9a b9
S D
s d
t →
` n
n
278 K
. Neusser Economics Letters 67 2000 273 –281
Corollary.
1. rank C 5 n 2 r 2. b9C 5 0 and C a 5 0
3. C F 5 F C 5 C
[ [
4. C and E 5 P P 5 PP 5 I 2 C are idempotent matrices
n
Under this circumstance, there exists a Beveridge–Nelson decomposition for hX j:
t
Theorem 3. Beveridge –Nelson decomposition: Under Assumptions 1 and 2 and a suitable distribution of X ,
hX j is the sum of a random walk hT j and a stationary process hS j:
t t
t t
[
X 5 T 1 S where T 5 CX 5 C o
Z 1 CX and S 5 I 2 C X 5 P PX 5 F 2 CS
1
t t
t t
t t 51
t t
n t
t t 21
[
P PZ
t
Proof. Define T 5 CX then T 5 C
FX 1 CZ 5 CX
1 CZ 5 T 1 CZ . Thus
hT j is a random
t t
t t 21
t t 21
t t 21
t t
walk and we immediately get the above representation. Similarly define S 5 X 2 T 5 I 2 C X 5
t t
t n
t [
P PX . Because I 2 C is idempotent and commutes with F, S 5 I 2 CI 2 CFX 1 I 2
t n
t n
n t 21
n
C Z 5 F 2 CF I 2 CX
1 I 2 CZ 5 F 2 CS
1 I 2 CZ . This defines a stationary
t n
t 21 n
t t 21
n t
21 n 2r
stochastic process because rJ , 1 implies that F 2 C 5 P
P has all its eigenvalues
s d
S D
J smaller than one.
h The Beveridge–Nelson decomposition demonstrates the effect of the cointegration vectors in b on
X : hb9X j is stationary as the random walk component is knocked out because b9C 5 0. Note that b9
t t
also eliminates the dependence on the initial value CX . Because rank C 5 n 2 r, hT j is made up of
t
n 2 r independent random walks, sometimes called common trends, whereas the dimension of hS j is
t
r. In case of an I2-process, Ind
P 52 and F can be written according to Theorem 1 as I
2 N I
N
21 21
21 n 2p
n 2p
F 5 P P
5 P P
2 P P
5 F 2 F
S D
S D
S D
C N
J J
J
t t
where N is idempotent of index 2. Therefore F 5 F 2 tF . For a suitable choice of an initial
C N
distribution, a representation of hX j can be obtained by repeated substitutions:
t t 21
t 21 t
t
X 5 F X 2 tF X 1
O
F Z 2
F
O
tZ
t C
N C
t 2 t
N t 2
t t 50
t 51 t 21 t 21
5 T 1 S 2 t F X 2 F
O O
Z
t t
N N
t 2 t
t 5s s 51
where the random walk T and the stationary component S are defined as in Theorem 3 but replacing
t t
F by F . The third expression is just a linear time trend whereas the last expression represents an
C
integrated random walk which is I2.
K . Neusser Economics Letters 67 2000 273 –281
279
3. Some algebraic properties
