Economics Letters 67 2000 273–281
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An algebraic interpretation of cointegration
Klaus Neusser
University of Berne , Department of Economics, Gesellschaftsstrasse 49 3012 Berne, Switzerland
Received 24 August 1999; accepted 16 December 1999
Abstract
The order of integration in a VAR equals the index of its long-run impact matrix. If this index equals one the matrix belongs to a multiplicative subgroup. Its Drazin inverse then allows to characterize the set of
cointegration vectors.
2000 Elsevier Science S.A. All rights reserved.
Keywords : Integration; Cointegration; Drazin and group inverses
JEL classification : C32
1. Introduction and basic definitions
The notion of cointegration introduced by Engle and Granger 1987 stimulated a vast amount of theoretical and empirical research Stock and Watson, 1988; Johansen, 1991, 1995; Neusser, 1991;
Phillips, 1991; Banerjee et al., 1993; Watson, 1994; and many others. Most of this research is concerned with the estimation and testing of cointegration. Relatively little notice has been given to
the algebraic properties of cointegration. This note tries to fill this gap. Although some of the results presented below can be found in the literature, they have not been presented from an algebraic point
of view.
My approach is based on the concept of the Drazin inverse of a singular matrix Drazin, 1958. Although this concept is central in the analysis of singular linear systems of difference equations
1
Campbell and Meyer, 1979, it has not yet been exploited to analyze cointegrated systems. The
Tel.: 141-31-631-4776; fax: 141-31-631-3992. E-mail address
: klaus.neusservwi.unibe.ch K. Neusser
1
Following Yoo 1986 and Engle and Yoo 1991, most papers use the Smith–McMillan form to represent cointegrated systems see for example Banerjee et al., 1993. Gregoir and Laroque 1991 and Johansen 1995 follow alternative
approaches. The analyses of d’Autume 1992 and Archontakis 1999 using the Jordan canonical form come closest to my approach.
0165-1765 00 – see front matter
2000 Elsevier Science S.A. All rights reserved. P I I : S 0 1 6 5 - 1 7 6 5 0 0 0 0 2 1 5 - 9
274 K
. Neusser Economics Letters 67 2000 273 –281
Drazin inverse provides an elegant algebraic tool which not only leads to fresh insights into results already obtained in the literature, but also generates new ones.
n 3 n
Throughout the paper I use the following notation. R is the set of all n 3 n real matrices.
rA
n 3 n
and sA denote the spectral radius and the set of all eigenvalues of A[R
. RA and NA are range and the null space of A. Moreover, I adopt the convention that A 5 I .
n
I present my argument by assuming that the n-dimensional stochastic process hX , t 5 0,1, . . . j is
t
generated by a first order stochastic difference equation: X 5
FX 1 Z
t t 21
t
where Z | WN0, S
1
t
D 5 2 I 2 F X 1 Z 5 2
PX 1 Z
s d
t n
t 21 t
t 21 t
with X given. The assumption that hZ j is white noise is made just for expositional convenience. The
t
analysis can easily be extended to higher order VARs, as in Johansen 1995, by rewriting the system in companion form and to include deterministic components.
Definition 1. Integration and cointegration: The process defined in Eq. 1 is said to be integrated of
t
order zero, denoted by hX j | I0, if hX j is stationary but S 5 o
X is nonstationary. The process
h j
h j
t t
t t 51
t d
is integrated of order d, denoted by hX j | Id, d 5 1, 2,. .., if hD X j | I0. The system is said to be
t t
cointegrated if hX j is integrated of order one and if there exists a n-vector b ± 0 such that b9X is
t t
stationary for a suitable choice of its initial distribution. Throughout the paper, I maintain the following assumption:
Assumption 1. unit root
1. rF 1 and 1 [ sF
2. l [ sF and
ulu 5 1 ⇒
l 5 1 Part a of Assumption 1 states that there is at least one unit root, i.e. one eigenvalue equal to one.
This implies that hX j is nonstationary and excludes the trivial and uninteresting case that hX j is
t t
stationary. Part b focuses the analysis on the case where one is the only eigenvalue on the unit circle and rules out, for example, seasonal unit roots.
The unit root assumption implies that P is singular with rank P 5 r , n. P therefore admits a
full-rank factorization P 5 ab9 with rank P 5 rank a 5 rank b 5 r
2 where a and b are n 3 r matrices. The r linearly independent n-vectors of b are usually called
21
cointegration vectors if the system is I1. Because P 5 aR
bR99 is also a valid full-rank factorization for any nonsingular r 3 r matrix R, the cointegration vectors are not unique; although the
space they generate is. Assumption 1 does not restrict the order of integration for
hX j: an additional assumption is
t
necessary. The statement of this assumption requires the notions of the index of a matrix and the Drazin inverse, respectively the group inverse, which are defined below.
K . Neusser Economics Letters 67 2000 273 –281
275
n 3n
Definition 2. Index of a matrix: For
P [ R , the smallest non-negative integer k such that rank
k k 11
n k
k
P 5 rank P , or equivalently, such that R is the direct sum of R
P and NP , is called the index of
P and is denoted by IndP . If P is invertible IndP 5 0.
D n 3n
Definition 3. Algebraic definition:
P is called the Drazin inverse of P [ R with Ind
P 5 k if and only if it satisfies the following conditions:
D D
D
1. P
P P 5 P
D D
2. P P 5 P
P
k 11 D
k
3. P
P 5 P As k 5 Ind
P is well defined for any matrix, it can be shown that the Drazin inverse always exists
k k 11
k
and that it is unique Campbell and Meyer, 1979: 123. Because P RP 5 RP
5 R P , P
k
restricted to R P is nonsingular. This leads to a geometric or functional definition of the Drazin
n
inverse, equivalent to the algebraic definition given above. For any vector x [ R , write x 5 u 1 v
k k
D D
21
with u [ R P and v [ NP . Then P
is defined by P x 5 A
u where A is the
k k
R
s
P
d
R
s
P
d
k
transformation P restricted to RP .
D [
If Ind P happens to be 1, P
is called the group inverse denoted by P . Combining 1 and 2
D [
in the definition of P
implies that PP P 5 P. Note that both the Drazin and the group inverse if
2
it exists are in general different from the Moore–Penrose generalized inverse.
D
For the subsequent exposition it is important to notice that P and P can be represented in what is
known as the canonical form Campbell and Meyer, 1979: 122:
n 3n k
Theorem 1. Canonical form representation: For
P [ R with Ind
P 5 k . 0 and rank P 5 p, there exists a non-singular matrix P such that
N
21
P 5 P P
S D
B
p 3p n 2p3n 2p
where B [ R is nonsingular and N [ R
is nilpotent of index k. Furthermore, if P, B and N are any matrices satisfying the above conditions, then
n 2p D
21.
P 5 P P
S D
21
B
k k
Note that p 5rank P r , n because RP RP . Defining J 5 I 2 B, Assumption 1 and the
p
nonsingularity of B imply that J is convergent, i.e. rJ,1, and that I 2 J is nonsingular. With these
p
definitions and preliminaries, the order of integration of a process hX j is characterized by the index of
t
the matrix P :
Theorem 2. Under Assumption 1,
hX j is integrated of order d if and only if IndP 5 d.
t d
Proof.
⇒ :If
hX j is Id, hD X j is stationary and there exists matrices C such that
t t
j
2
The Drazin and the Moore–Penrose generalized inverses coincide if and only if RA 5 RA9.
276 K
. Neusser Economics Letters 67 2000 273 –281
` `
d 2
D X 5
O
C Z with C 5 I and
O
iC i , `
t j
t 2j n
j j 50
j 50 2
9
where the matrix norm i?i. is defined as iC i 5 tr C C . Because hX j also satisfies the stochastic
s d
j j
j t
difference Eq. 1, we have that
2 d
1 2 L I 5 I 1 C L 1 C L 1 ? ? ? I 2 FL
s d
s d
s d
n n
1 2
n
where L denotes the lag operator. The matrices C are therefore determined recursively
j
d C 5
F 1 2 1 I
s d
S D
1 n
d 2 1 d
d
2 2
C 5 F 1 2 1
F 1 2 1 I
s d
S D
s d
S D
2 n
d 2 1 d 2 2
? ? ? d
d
d d 21
d 22 2
C 5 F 1 2 1
F 1 2 1
F 1 ? ? ?
s d
S D
s d
S D
d
d 2 1 d 2 2
d d
d 21 d
d
1 2 1 F 1 2 1
I 5 F 2 I
s d
S D
s d
S D
s d
n n
1 C 5 C
F j . d
j j 21
d where
denotes the binomial coefficient d id 2 i . Representing P 5 I 2 F and F by their
S D
n
i canonical form, C can be written, for j . d, as
j d
j 2d
N I
2 N
s d
n 2p j 2d
21 d
d
C 5 F 2 I F
5 2 1 P P
s d
s d
j n
j 2d d
1 21
2
I 2 J J
s d
p d
j 2d
As hD X j is stationary, C must converge to zero as j
→ `. Whereas J
vanishes asymptotically
t j
j 2d
because rJ , 1, I
2 N does not converge to zero. N must therefore be nilpotent of index d,
n 2p d
i.e. N 50. But this is equivalent to Ind P 5 d by Theorem 1.
⇐ : Substitute recursively into Eq. 1 to obtain:
d d
d
d 2
d
I
D X 2 Z 2 F 2 I
Z 2
F 2 F 1
I Z
2 ? ? ? 2 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
j 11 d
d
2 F 2 I FZ
2 ? ? ? 2 F 2 I F Z
I
5 iF 2 I F
X i
s d
s d
n t 2d 21
n t 2d 2j
n t 2d 2j 21
Ind P 5 d then implies that
d j 11
N I 2 N
s d
n 2p j 11
21 21
d d
d
F 2 I F 5 2 1 P
P 5 2 1 P
P
s d
s d
s d
n j 11
j 11 d
d
1 2
1 2
I 2 J J I 2 J J
s d
s d
p p
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
