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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.neusser@vwi.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.
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.
n3n
Throughout the paper I use the following notation.R is the set of all n3n real matrices. r(A)
n3n
and s(A) denote the spectral radius and the set of all eigenvalues of A[R . R(A) and N(A) are
0
range and the null space of A. Moreover, I adopt the convention that A 5I .n
I present my argument by assuming that the n-dimensional stochastic processhX , tt 50,1, . . .j is generated by a first order stochastic difference equation:
Xt5FXt211Zt
where Z |WN(0,S) (1)
t
Dt5 2sIn2FdXt211Zt5 2PXt211Zt
with X given. The assumption that0 hZtjis white noise is made just for expositional convenience. The 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 byhXj|I(0), if hXjis stationary but Sh j5
h
o X is nonstationary. The processj
t t t t51 t
d
is integrated of order d, denoted by hXj|I(d ), d51, 2,. .., if hD Xj|I(0). The system is said to be
t t
cointegrated if hXj 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. r(F)#1 and 1[s(F) 2. l[s(F) and ulu51⇒l51
Part (a) of Assumption 1 states that there is at least one unit root, i.e. one eigenvalue equal to one. This implies that hXtj is nonstationary and excludes the trivial and uninteresting case that hXtj is 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 P5r,n. P therefore admits a full-rank factorization
P5ab9 with rankP5rank a5rank b5r (2)
where a and b are n3r matrices. The r linearly independent n-vectors of b are usually called
21
cointegration vectors if the system is I(1). Because P5(aR ) (bR9)9 is also a valid full-rank factorization for any nonsingular r3r matrix R, the cointegration vectors are not unique; although the
space they generate is.
n3n
Definition 2. (Index of a matrix): ForP [ R , the smallest non-negative integer k such that rank
k k11 n k k
P 5rank P , or equivalently, such that R is the direct sum of R(P ) and N(P ), is called the
index of P and is denoted by Ind(P). If P is invertible Ind(P)50.
D n3n
Definition 3. (Algebraic definition):P is called the Drazin inverse ofP [ R with Ind(P)5k
if and only if it satisfies the following conditions:
D D D
As k5Ind(P) is well defined for any matrix, it can be shown that the Drazin inverse always exists
k k11 k
and that it is unique (Campbell and Meyer, 1979: 123). Because P R(P )5R(P )5R(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 x5u1v
k k D D 21
with u[R(P ) and v[N(P ). Then P is defined by P x5ARsPkdu where ARsPkd is the
k
transformation P restricted to R(P ).
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 ofP implies thatPP P5P. 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 thatP andP can be represented in what is known as the canonical form (Campbell and Meyer, 1979: 122):
n3n k
Theorem 1. (Canonical form representation): For P[R with Ind(P)5k.0 and rank P 5p,
there exists a non-singular matrix P such that
N 0 21
P5P
S D
P0 B
p3p (n2p)3(n2p)
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
0n2p 0
nonsingularity of B imply that J is convergent, i.e.r(J ),1, and that Ip2J is nonsingular. With these
definitions and preliminaries, the order of integration of a processhXtj is characterized by the index of the matrix P:
Theorem 2. Under Assumption 1, hXtj is integrated of order d if and only if Ind(P)5d.
d
Proof. (⇒):If hXtj is I(d ), hD Xtj is stationary and there exists matrices C such thatj
2
` `
where L denotes the lag operator. The matrices C are therefore determined recursivelyj
d
As hD Xtj is stationary, C must converge to zero as jj →`. Whereas J vanishes asymptotically
j2d
because r(J ),1, (In2p2N ) does not converge to zero. N must therefore be nilpotent of index d,
d
i.e. N 50. But this is equivalent to Ind(P)5d by Theorem 1.
(⇐): Substitute recursively into Eq. (1) to obtain:
d d d
Ind(P)5d then implies that
converges to zero. Therefore
d d d
d 2 d
D Xt5Zt1
S S D D
F2 In Zt211S S D S D D
F 2 F1 In Zt221 ? ? ? 1sF2Ind Zt2d1 1 2
`
j d
1
O
sF2Ind F Zt2d2jj51
d
is a stationary representation of hD Xtj. h
2. Representations and basic properties of the group inverse
In most of the following I restrict myself to the case where hXj|I(0), thus assuming:
t
Assumption 2. (Integration of order one)
2
rank(P)5rank(P ) or Ind(P)51
This implies r5p and N50n2r. In the literature Assumption 2 is not always made explicit and its 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 r3r matrix b9a is nonsingular and P 5a (b9a) b9.
3. F has a full set of eigenvectors corresponding to the eigenvalue 1.
4. The canonical form representations specialize to:
0n2r 0 21 P5P
S
D
P ,0 Ir2J
In2r 0 21 [ 0n2r 0 21
F5P
S D
P and P 5PS
21D
P .0 J 0 sIr2Jd
5. R(P) and N(P) are complementary subspaces. 0n2r 0
[ [ 21
6. E5P P5PP 5P
S
D
P is a projection on R(P) along N(P).0 Ir
I 0
t n2r 21 [ 21
Lemma 2. F is semiconvergent: lim F 5C5P
S
D
P 5I 2P P5I 2a bs 9ad b9Corollary.
1. rank C5n2r
2. b9C50 and C a50 3. C F5F C5C
[ [
4. C and E5P P5PP 5In2C are idempotent matrices
Under this circumstance, there exists a Beveridge–Nelson decomposition for hXtj:
Theorem 3. (Beveridge –Nelson decomposition): Under Assumptions 1 and 2 and a suitable
distribution of X ,0 hXtj is the sum of a random walk hTtj and a stationary process hStj:
t [
Xt5Tt1S where Tt t5CXt5C ot51 Zt1CX and S0 t5(In2C )Xt5P PXt5(F2C )St211
[
P PZt
Proof. Define Tt5CX then Tt t5CFXt211CZt5CXt211CZt5Tt211CZ . Thust hTtj is a random walk and we immediately get the above representation. Similarly define St5Xt2Tt5(In2C )Xt5
[
P PX . Because (It n2C ) is idempotent and commutes with F, St5(In2C )(In2C )FXt211(In2 C )Zt5(F2CF)(In2C )Xt211(In2C )Zt5(F2C )St211(In2C )Z . This defines a stationaryt
0n2r 0 21
stochastic process because r(J ),1 implies that sF2Cd5P
S
D
P has all its eigenvalues0 J
smaller than one. h
The Beveridge–Nelson decomposition demonstrates the effect of the cointegration vectors in b on
X :t hb9Xtj is stationary as the random walk component is knocked out because b9C50. Note that b9
also eliminates the dependence on the initial value CX . Because rank C0 5n2r,hTtj is made up of
n2r independent random walks, sometimes called common trends, whereas the dimension ofhStj is
r.
In case of an I(2)-process, Ind(P)52 and F can be written according to Theorem 1 as
In2p2N 0 21 In2p 0 21 N 0 21
F5P
S
D
P 5PS
D
P 2PS D
P 5FC2FN0 J 0 J 0 J
t t
where N is idempotent of index 2. Therefore F 5FC2tFN. For a suitable choice of an initial distribution, a representation of hXtj can be obtained by repeated substitutions:
t21 t21
t t
Xt5FCX02tFNX01
O
FCZt2t2FNO
tZt2tt50 t51
t21 t21 5Tt1St2tFNX02FN
O O
Zt2tt5s s51
3. Some algebraic properties
n3n
From now on I view the set of all n3n matrices, R , as a semigroup with respect to the matrix
3 n3n n3n
multiplication. The identity element inR is I . For any given E[R consider the subset& of
n E
that E is idempotent and E[& . It is also straightforward to see that & is closed under
E E
n3n
multiplication and that EBE[& is the inverse of A in& . In other words& is a subgroup of R
E E E
with identity element given by E. The subgroup is maximal because every subgroup * containing E is in fact a subgroup of& . This can be seen by noting that E must be the identity of any group * in
E
which it lies, so that any A[* satisfies the defining conditions for& (Drazin, 1958: 513). This leads
E
4
to a family of disjoint maximal subgroups h& j where E runs over all idempotent matrices. This
E n3n
family does not form a partition of R because, as is shown in Theorem 4, matrices with index
strictly greater than one do not belong to any subgroup.
Theorem 4. Under Assumption 1, Ind(P)51 if and only if P belongs to some subgroup.
g
Proof. (⇐): Suppose P belongs to some subgroup &. Then P must have an inverse P in &. This
g g g g g g g [
inverse satisfies PP 5P P, P PP 5P , PP P5P. Therefore P is nothing but P , the
5
group inverse of P. Therefore Ind(P)51, given Assumption 1.
n3n r3r
Proof. Any element in *PP[ satisfies the definition of &PP[. Moreover, PP is idempotent and
[
PP [* [, thus * [ is a subgroup of & [.
PP PP PP
3
A semigroup is just a set together with an associative binary operation. 4
The fact that the groups are disjoint can be established as follows. Let G and G be two subgroups such that EE F ±F and GE>GF±[. Then for any A[GE>G , there exists B and C such that EFF 5BAF5BA5E5AB5FAB5FE and EF5EAC5AC5F5CA5CAE5FE. This implies that F[G and EE [G . The two groups must therefore be equalF
because they are both maximal subgroups.
5 [
0 0 21 q3q [ [
If A is any A[&PP[, A5Q
S D
Q with B[R and rank B5q. Because AA 5PP0 B
these matrices are similar and therefore r5q. Furthermore there exists an invertible matrix R such
[ [
that Q5PR. Observing that the definition of& [ impliesPP A5A and APP 5A, it is easy to
PP
verify that A can be written as
0 0 21
A5P
S
21D
P0 R BR22 22
where R22 is the (2,2)-element of the appropriately partitioned matrix R. Thus A[*PP[. h
[ 2 [2
Note that the set of matrices hE,P,P ,P ,(P ) , . . .j constitutes a commutative (Abelian)
[ [
subgroup of & . The identity is again the matrix E5P P5PP . The subgroup is cyclic and
E
generated by the positive and negative powers ofP where negative powers are interpreted as powers
[
of P .
Following Grillet (1995: 27), we introduce a partial order on the set of idempotent matrices:
Theorem 6. (Partial order): The binary relation # defined by E#F⇔EF5FE5E represents a partial order on the set of all idempotent matrices.
Proof. The reflexivity follows from E being idempotent. Symmetry is also immediate from the
definition. Finally, if E#F and F#G then EG5EFG5EF5E5FE5GFE5GE. Thus E#G
and # is also transitive.
Let us translate this definition into our context. The definition of E#F implies that E and F
commute. Moreover, E and F are diagonizable matrices, consequently they must share the same 0 0 21
eigenvectors (Strang, 1988: 259). E and F are therefore both of the form P
S D
0 I P . They mayr
differ only with respect to r. Thus for any two idempotent matrices E and F, E#F is equivalent to E
and F share the same eigenvectors and rE5rank E#rF5rank F. The cointegration space corresponding to E is therefore included in the one corresponding to F.
4. Conclusion
This note showed how the concept of the Drazin and group inverse can be fruitfully applied for the analysis of integrated systems. Although some results are known in the literature, this algebraic approach provides fresh insights. In the context of systems integrated of order one, I have established a one-to-one correspondence between the cointegration vectors, respectively the space they generate,
[
and matrix subgroups identified by unity element PP .
The methods presented in this paper should be easily accessible because algorithms to compute the index of a matrix and its Drazin inverse are readily available and simple to implement (Campbell and
6
Meyer, 1979: 255–60). The analysis can easily be extended to higher order VARs or more complicated error processes hZtj.
6
Acknowledgements
I thank Søren Johansen for his comments. The usual disclaimer applies.
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