Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol67.Issue3.Jun2000:

www.elsevier.com / locate / econbase

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.

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

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 , t_t 50,1, . . .j is generated by a first order stochastic difference equation:

X_t5FX_t211Z_t

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} h_Xj_{is stationary but S}h j₅

h

o _{X is nonstationary. The process}

j

t t t t51 t

d

is integrated of order d, denoted by hXj|_{I(d ), d51, 2,. .., if} h_{D X}j|_{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 hX_tj is nonstationary and excludes the trivial and uninteresting case that hX_tj 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.

Assumption 1 does not restrict the order of integration for hX_tj: an additional assumption is 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.

n3n

Definition 2. (Index of a matrix): ForP [ R _{, the smallest non-negati}ve 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

1. P P P 5P

D D

2. P P 5P P

k11 D k

3. P P 5P

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 x5u1}v

k k D D 21

with u[R(P ) and v[N(P ). Then P is defined by P x5A_RsPk_du where A_RsPk_d 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

P

0 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

0_n2p 0

D 21.

P 5P

S

₂₁

D

P

0 B

k k

Note that p5rank P #r,n because R(P )#R(P). Defining J5Ip2B, Assumption 1 and the

nonsingularity of B imply that J is convergent, i.e.r(J ),1, and that I_p2J 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, hX_tj is integrated of order d if and only if Ind(P)5d.

d

Proof. (⇒):If hX_tj is I(d ), hD X_tj is stationary and there exists matrices C such that_j

2

` `

d 2

D X_t5

O

C Z_{j t2j} with C₀5I and_n

O

iC_ji, `

j50 j50

2

₉

where the matrix norm i?i. is defined as iCji 5tr C C . Because

s

j j

d

hXtj also satisfies the stochastic difference Eq. (1), we have that

2

d

12L I 5I 1C L1C L 1 ? ? ? I 2FL s d n

s

n 1 2

d

sn d

where L denotes the lag operator. The matrices C are therefore determined recursively_j

d C₁5F1 2s 1d

S D

I_n

d21

d d

2 ₂

C25F 1 2s 1d

S D

_d21 F1 2s 1d

S D

_d22 In

? ? ?

d d

d d21 ₂ d22

Cd5F 1 2s 1d

S D

_d21 F 1 2s 1d

S D

_d22 F 1 ? ? ?

d d

d21 d d

1 2s 1d

S D

₁ F1 2s 1d

S D

₀ In5sF2Ind

Cj5Cj21F j.d

d

where

S D

_i denotes the binomial coefficient d! /(i!d2i )!. Representing P5In2F and F by their canonical form, C can be written, for j_j .d, as

d _j2d

N 0

s

I_n2p2N

d

0

j2d 21

d d

Cj5sF2Ind F 5 2s 1 Pd

1

_d

21

j2d

2

P

0

s

I_p2J

d

0 J

d j2d

As hD X_tj is stationary, C must converge to zero as j_j →`. Whereas J vanishes asymptotically

j2d

because r(J ),1, (I_n2p2N ) 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

d 2 _d

I

D Xt2Zt2

S S D D

F2 ₁ In Zt212

S S D S D D

F 2 ₁ F1 ₂ In Zt222 ? ? ? 2sF2Ind Zt2d

j d j11

d d

2sF2Ind FZt2d212 ? ? ? 2sF2Ind F Zt2d2j

I

5i(F2I )n F Xt2d2j21i

Ind(P)5d then implies that

d _j11

N I

s

n2p2N

d

0 0 0

j11 21 21

d d d

F2I F 5 2s 1 Pd P 5 2s 1 Pd P

s nd

1

_d j11

2

1

_d j11

2

converges to zero. Therefore

d d d

d 2 _d

D X_t5Z_t1

S S D D

F2 I_n Z_t211

S S D S D D

F 2 F1 I_n Z_t221 ? ? ? 1sF2I_nd Z_t2d

1 1 2

`

j d

1

O

sF2I_nd F Z_t2d2j

j51

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 N50_n2r. 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:

0_n2r 0 ₂₁ P5P

S

D

P ,

0 Ir2J

I_n2r 0 _{21 [} 0_n2r 0 ₂₁

F5P

S D

P and P 5P

S

₂₁

D

P .

0 J 0 sI_r2Jd

5. R(P) and N(P) are complementary subspaces. 0_n2r 0

[ [ 21

6. E5P P5PP 5P

S

D

P is a projection on R(P) along N(P).

0 I_r

I 0

t n2r 21 [ 21

Lemma 2. F is semiconvergent: lim F 5C5P

S

D

P 5I 2P P5I 2a bs 9ad b9

Corollary.

1. rank C5n2r

2. b9C50 and C a50 3. C F5F C5C

[ [

4. C and E5P P5PP 5I_n2C are idempotent matrices

Under this circumstance, there exists a Beveridge–Nelson decomposition for hX_tj:

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 [

X_t5T_t1S where T_{t t}5CX_t5C o_t51 Z_t1CX and S_{0 t}5(I_n2C )X_t5P PX_t5(F2C )S_t211

[

P PZt

Proof. Define Tt5CX then Tt t5CFXt211CZt5CXt211CZt5Tt211CZ . Thust hTtj is a random walk and we immediately get the above representation. Similarly define S_t5X_t2T_t5(I_n2C )X_t5

[

P PX . Because (I_{t n}2C ) is idempotent and commutes with F, S_t5(I_n2C )(I_n2C )FX_t211(I_n2 C )Z_t5(F2CF)(I_n2C )X_t211(I_n2C )Z_t5(F2C )S_t211(I_n2C )Z . This defines a stationary_t

0n2r 0 21

stochastic process because r(J ),1 implies that sF2Cd5P

S

D

P has all its eigenvalues

0 J

smaller than one. h

The Beveridge–Nelson decomposition demonstrates the effect of the cointegration vectors in b on

X :_t hb9X_tj 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,hT_tj 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 5P

S

D

P 2P

S D

P 5FC2FN

0 J 0 J 0 J

t t

where N is idempotent of index 2. Therefore F 5F_C2tF_N. For a suitable choice of an initial distribution, a representation of hXtj can be obtained by repeated substitutions:

t21 t21

t t

Xt5FCX02tFNX01

O

FCZt2t2FN

O

tZt2t

t50 t51

t21 t21 5T_t1S_t2tF_NX₀2F_N

O O

Z_t2t

t5s s51

where the random walk T and the stationary component S are defined as in Theorem 3 but replacing_{t t} F byFC. The third expression is just a linear time trend whereas the last expression represents an integrated random walk which is I(2).

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

n3n

R _{defined in the following way:}

n3n

u

& ₅

h

_{A AE5EA5A and 'B} [ R _{: AB5BA5E}

j

E

n3n 2

If& _{is nonempty, there exists A}[& _{and B}[R _{such that AB5E. Thus E 5EAB5AB5E so}

E 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

(⇒): If Ind(P)51, there exist invertible matrices P[ R _{and I 2J} [ R _{such that}

r

0_n2r 0 ₂₁

P5P

S

D

P

0 Ir2J

Define the set of matrices*_PP[ _as

0n2r 0 21_u r3r

*_PP[₅

H S

_P

D

_{P M} [ R _{, rank M}s d5r

j

.

0 M

It is easy to see that P[*_PP[ _{and that} *_PP[ _{is a group. h}

Theorem 5. &_PP[₅*_PP[

[

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 E_{E 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 E_E [G . The two groups must therefore be equal_F

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 5PP}

0 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

21

D

P

0 R BR_{22 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 may

r

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 r_E5rank E#r_F5rank 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.

References

Archontakis, F., 1999. Jordan matrices on the equivalence of I(1) conditions for var systems. In: EUI Working Paper ECO, Vol. No. 99 / 12, European University Institute.

Banerjee, A., Dolado, J., Galbraith, J.W., Hendry, D.F., 1993. Co-Integration, Error-Correction, and the Econometric Analysis of Non-Stationary Data, Oxford University Press, Oxford.

Campbell, S.L., Meyer, Jr. C.D., 1979. Generalized Inverses of Linear Transformations, Dover Publications, New York. ´ d’Autume, A., 1992. Deterministic Dynamics and Cointegration of Higher Orders, Cahiers ECOMATH 92.43, Universite de

Paris I.

Drazin, M.P., 1958. Pseudo-inverses in associative rings and semigroups. American Mathematical Monthly 65, 506–514. Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: representation, estimation and testing. Econometrica

55, 251–276.

Engle, R.F., Yoo, B.S., 1991. Cointegrated economic time series: an overview with new results. In: Engle, R.F., Granger, C.W.J. (Eds.), Long Run Economic Relationships, Oxford University Press, Oxford, pp. 237–266.

Gregoir, S., Laroque, G., 1991. A polynomial error correction representation theorem. Econometric Theory 9, 329–342. Grillet, P.A., 1995. Semigroups. An Introduction to the Structure Theory, Marcel Dekker, New York.

Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–1580.

Johansen, S., 1995. Likelihood Inference in Cointegrated Vector Autoregressive Models, Oxford University Press, Oxford. Neusser, K., 1991. Testing the long-run implications of the neoclassical growth model. Journal of Monetary Economics 27,

3–37.

Phillips, P.C.B., 1991. Optimal inference in cointegrated systems. Econometrica 59, 283–306.

Stock, J.H., Watson, M.W., 1988. Testing for common trends. Journal of the American Statistical Association 83, 1097–1107. Strang, G., 1988. Linear Algebra and its Application, 3rd edition, Harcourt Brace Jovanovich, San Diego.

Watson, M.W., 1994. Vector autoregressions and cointegration. In: Engle, R.F., McFadden, D.L. (Eds.), Handbook of Econometrics, Vol. 4, North-Holland, Amsterdam, pp. 2843–2915.

` `

d 2

D X_t5

O

C Z_{j t2j} with C₀5I and_n

O

iC_ji, `

j50 j50

2

₉

where the matrix norm i?i. is defined as iCji 5tr C C . Because

s

j j

d

hXtj also satisfies the stochastic difference Eq. (1), we have that

2 d

12L I 5I 1C L1C L 1 ? ? ? I 2FL

s d n

s

n 1 2

d

sn d

where L denotes the lag operator. The matrices C are therefore determined recursively_j

d C₁5F1 2s 1d

S D

I_n

d21

d d

2 ₂

C25F 1 2s 1d

S D

_d21 F1 2s 1d

S D

_d22 In

? ? ?

d d

d d21 ₂ d22

Cd5F 1 2s 1d

S D

_d21 F 1 2s 1d

S D

_d22 F 1 ? ? ?

d d

d21 d d

1 2s 1d

S D

₁ F1 2s 1d

S D

₀ In5sF2Ind Cj5Cj21F j.d

d

where

S D

_i denotes the binomial coefficient d! /(i!d2i )!. Representing P5In2F and F by their canonical form, C can be written, for j_j .d, as

d _j2d

N 0

s

I_n2p2N

d

0

j2d 21

d d

Cj5sF2Ind F 5 2s 1 Pd

1

_d

21

j2d

2

P

0

s

I_p2J

d

0 J

d j2d

As hD X_tj is stationary, C must converge to zero as j_j →`. Whereas J vanishes asymptotically

j2d

because r(J ),1, (I_n2p2N ) 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

d 2 _d

I

D Xt2Zt2

S S D D

F2 ₁ In Zt212

S S D S D D

F 2 ₁ F1 ₂ In Zt222 ? ? ? 2sF2Ind Zt2d

j d j11

d d

2sF2Ind FZt2d212 ? ? ? 2sF2Ind F Zt2d2j

I

5i(F2I )n F Xt2d2j21i

Ind(P)5d then implies that

d _j11

N I

s

_n2p2N

d

0 0 0

j11 21 21

d d d

F2I F 5 2s 1 Pd P 5 2s 1 Pd P

s nd

1

_d j11

2

1

_d j11

2

converges to zero. Therefore

d d d

d 2 _d

D X_t5Z_t1

S S D D

F2 I_n Z_t211

S S D S D D

F 2 F1 I_n Z_t221 ? ? ? 1sF2I_nd Z_t2d

1 1 2

`

j d

1

O

sF2I_nd F Z_t2d2j j51

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 N50_n2r. 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:

0_n2r 0 ₂₁

P5P

S

D

P ,

0 Ir2J

I_n2r 0 _{21 [} 0_n2r 0 ₂₁

F5P

S D

P and P 5P

S

₂₁

D

P .

0 J 0 sI_r2Jd

5. R(P) and N(P) are complementary subspaces. 0_n2r 0

[ [ 21

6. E5P P5PP 5P

S

D

P is a projection on R(P) along N(P). 0 I_r

I 0

t n2r 21 [ 21

Lemma 2. F is semiconvergent: lim F 5C5P

S

D

P 5I 2P P5I 2a bs 9ad b9

Corollary.

1. rank C5n2r

2. b9C50 and C a50 3. C F5F C5C

[ [

4. C and E5P P5PP 5I_n2C are idempotent matrices

Under this circumstance, there exists a Beveridge–Nelson decomposition for hX_tj:

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 [

X_t5T_t1S where T_{t t}5CX_t5C o_t51 Z_t1CX and S_{0 t}5(I_n2C )X_t5P PX_t5(F2C )S_t211

[

P PZt

Proof. Define Tt5CX then Tt t5CFXt211CZt5CXt211CZt5Tt211CZ . Thust hTtj is a random walk and we immediately get the above representation. Similarly define S_t5X_t2T_t5(I_n2C )X_t5

[

P PX . Because (I_{t n}2C ) is idempotent and commutes with F, S_t5(I_n2C )(I_n2C )FX_t211(I_n2

C )Z_t5(F2CF)(I_n2C )X_t211(I_n2C )Z_t5(F2C )S_t211(I_n2C )Z . This defines a stationary_t

0n2r 0 21

stochastic process because r(J ),1 implies that sF2Cd5P

S

D

P has all its eigenvalues

0 J

smaller than one. h

The Beveridge–Nelson decomposition demonstrates the effect of the cointegration vectors in b on

X :_t hb9X_tj 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,hT_tj 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 5P

S

D

P 2P

S D

P 5FC2FN

0 J 0 J 0 J

t t

where N is idempotent of index 2. Therefore F 5F_C2tF_N. For a suitable choice of an initial distribution, a representation of hXtj can be obtained by repeated substitutions:

t21 t21

t t

Xt5FCX02tFNX01

O

FCZt2t2FN

O

tZt2t t50 t51

t21 t21

5T_t1S_t2tF_NX₀2F_N

O O

Zt2t t5s s51

where the random walk T and the stationary component S are defined as in Theorem 3 but replacing_{t t} F byFC. The third expression is just a linear time trend whereas the last expression represents an integrated random walk which is I(2).

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

n3n

R _{defined in the following way:}

n3n

u

& ₅

h

_{A AE5EA5A and 'B} [ R _{: AB5BA5E}

j

E

n3n 2

If& _{is nonempty, there exists A}[& _{and B}[R _{such that AB5E. Thus E 5EAB5AB5E so}

E 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

(⇒): If Ind(P)51, there exist invertible matrices P[ R _{and I 2J} [ R _{such that} r

0_n2r 0 ₂₁

P5P

S

D

P

0 Ir2J

Define the set of matrices*_PP[ _as

0n2r 0 21_u r3r

*_PP[₅

H S

_P

D

_{P M} [ R _{, rank M}s d5r

j

.

0 M

It is easy to see that P[*_PP[ _{and that} *_PP[ _{is a group. h} Theorem 5. &_PP[₅*_PP[

[

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 E_{E 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 E_E [G . The two groups must therefore be equal_F

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 5PP}

0 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

21

D

P

0 R BR_{22 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 may

r

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 r_E5rank E#r_F5rank 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.

References

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