# Solutions of Yule‐Walker equations for singular AR processes

**ABSTRACT** A study is presented on solutions of the Yule‐Walker equations for singular AR processes that are stationary outputs of a given AR system. If the Yule-Walker equations admit more than one solution and the order of the AR system is no less than two, the solution set includes solutions which define unstable AR systems. The solution set also includes one solution, the minimal norm solution, which defines an AR system whose characteristic polynomial has either only stable zeros (implying that only one stationary output exists for this system and it is linearly regular) or has stable zeros as well as zeros of unit modulus, (implying that stationary solutions of this system are a sum of a linearly regular process and a linearly singular process). The numbers of stable and unit circle zeros of the characteristic polynomial of the defined AR system can be characterized in terms of the ranks of certain matrices, and the characteristic polynomial of the AR system defined by the minimal norm solution has the least number of unit circle zeros and the most number of stable zeros over all possible solutions.

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**ABSTRACT:**We deal with singular multivariate AR systems and the corresponding AR processes. An AR system is called singular if the variance of the white noise innovation is singular. AR processes are the stationary solutions of AR systems. In the singular case AR processes consist of a linearly regular and a linearly singular component. The corresponding Yule-Walker equations and in particular the possible non-uniqueness of their solutions are discussed. A particular canonical form is presented. Singular AR systems naturally arise as models for latent variables in dynamic factor analysis.01/2011; 11:225-236.

Page 1

Solutions of Yule-Walker equations for

singular AR processes

Weitian Chena,*,†, Brian D.O. Andersona, Manfred Deistlerband

Alexander Fillerc

A study is presented on solutions of the Yule-Walker equations for singular AR processes that are stationary

outputs of a given AR system. If the Yule-Walker equations admit more than one solution and the order of the AR

system is no less than two, the solution set includes solutions which define unstable AR systems. The solution set

also includes one solution, the minimal norm solution, which defines an AR system whose characteristic polynomial

has either only stable zeros (implying that only one stationary output exists for this system and it is linearly

regular) or has stable zeros as well as zeros of unit modulus, (implying that stationary solutions of this system are a

sum of a linearly regular process and a linearly singular process). The numbers of stable and unit circle zeros of the

characteristic polynomial of the defined AR system can be characterized in terms of the ranks of certain matrices,

and the characteristic polynomial of the AR system defined by the minimal norm solution has the least number of

unit circle zeros and the most number of stable zeros over all possible solutions.

Keywords: Singular AR process; minimal norm solution; stationary process.

JEL classifications: C32; C67.

1. INTRODUCTION

In this article we consider multivariable AR systems of the form

zt¼ e1zt?1þ ... þ epzt?pþ mt;

ð1Þ

where ej2 Rr?r; j ¼ 1;...;p and mtis a white-noise r · 1 vector. We are only interested in stationary solutions of (1). In addition, we

assume that mtis uncorrelated with the past zs, s < t. In that case, mtis the innovation of zt. Our emphasis is on the case where

Rm ¼ E½mtmT

According to the Wold decomposition ( e.g. Hannan and Deistler (1988)) a stationary process ztcan be decomposed into a sum of a

linearly regular part zr

twhich are mutually orthogonal, i.e.

zt¼ zr

For any stationary solution ztof (1), if its corresponding covariance matrices are denoted as cj¼ E½ztzT

associated Yule-Walker equations are of the form

t? is singular.

tand linearly singular part zs

tþ zs

t:

ð2Þ

t?j?; j ¼ 0;1;2;???; then its

½e1... ep?Cp¼ ½c1... cp?ð3Þ

Rm¼ c0? ½e1... ep?½c1... cp?T;

ð4Þ

where

Cp¼

c0

...

...

???

c0

???

cp?1

...

...

c0

..

.

cT

p?1

??????

2

66664

3

77775

:

Here we are interested in the case where Rmis singular and we will call the corresponding AR system singular. Singular AR systems (1)

can be represented as

eðzÞzt¼ mt¼ fet;

ð5Þ

aAustralian National University

bVienna University of Technology & Institute of Advanced Studies

cVienna University of Technology

*Correspondence to: Weitian Chen, Research School of Information Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia.

†E-mail: weitian.chen@anu.edu.au

J. Time Ser. Anal. 2011, 32 531–538

? 2011 Blackwell Publishing Ltd.

Original Article

First version received June 2010 Published online in Wiley Online Library: 27 January 2011

(wileyonlinelibrary.com) DOI: 10.1111/j.1467-9892.2010.00711.x

531

Page 2

where z is used for the backward shift operator as well as a complex variable, e(z) ¼ I ? e1z ?...? epzp, and f 2 Rr?qwhere Rm¼ ffT

andrk(Rm) ¼ q < r.Inthisarticle,wecalldet(e(z))thecharacteristicpolynomialofanARsystemoftheform(5).Aparticularexamplearises

when f ¼ 0 and thus Rm¼ 0. The AR system is then degenerate, the solutions of which are simply a linearly singular process.

In the standard case where the innovation covariance matrix Rmis non-singular, it is well known that a stationary solution exists for

(1) if and only if the roots of det(e(z)) lie outside the unit circle. In that case the stationary solution is unique, as are the covariance

matrices cj, and these matrices and the matrices ejand Rmare linked by the Yule-Walker equations given by (3) and (4). Moreover, the

matrix Cpis non-singular. Further, given the matrix sequence of cjsuch that the matrix Cpis non-singular, the matrices ejcan be

uniquely determined using the Yule-Walker equations.

However, when Rmis singular, the simple conclusions mentioned in the previous paragraph no longer hold in general and things

can become rather complicated. To appreciate this, we consider a simple example.1

EXAMPLE 1:Consider the following AR system.

ðI ? ezÞzt¼ fet;

ð6Þ

where e is a 2 · 2 non-singular matrix with two distinct eigenvalues, f is a 2 · 1 vector, and etis scalar.

Since e is a 2 · 2 non-singular matrix with two distinct eigenvalues, there exits a non-singular matrix M and a diagonal matrix d

d1

0

0

d2

Define xt ¼ x1t

ðI ? dzÞxt¼ get

which can be rewritten as

such that d ¼

??

?T¼ Mztand g ¼ g1

¼ MeM?1.

x2t

½

g2

½?T¼ Mf, then it follows from (6) that

ð7Þ

ð1 ? d1zÞx1t¼ g1et;

ð1 ? d2zÞx2t¼ g2et:

ð8Þ

The following observations can be made on (8):

? If |d1| < 1 and |d2| < 1, then there exists only one stationary solution for (6), which is linearly regular.

? If |d1| < 1 and |d2| > 1, then stationary solutions for (6) exist only if g2¼ 0. When g2¼ 0, the solution is unique:

x1t¼ (1 ? d1z)?1g1et, x2t¼ 0, which is linearly regular.

? If |d1| < 1 and |d2| ¼ 1, then stationary solutions for (6) exist only if g2¼ 0. When g2¼ 0, the stationary solutions are:

x1t¼ (1 ? d1z)?1g1et, x2t¼ a, where a is any stochastic variable uncorrelated with esfor all s. The only linearly regular solution

in this stationary solution set corresponds to a stochastic variable a that equals to zero almost everywhere. For the linearly

regular solution, it is obvious that

C1¼ c0¼ E½ztzT

t? ¼ M?1E½xtxT

t?M?T¼ M?1 E½x2

1t?

0

00

??

M?T;

which implies that C1is singular. For a solution which additionally has a singular component, the above matrix C1is replaced by

?

This illustrates that the covariance appearing in the Yule-Walker equations actually depends on the particular admissible solution

for (6) [or (1)].

The last two observations imply that unlike regular AR systems, (i) singular AR systems may admit stationary solutions even if the

roots of det(e(z)) lie inside or on the unit circle; (ii) singular AR systems can have multiple stationary solutions; and (iii) the matrix Cp

corresponding to a stationary solution can be singular.

Our interest in singular AR systems was triggered by the analysis of Generalized Dynamic Factor Models (see e.g. Forni et al. (2000),

Forni and Lippi (2001), Anderson and Deistler (2009), and Deistler et al. (2010)). In a certain sense these systems are more complicated

compared to the regular AR case. This in particular applies to the problem of identifiability see Anderson and Deistler (2008),

Anderson et al. (2009). The elements of the equivalence class of left coprime (e(z), f) are related by left multiplication by unimodular

matrices, see Hannan and Deistler (1988).

A detailed analysis of singular AR systems has been given in Inouye (1983) which is the main reference for our article.

M?1 E½x2

1t?

0

0

E½a2?

?

M?T:

2. SOLUTIONS OF YULE-WALKER EQUATIONS

For AR systems (1), if rk(Rm) > 0 holds, a stationary solution (if it exists) has a linearly regular part

zr

t¼ e?1ðzÞfet

ð9Þ

1We are grateful to a reviewer for suggesting this example.

W. CHEN ET AL.

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and the corresponding transfer function e?1(z)f has no poles inside and on the unit circle. Note that this is not the same as asserting

that the characteristic polynomial det (e(z)) has no zeros inside or on the unit circle, because (e(z), f) is not necessarily left coprime.

However, there always exists a left coprime pair ð? eðzÞ;fÞ with the same transfer function such that detð? eðzÞÞ has all zeros outside the

unit circle. The linearly singular part of the stationary solution is a solution of the homogeneous equation

eðzÞzs

t¼ 0

ð10Þ

and is obtained by a suitable choice of initial values and corresponds to the zeros of det(e(z)) on |z| ¼ 1. Note that for singular AR

systems the matrix Cp+1must be singular and Cpmay be singular (for a more detailed discussion see Deistler et al. (2010)). If Cpis

non-singular, the Yule-Walker equations have a unique solution. If Cpis singular then the Yule-Walker equations are still always

solvable and the solution set for each row of [e1... ep] is an affine subspace obtained by a particular solution plus an arbitrary vector

in the kernel of Cp.

The (row-wise) minimum norm solution in the solution set is given by

½^ e1... ^ ep? ¼ ½c1... cp?C#

p;

ð11Þ

where C#

pis the Moore-Penrose pseudo-inverse of Cp, which is defined as

C#

p¼ OT

p

K?1

1

0

0

0

??

Op;

ð12Þ

where OT

Making use of the minimal norm solution, it can be shown that all solutions of the Yule-Walker equations given by (3) and (4) can

be written in the following form

pOp ¼ I, and K1is diagonal and non-singular and consists of all nonzero eigenvalues of Cp.

½e1... ep? ¼ ½^ e1... ^ ep? þ ½0 U?Op

ð13Þ

ep].

with U being any matrix with r rows and pr?rk(Cp) columns. Note also that Rmin (4) is the same for all solutions [e1

...

3. MAIN RESULTS

Consider the system

^ eðzÞzt¼ mt¼ fet

ð14Þ

defined by the minimal norm solution (11) and by ffT¼ Rm.

The AR system given by (1) or (5) is said to be stable if its characteristic polynomial satisfies det(e(z)) 6¼ 0, |z| £ 1. It is said to be

unstable if its characteristic polynomial has a zero inside the unit circle. Thus, in this sense, if the characteristic polynomial has zeros

outside and on the unit circle, the AR system is neither stable nor unstable.

As has been shown in Deistler et al. (2010), the solution set of the Yule-Walker equations contains a solution [e1...ep] that

corresponds to a stable AR system (1) if and only if the minimum norm solution (11) defines a stable AR system (For non-singular Cp,

this is of course trivial; it is quite nontrivial when Cpis singular). Much of this paper explores the consequences of this condition not

being satisfied.

It is straightforward to transform (1) into an AR(1) system of the form

xt¼ Ext?1þ Get;

ð15Þ

where xt ¼

zT

t

???

zT

t?pþ1

??T, and

E ¼

e1

I

0

...

0

e2

0

I

...

0

...

...

...

ep?1

0

0

ep

0

0

...

0 ...I

2

66664

3

77775

; G ¼

f

0

...

0

2

664

3

775:

ð16Þ

Then for stationary zt, Cp ¼ ExtxT

tand the following Lyapunov equation (e.g. Gajic and Qureshi (1995)) holds

Cp? ECpET¼ Q;

ð17Þ

where Q ¼

Rm

0

...

0

0

0

...0

0

2

664

3

775.

For the minimal norm solution ½^ e1... ^ ep?, define^E analogously as E; there holds

Cp?^ECp^ET¼ Q:

ð18Þ

SOLUTIONS OF YULE-WALKER EQUATIONS

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The matrix pair [E, G] is said to be reachable if [kI ? E G] is of full row rank for all k 2 C, or equivalently the matrix [G EG...Erp?1G] is

of full row rank. Given any eigenvalue k0of E, if [k0I ? E G] is of full row rank, then k0is said to be a reachable mode. Otherwise, it is

called an unreachable mode (e.g. Kailath (1980) or Gajic and Qureshi (1995)). One can derive the following results, cast in the language

of linear system theory.

LEMMA 1.

with Rm¼ ffT. Then

[i] If the pair [E, G] is reachable, then Cpis positive definite.

[ii] If |ki(E)| < 1 and Cpis positive definite, then the pair [E,G] is reachable.

[iii] If Cpis positive definite and reachability does not hold, then E has one or more eigenvalues on the unit circle corresponding to

unreachable modes.

[iv] Suppose (17) for fixed E and Rmhas a nonnegative solution?Cp. Define Cmin ¼P1

Assume that ztis a stationary solution of (1), define xt ¼ ½zT

t??? zT

t?pþ1?T, let E,G be defined as in (16) and consider (17)

j¼0EjGGTðETÞj. Then Cminexists and satisfies (17),

and any nonnegative solution Cpof (17) satisfies Cp? Cmin.

PROOF.

where zr

xr

respectively. Since the linearly regular part xr

function can only have poles outside the unit circle. We have xr

[i] Note that xt ¼ ½zT

tand zs

t¼ ½ðzr

t??? zT

t?pþ1?Tis stationary as ztis stationary by assumption. It is shown in Section 2 that zt ¼ zr

tare the linearly regular part and the linearly singular part respectively. Accordingly, we have xt ¼ xr

tÞT??? ðzr

tof the stationary process xt, corresponds to the transfer function [I?Ez]?1G, this transfer

t¼ ½GEGE2G...?½eT

tþ zs

t, where

t,

tþ xs

t?pþ1ÞT?Tand xs

t¼ ½ðzs

tÞT??? ðzs

t?pþ1ÞT?Tand are the linearly regular part and the linearly singular part of xt

teT

t?1eT

t?2????Tand thus by reachability

Cr

p¼ ½G EG E2G ...?½G EG E2G ...?T

ð19Þ

has full rank. Therefore the same holds for Cpas it is the sum of Cr

[ii] Under the assumption |ki(E)| < 1, Cpis equal to its linearly regular part Cr

[iii] If reachability does not hold, then Cr

which corresponds to eigenvalues on the unit circle.

[iv] Define

pand Cs

p, the latter being the covariance of xs

p. Thus the result follows immediately from (19).

p6¼ 0 and corresponds to the linearly singular component

t.

pis singular and thus Cp? Cr

CðiÞ¼

X

i

j¼0

Ejdiag ½R;0;...;0?ðETÞj;

ð20Þ

and Q ¼ diag [R, 0, ..., 0].

We claim that?Cp ? Cðiþ1Þ? CðiÞ8i. Observe that?Cp ¼ E?CpETþ Q ? Q ¼ C0. Assume for an induction that?Cp ? CðiÞ. Then

?Cp¼ Q þ E?CpET? Q þ ECðiÞET¼ Cðiþ1Þ

ð21Þ

Hence Cminis well defined as limi!1C(i), and it satisfies the Lyapunov equation (17), as well as?Cp? Cmin. Furthermore, since the

above arguments can be carried over to any nonnegative solution Cpof the Lyapunov equation (17), we have Cp? Cmin. This

completes the proof of conclusion [iv].

The main results are presented in the following four theorems, where the first theorem deals with the case that Cpis non-singular

and the remaining three theorems deal with the case that Cpis singular.

THEOREM 1.

denote the unique solution of the Yule-Walker equations (3) and (4) and let E be the corresponding block companion matrix, f be such that

ffT¼ Rm, and G be defined by (16). Then the associated AR system (14) is stable if and only if rk([G EG...Erp?1G]) ¼ rk(Cp). When this

condition fails, all zeros of the characteristic polynomial e(z) lie outside or on the unit circle, with the number of unit circle zeros equal to

rk(Cp)?rk([G EG...Erp?1G]).

Consider a stationary solution of the AR system (1) and suppose that the corresponding Cpis non-singular. Let [e1...ep]

THEOREM 2.

Suppose that Cpis singular and p > 1. Then there always exists at least one solution [e1...ep] of the Yule-Walker equations (3) and (4)

such that the AR system (1) is unstable.

Suppose that the AR system (1) has one or more stationary solutions, and let Cpbe defined using one of these solutions.

THEOREM 3.

denote the minimal norm solution of the Yule-Walker equations (3) and (4) and let^E be the corresponding block companion matrix, f be

such that ffT¼ Rmand G be defined as in (16). Then the minimal norm solution defines a stable AR system (14) if and only if

rkð½G^EG ???^Erp?1G?Þ ¼ rkðCpÞ.

Consider a stationary solution of the AR system (1) and suppose that the corresponding Cpis singular. Let ^ ei; i ¼ 1;???;p

W. CHEN ET AL.

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THEOREM 4.

denote the minimal norm solution of the Yule-Walker equations (3) and (4) and let^E be the corresponding companion matrix. If the AR

system (14) is not stable, then the characteristic polynomial of the minimum norm solution has the following properties

Consider a stationary solution of the AR system (1) and suppose that the corresponding Cpis singular. Let ^ ei; i ¼ 1;...;p

[i] It can not have zeros inside the unit circle and must have zeros both outside and on the unit circle.

[ii] The number of unit circle zeros is equal to rkðCpÞ ? rkð½G^EG ???^Erp?1G?Þ.

[iii] It has the least number of unit circle zeros among all solutions of the Yule Walker equations.

[iv] It has the largest number of stable zeros among all solutions of the Yule Walker equations.

The proofs of these theorems are placed in the appendix.

So as to gain further insights into Theorem 4, a simple example is provided below to show that the minimal norm solution of the Yule-

Walker equations may be not stable when Cpis singular.

EXAMPLE 2:Consider the following AR system.

ðI ? e1z ? e2z2Þzt¼ fet;

ð22Þ

where zt ¼ z1t

z2t

½?T, etis scalar and E½e2

t? ¼ 1,

e1¼

0:5

0

0

1

??

; e2¼

?0:6

0

0

0

??

; f ¼

1

0

? ?

:

The system equation (22) can be rewritten as

ð1 ? 0:5z þ 0:6z2Þz1t¼ et;

ð1 ? zÞz2t¼ 0:

ð23Þ

It is obvious that the stationary solutions of (22) are: z1t¼ (1?0.5z + 0.6z2)?1et, z2t¼ a, where a is any stochastic variable

uncorrelated with esfor all s.

Now take z2tas any stochastic variable a, which is uncorrelated with esfor all s and whose variance is 1. Then this stationary

solution of (23) leads to

Cp¼

1:7316

0

0:5411

0

0

1

0

1

0:5411

0

1:7316

0

0

1

0

1

2

664

3

775;

c1

c2

½ ? ¼

0:5411

0

0

1

?0:7684

0

0

1

??

:

ð24Þ

Obviously Cpis singular as it has two rows that are the same. The minimal norm solution is computed as

^ e1^ e2

½? ¼ c1

c2

½?C#

p¼

0:5000

0

0

?0:6000

0

0

0:50000:5000

??

:

ð25Þ

It is easy to check that detð^ eðzÞÞ ¼ ð1 ? 0:5z þ 0:6z2Þð1 ? 0:5z ? 0:5z2Þ. The roots of detð^ eðzÞÞ are 0.4167 + 1.2219i,

0.4167 ? 1.2219i, ?2, 1, where the first three roots are outside the unit circle and the last one is on the unit circle. Since

detð^ eðzÞÞ has one root on the unit circle, the minimal norm solution of the Yule-Walker equations is indeed not stable.

Based on the above example, a remark can be made as follows: It has been shown that when Cpis singular and the minimum norm

solution is not stable, the minimum norm solution is the best possible solution in the sense of Theorem 4. In other words, it is

impossible to find a stable characteristic polynomial by solving the Yule-Walker equations associated with Cp. Recall also that when

the pair ð^ eðzÞ;fÞ is not coprime, there is a coprime pair ð? eðzÞ;fÞ with the same transfer function and such that detð? eðzÞÞ is stable. This

however would correspond to different covariances cp(which are associated with the linearly regular part zr

tof zt).

4. CONCLUSIONS

We have obtained a number of interesting properties of solutions of the Yule-Walker equations of a singular AR system with a

stationary output process. In case that the Yule-Walker equations have one unique solution, the associated AR system has a

characteristic polynomial whose zeros are confined to the exterior of the unit circle, or the unit circle itself. The number of unit circle

zeros is easily characterized by a rank test. In case that the Yule-Walker equations admit multiple solutions, it has been proved first

that there must exist a solution that defines an AR system that is unstable if the maximal lag of the AR system is larger than or equal

to two. Then, sufficient and necessary conditions have been derived for the minimal norm solution to define a stable AR system, or

equivalently, for there to exist any solution of the Yule-Walker equations defining a stable AR system. Next, when the minimal norm

solution cannot define a stable AR system, it has also been proved that it will always define an AR system whose characteristic

SOLUTIONS OF YULE-WALKER EQUATIONS

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polynomial cannot have unstable zeros but has both stable and unit circle zeros. Finally, the number of unit circle zeros of the

characteristic polynomial is characterized by a rank test that is the same as the rank test for the unique solution case, and the

characteristic polynomial of the AR system defined by the minimal norm solution is shown to have the most number of stable zeros

and the least number of unit circle zeros.

APPENDIX: PROOFS OF THEOREMS

PROOF OF THEOREM 1.

Cp ¼ Cr

condition rk([G EG ...

coprime, and thus, since the transfer function is stable, e(z) has to be stable.

If rk([G EG ... Erp?1G]) 6¼ rk(Cp), then rk([G EG ... Erp?1G]) < rk(Cp) and the pair [E,G] is not reachable. Then it is a well known fact

from linear systems theory (see Kailath (1980)) that there exists a non-singular matrix?T such that

?

where the pair ½?E11;g1? is reachable, every eigenvalue of?E22is an unreachable mode of the pair [E, G], and the number of rows

(columns) of?E11is equal to rk([G EG ... Erp?1G]).

As the pair ½?E11;g1] is reachable and the corresponding transfer function [I??E11z?1g1] has its poles outside the unit circle (as etis

persistently exciting and the output is stationary), all eigenvalues of?E11must be of modulus smaller than 1.

Let k be an eigenvalue of?E22. Since k is an unreachable mode of the pair [E,G], then its associated eigenvector x satisfies

x?E ¼ kx?, x?G ¼ 0. Using (18) it is immediate that (1 ? |k|2)x?Cpx ¼ 0. Since Cpis non-singular and thus positive definite, it follows

that |k| ¼ 1. This proves that all eigenvalues of?E22are on the unit circle.

Note that E and Cphas the same number of rows and columns and the number of rows (columns) of?E11 is equal to rk([G

EG...Erp?1G]), it follows that the number of rows (columns) of?E22is equal to rk(Cp) ? rk([G EG ... Erp?1G]). This fact together with

the facts that all eigenvalues of?E11are inside the unit circle and all eigenvalues of?E22are on the unit circle implies that all zeros of the

characteristic polynomial e(z) lie outside or on the unit circle, with the number of unit circle zeros equal to rk(Cp) ? rk([G

EG...Erp?1G]).

If the associated AR system (14) is stable, then the linearly singular part of the stationary solution is zero. Thus,

p. Now, by the assumption that Cpis non-singular, it follows from (19) that rk([G EG ... Erp?1G]) ¼ rk(Cp). Conversely, the

Erp?1G]) ¼ rk(Cp) says that the pair [E, G] is reachable, which is equivalent to saying that (kI ? E, G) is left

?TE?T?1¼

?E11

0

?E12

?E22

?

;?TG ¼

g1

0

??

;

ð26Þ

h

PROOF OF THEOREM 2.

proof Theorem 5 in Deistler et al. (2010) that

Define^F ¼ Op^EOT

pand?F ¼ Op?EOT

p, where OT

pOp ¼ I and OT

p

K1

0

0

0

??

Op ¼ Cp. It has been established in the

^F ¼

^F11

0

^F12

^F22

??

;

ð27Þ

where all eigenvalues of^F22are within the unit circle (no matter where the eigenvalues of^F11lie), and

?F ¼

?F11

0

?F12

?F22

??

;

ð28Þ

where?F11 ¼^F11. Note that^F22and?F22have the same number of rows and columns as the number of zero eigenvalues of Cp.

Partition Opaccording to^F as

?

Using (13) and noting that^F22and U have the same number of columns, it is easy to show that

?

We claim that O216¼ 0. To obtain a contradiction, assume that O21¼ 0; then it follows from OT

2

Op¼

O11

O21

O12

O22

?

:

ð29Þ

?F ¼

^F11

0

^F12þ O11U

^F22þ O21U

?

:

ð30Þ

pOp ¼ I that O12¼ 0.

This implies that Op ¼

O11

0

0

O22

??

. Then

OT

11K1O11

0

0

0

??

¼

c0

...

...

???

c0

???

cp?1

...

...

c0

..

.

cT

p?1

??? ???

66664

3

77775

. If pr ? rk(Cp) ? r, then the bottom right

zero block of Cphas at least r rows and columns and thus c0¼ 0 by looking at the matrix c0sitting at the bottom right corner of Cp,

which is a contradiction to the definition of the AR system (1). If pr ? rk(Cp) < r and p > 1, the bottom right zero block of Cphas

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Page 7

strictly less than r rows and columns and thus c0sitting at the right corner of Cpmust have the form c0¼

positive definite. However, since OT

11K1O11 is positive definite and its dimension is equal to rk(Cp) (which satisfies that

rk(Cp) > (p?1)r ? r), c0, which sits at the top left corner of Cpmust sit at the top left corner of OT

definite, another contradiction results. This proves the claim.

Since O216¼ 0, then the matrix pair^F22;O21has at least one reachable mode. Therefore, there always exists a matrix U such that

^F22 þ O21U will have at least one eigenvalue that lies outside the unit circle. This completes the proof.

c01

0

0

0

??

, which is not

11K1O11, and thus must be positive

PROOF OF THEOREM 3.

shown that rkð½G^EG ???^Erp?1G?Þ ¼ rkðCpÞ.

Conversely, suppose that rkð½G^EG ???^Erp?1G?Þ ¼ rkðCpÞ, it is left to show that the minimal norm solution defines a stable AR

system (14). As mentioned in the proof of Theorem 2, it has been established in the proof Theorem 5 in Deistler et al. (2010) that

If the minimal norm solution defines a stable AR system (14), then as in the proof of Theorem 1, it can be

^F ¼ Op^EOT

p¼

^F11

0

^F12

^F22

??

;

ð31Þ

where all eigenvalues of^F22are within the unit circle (no matter where the eigenvalues of^F11lie).

It has also been established in the proof of Theorem 5 in Deistler et al. (2010) that the matrix^F satisfies:

?

where rk(K1) ¼ rk(Cp) and the first and last matrices in the above equation are partitioned the same way as^F.

Let^G ¼ OpG; it is easy to show that^GT¼

Since

K1

0

0

0

?

?^F

K1

0

0

0

??

^FT¼ OpGGTOT

p¼

R1

0

0

0

??

;

ð32Þ

^GT

1

0

??; R1 ¼^G1^GT

1.

rkð½G^EG ???^Erp?1G?Þ ¼ rkðOp½G^EG ???^Erp?1G?Þ ¼ rkð½^G1^F11^G1 ???^Frp?1

11

^G1?Þ;

there holds

rkð½^G1^F11^G1 ???^Frp?1

11

^G1?Þ ¼ rkðCpÞ ¼ rkðK1Þ:

This equation implies that the pair ½^F11;^G1? is reachable. Then it follows from K1?^F11K1^FT

within the unit circle.

11¼^G1^GT

1that all eigenvalues of^F11are

h

PROOF OF THEOREM 4.As in the proof of Theorem 3, we have

^F ¼ Op^EOT

p¼

^F11

0

^F12

^F22

??

ð33Þ

and

K1

0

0

0

??

?^F

K1

0

0

0

??

^FT¼

^G1

0

??

^G1

0

??T

;

ð34Þ

where all eigenvalues of^F22are within the unit circle and K1is positive definite, rk(K1) ¼ rk(Cp) and the first and last matrices in the

above equation are partitioned the same way as^F. Since Cpis singular,^F22always exists and has at least one eigenvalue inside the

unit circle. This proves that the characteristic polynomial of the AR system defined by the minimal norm solution will always have a

stable zero.

If the pair ½^F11;^G1? is reachable, all eigenvalues of^F11must lie within the unit circle. This implies that^E can only have eigenvalues

inside the unit circle, which contradicts the assumption that the minimal norm solution does not define a stable AR system.

Hence, the pair ½^F11;^G1? is not reachable. Then there exists a non-singular matrix?T such that

?T^F11?T?1¼

^F11;1

0

^F11;2

^F11;3

??

;?T^G1¼

g1;T

0

??

;

ð35Þ

where ½^F11;1;g1;T? is reachable. Thus, as the corresponding transfer function [I ?^F11;1z??1g1;Thas its poles outside the unit circle, all

eigenvalues of^F11;1must be of modulus smaller than 1.

Again, partition?TK1?TTconformally as

?TK1?TT¼

?K11

?KT

?K12

?K22

12

??

;

ð36Þ

and define?Rijas the associated submatrix of?T^G1^GT

1?TT.

SOLUTIONS OF YULE-WALKER EQUATIONS

J. Time Ser. Anal. 2011, 32 531–538

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Noting that?T^G1^GT

1?TT¼

?R11

0

0

0

??

. It follows from K1?^F11K1^FT

11¼^G1^GT

1that

?K11

?KT

?K12

?K22

12

??

?

^F11;1

0

^F11;2

^F11;3

??

?K11

?KT

?K12

?K22

12

??

^F11;1

0

^F11;2

^F11;3

??T

¼

?R11

0

0

0

??

;

ð37Þ

which implies

?K22?^F11;3?K22^FT

11;3¼ 0:

Let x be an eigenvector of^F11;3corresponding to eigenvalue k. It is immediate that ð1 ? jkj2Þx??K22x ¼ 0. Since?K22is positive

definite, it follows that we must have |k| ¼ 1. Since the eigenvalues of^E are the union of the eigenvalues of^F11;1,^F11;3and^F22,

conclusion [i] is proved.

It has been shown that the number of eigenvalues of^E that lie on the unit circle is equal to number of columns (or rows) of the

matrix

^F11;3, whichis equal torkðK1Þ ? rkð½g1;T^F11;1g1;T???^Frp?1

rkðK1Þ ? rkð½g1;T^F11;1g1;T???^Frp?1

Using (30) and the facts that?F11 ¼^F11 and the eigenvalues of^F22 are all within the unit circle, conclusions [iii] and [iv] are

immediate.

11;1g1;T?Þ. conclusion [ii]followsfrom the factthat

11;1g1;T?Þ ¼ rkðCpÞ ? rkð½G^EG ???^Erp?1G?Þ.

h

Acknowledgements

Support by the ARC Discovery Project Grant DP1092571, by the FWF (Austrian Science Fund) under contracts P17378 and P20833/

N18 and by the Oesterreichische Forschungsgemeinschaft is gratefully acknowledged. The authors are indebted to Per Enqvist at

KTH, Stockholm for drawing their attention to the article by Inouye (1983).

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