Stationary uncertainty frontiers in macroeconometric models and existence and uniqueness of solutions to matrix Riccati equations

Journal of Fkonomic Dynamics and Control 11 (1987) 93-116. North-Holland

STATIONARY UNCERTAINTY FRONTIERS IN MACROECONOMETRIC MODELS AND EXISTENCE AND

UNIQUENESS OF SOLUTIONS TO MATRIX RICCATI EQUATIONS

Cuong LE VAN* CNRS-CEPREMAP, 75013 Paris, France

Received May 1985, final version received August 1986

In this paper, we characterise the stationary uncertainty frontiers in dynamic macroeconometric models. This frontier, the definition of which is due to Deleau and Malgrange (1979), is the set of the least positive semi-definite covariance matrices of the objective variables stabilised by stationary policies. We prove that this frontier coincides with the set of the covariance matrices stabilised by optimal stationary non-singular policies. We prove also that solving the matrix Riccati equations with stable feedback controls is equivalent to minimising a linear form in a closed convex set of covariance matrices of the objective variables. As corollaries of this proposition, we have results on the existence and uniqueness of solutions of matrix Riccati equations.

1. Introduction

The application of optimisation methods to macroeconometric models is now current practice. The early 1970’s saw a ‘flurry’ of papers dealing, on the one hand, with the stability properties of optimal dynamic control [e.g., Chow (1970), Rappaport and Silverman (1971), Chow (1972), Aoki (1973), Tumovsky (1974)], on the other hand, with the existence and uniqueness of solutions of matrix Riccati equations which arise in optimal dynamic control with infInite time horizon [e.g., Wonham (1968), Kucera (1972) Payne and Silverman (1973)]. Deleau and Malgrange (1979) introduced the concept of the stalionary uncertainty frontier which is the set of least positive semi-definite covariance matrices of the endogenous variables stabilised by the stationary policies. In order to characterise this uncertainty frontier, they implicitly assumed that it coincided with the set of covariance matrices of objectives stabilised by various quadratic welfare functions. Hence they used results on existence and uniqueness of solutions of matrix Riccati equations.

*I would like to thank Pierre Malgrange who introduced me to the subject. I am also indebted to an anonymous referee who contributed to improve some results of this paper.

0165-1889/87/$3.5001987, Elsevier Science Publishers B.V. (North-Holland)

94 C. Le Van, Sfafionary uncerraing frontiers

Our analysis proceed in the opposite direction. We characterise the stationary uncertainty frontier independently of the quadratic cost function and of results on matrix Riccati equations. Then, we show that the uncertainty frontier coincides with the set of covariance matrices of objectives stabilised by quadratic welfare functions which are called by us ‘non-singular’ (the definition will be given in section 2). We show also that solving the matrix Riccati equation yielding a stationary feedback control is equivalent to minimising a linear form on a closed convex set (contained in the cone of positive semi-definite matrices) whose solutions belong to the uncertainty frontier. One can deduce very easily results on the existence of solutions to matrix Riccati equations.

The solution of this kind of equation uses the well-known ‘backward’ algorithm. In this paper, we study the convergence of this algorithm and the nature of the solution it yields. We give also the necessary and sufficient conditions for the existence and uniqueness of solutions of matrix Riccati equations corresponding to control costs associated with the objective and instrument variables. These results are standard; but we think that our way of proving them is relatively new and simpler than those one can find in the optimal control literature.

Our paper is organised as follows. In section 2, we characterise the stationary uncertainty set (which is the set of covariance matrices of targets stabilised by stationary feedback controls) and the stationary uncertain frontier; a link will be made between this frontier and the covariance matrices of objective variables stabilised by optimal feedback controls. In section 3, we prove the equivalence between two problems: solving matrix Riccati equations with stationary linear feedback control and minimising a linear form on the uncertainty set; then we deduce results on the existence of solutions of matrix Riccati equations; we also study the usual algorithm for solving this equation; and we give necessary and sufficient conditions for the existence and uniqueness of solutions of matrix Riccati equations corresponding to matrix costs associated with the objectives and the instruments.

2. The stationary uncertainty set and the stationary uncertainty frontier

2.1. Definitions and assumptions

Let us consider the following linear econometric dynamic model in state variable form:

x,=/ix,-,+Bd,+s,, 0)

C. Lx Van, Stationary uncertain~ frontiers 95

where x, is the n-dimensional vector of objective variables at period t, d, is the p-vector of instruments (variables subject to control), with p 2 n, and S, is the vector of n random variables.

Model (1) can be considered as a linearisation of a non-linear model around a steady state (reference trajectory). The following assumptions are usually made:

I-4. The rank of matrix B is equal to p.

H,. E(s,) = 0,Vt; E( s,,s,!) = 0, Vt, # t,; E(s,s,‘) = S; rank S = n.

It is well-known that the feedback control associated with a quadratic objective function is linear with respect to x,-~. Following Deleau and Malgrange (1979), we use the following feedback control policies:

d, = T,(Ax,-, + s,).

Then covariance matrices 2, of x, verify

2’,= (A + BT,A)2,&4 + BT,A)‘+ (I+ BT,)S(I+ Br,)‘,

If we assume that the policy is stable, i.e., T, converges to a matrix T yielding a stable closed-loop system, then the matrices 2, converge to Z verifying

2=(A+BTA)Z(A+BTA)‘+(I+BT)S(I+BT)’, (2)

where the matrix (A + BTA) is stable. In this case, the feedback control T will be called stationary. Let us define the stationay uncertainty set, C, as the set of covariance matrices 2, associated with the whole set of stationary feedback controls.

The uncertainty fr@er, 9 [the definition of which is due to Deleau and Malgrange (1979)], is the set of elements of C which are the ‘least positive semi-definite’, i.e., those for which there is no matrix r in C such that X - r is positive semi-definite. 9 can be viewed as the Pareto set of 8 with the partial preorder 2 defined on 8 as follows:

2 2 r if 2 - r is positive semi-definite.

Define -Y as the set of diagonal elements of matrices 2 of 6’.

96 C. Le Van, Stationary uncertainty frontiers

It will be proved that 8 is non-empty by the following assumption which is well-known in optimal control theory:

I-b There exists a matrix G such that the matrix A + BG is stable.

In that case, we say that (A, B) is stabilisable Throughout this paper, the notations

A-B2Q or ArB

for symmetric matrices A and B mean that

A - B is non-negative definite.

Given a matrix C, one can define a linear form, on the space of matrices, which associates to every matrix A the number denoted (C, A):

(C, A) = c CijAij. i.j

If C and A are symmetric, then

(C, A) = trace (CA).

If C and A are symmetric, non-negative definite, then the eigenvalues of C and A are positive, and consequently

CkO, ArO * (C,A)rO.

A’ denotes the transpose of A and A* the conjugate transpose of A (if A is a complex matrix).

Remark 1. The use of our feedback rule assumes that we have full information. But one can characterise the uncertainty frontier corresponding to the conventional feedback (without information) by changing slightly the statement of Proposition 1 below. It will be pointed out that a covariance matrix corresponding to a feedback without information can be viewed as the one corresponding to a feedback with full information (Remark 3). To make things simpler, we would rather deal, throughout this paper, with the uncertainty frontier with full information. As in the usual case, the matrix A is not assumed to be non-singular.

2.2. Characterisation of stationary uncertainty set and uncertainty frontier

Proposition 1. Let R be a matrix (n -p) X n with rank (n -p) such that RB = 0. Under assumptions HI and Hz, 8 is the set of positive semi-definite

C. Ix Van, Stationary uncertainty frontiers 97

symmetric matrices verifying

RIR’ = RAXA’R’ + RSR’. (3)

Proof. The existence of a matrix R such that RB = 0 is ensured by assumption Hi. Obviously, a matrix in C verifies relation (3).

Conyersely, let Z be a symmetric positive semi-definite matrix verifying (3). We shall show that there exist, first, a matrix N such that RN = R, secondly, a matrix T such that N = I + BT, and that the matrix NA is stable. Then Z is the stationary covariance matrix associated with the feedback control defined by d, = T( Ax,- 1 + s,).

We observe that there exist matrices u and + such that

zl=CMt, AZA’ + S = @‘.

The relation (3) can be rewritten as

Raa’R’ = R++‘R’.

Then we have

where # is an orthogonal matrix. It is easily proved that, if we define

N = CT+-‘$I-‘,

then we have

2 = NAZA’N I + NSN’, RN=R.

Since RB = 0 and the rank of B is p, there exists a matrix T such that N-I=BT.

We prove now that NA is stable. If not, there exist a vector x # 0 and a complex number X such that

A’N’x = Xx, IhI 2 1.

Then we have

x*2x = Ih12x*2x + x*NSN’x.

By assumption H,, one has N’x=O, then Xx=0, i.e., x=0. q

98 C. Le Van. Stationary uncertaitgv frontiers

Remark 2. The rank of every matrix in 8 is greater than (or equal to) n -p.

Assumption H, is not necessary for the proof of Proposition 1. The existence of 2 verifying (3) implies that there exist G = TA such that A + BG is stable, as it is pointed out in the proof.

Corollary 1. Under assumptions H,, H2 and H,, d and V are non-empty, closed and convex. m

It is obvious that 8 is convex and closed. Y is convex as the projection of 8 in the subspace of diagonal elements of matrices of 6. We prove now that Y is closed. Let {(Zyi)},, be elements of Y converging to (ri). For v large enough, one has

Vu, OIZ~~I+T~+E for i=l,..., n,

since

the matrices 1’ are bounded for v large enough. Then one can find a subsequence (2”“) which converges to a matrix B E 8. And obviously Zii = ri for i=l,...,n.

We prove now that 6’ (and a fortiori, “v) is non-empty under assumption H,. Let G be a matrix such that A + BG is stable. Then there exists a unique non-negative symmetric matrix verifying

X = (A + BG)X( A + BG)’ + S.

If R is a matrix (n - p) x n with rank (n -p) such that RB = 0, then one has

RXR’ = RABA’R’ + RSR’.

By Proposition 1, we find matrices N and T verifying

N=I+BT,

such that NA is stable, and

x=(A+BTA)Z(A+BTA)‘+(I+BT)S(I+BT)’;

in other words, 2 belongs to 8. •I

The following properties are demonstrated in the appendix.

C. L.e Vat,, Slalionaty uncertainty frontiers 99

Property 1. Zf X and r (I # T) belong to Q, then V’h ~10, l[, Zx = AZ + (1 - h)~ (which belongs to 8) is of rank strictly greater than (n - p).

Property 2. Zf Z E 8 and has rank greater than n -p, then there exists r E 8, r + 2, such that JI 2 7.

Property 3. Zf Z and r are two diflerent matrices of 8 with rank equal to n - p, then they are not comparable.

Property 4. Even symmetric matrix of rank n - p verifying RZR’ positive definite is non-negative definite (R is defined as in Proposition 1).

Proposition 2. Under assumptions H, and H,, 9 is the set of symmetric matrices of rank n - p verifying

RZR’ = RAZA’R’ + RSR’,

RER’ is positive definite.

R is defined as in Proposition 1. In that case, the matrix N, such that 2 = NAZA’N’ + NSN’ is uniquely

determined by 1.

Proof. It results from Properties 2, 3, and 4. The proof of the last statement is given in the appendix.

Remark 3. If we use the feedback rule without information defined by

d, = Gx,-~,

then the covariance matrix Z verifies

2 = (A + BG)z( A + BG)’ + S.

In that case, define a symmetric non-negative definite matrix 2 as

2=2-S.

Proposition 1 can be restated as follows:

Proposition 1 I. Let R be a matrix (n -p) X n with rank (n - p) such that RB = 0. Under assumptions HI and Hz, the uncertainty set without information, 6”, is the set of symmetric, non-negative matrices X verifying (i)

z=2+s.

100 C. Lx Van, Slalionaty uncertainly frontiers

and (ii) 2 is non-negative definite symmetric,

such that RZR’ = RA(e + S)A’R’.

In the proof of the non-emptiness of &, we have pointed out that every covariance matrix corresponding to a feedback without information is also the covariance matrix corresponding to another feedback with full information.

2.3. Stationary uncertainty frontier and optimal control

Let us consider the minimisation problem, in finite time horizon, of the welfare function

c-1

under the constraint [which is model (l)]

x, = Ax,el + Bd, + s,,

with positive (semi)definite matrix K. This problem has been studied by several authors [for example, Chow

(1972), Tumovsky (1974), Preston (1979)] in deterministic or stochastic con- texts (i.e., S, = 0 vs S, # 0).

We retain here the approach of Chow (1972). One has to minimise

,cl (KY 2,) E [fl s( Kijz,,ij) = i trace Kz,, E ‘,’ r-1

under the constraint

I,=(l+BT,)AZ,-,A’(I+BT,)‘+(I+BT,)S(I+BT,),

where 2, is covariance matrix of objective variables associated with the feedback control

d,= T,(Ax,-~+s,).

S is covariance matrix of s,. The Lagrangian expression, with H, denoting the symmetric matrices of Lagrange multipliers, is

L= 2 (K,X,) - i trace [H,(Z,- (I+BT,)A~,-,A!(I+BT,)’ t-l 1-l

- (I+BT,)S(I+BT,)‘)].

C. Le Van, Stationary uncertainy frontiers 101

writing 8L

ap, =0 and ;=O,

I

and keeping in mind the constraint, we are lead to the following system:

H,=K+A’(I+BT,)‘H,+,(z+BT,)A,

B/H,+ (B’H,B)T,=O,

HT= K,

z,= (I+BT,)AZ,-,A’(z+BT,)‘+ (z+Br,)s(z+BT,)‘,

x,=0.

In the steady state of the system, we have

x=(z+BT)AZA’(z+BT)‘+(z+BT)S(z+BT)’,

H=K+A’(Z+BT)‘H(Z+BT)A,

B’H + (B’HB)T= 0.

One says that H is a solution of matrix Riccati equation if H and T verify the two last equations. If H is such that B’HB is invertible, then T is said to be non-singular optimal policy; moreover if (I + BT)A is stable, then T is called stationary, and in that case there is a unique solution of the first equation which is non-negative.

Proposition 3. Under assumptions HI and Hz, X E 9, if and or+ if there exists a non-negative symmetric weighting matrix K which yields a non-singular sta- tionav optimal policy and such that the covariance matrix associated is 2.

Proof. If T is non-singular stationary optimal policy, then it is obvious that the covariance matrix associated with T belongs to 9; indeed if T= - (B’HB)-‘B’H, one has (I + BT) B = 0. By assumption Hi, Z + BT is of rank n -p. So is t: which belongs to 9 by Proposition 2.

Suppose now that I: ~3’. By Proposition 2, there exist a (n X n) matrix N with rank n -p, a matrix T (p x n), such that

N=I+BT,

RN=R,

NA is stable,

102 C. Lx Van, Stationary uncertainty frontiers

and

Z=NAXA'N'+NSN'.

By Proposition 3’ in Deleau and Malgrange (1979) there exists a matrix L (p xn) with rank p such that

LN=O,

LB is invertible, and

N=I-B(B'KB)-'B'K with K=L'L.

One can easily verify that K is a solution of the matrix Riccati equation associated with itself. 0

3. Existence and uniqueness of solutions of matrix Riccati equations

By Proposition 3 one can construct the stationary uncertainty frontier of a given dynamic linear model. But it is necessary to solve a non-linear matrix equation, called matrix Riccati equation. The existence and uniqueness of solutions to this kind of equations have been studied by Wonham (1968), Kucera (1972), Aoki (1973), Payne and Silverman (1973). For a survey of these results one can refer to Preston (1979). The techniques for proving the existence of the solutions are somewhat complicated. In our paper, the existence results can be deduced easily from a proposition proving that solving the matrix Riccati equation with stationary feedback control is equivalent to minimising a linear form on the stationary uncertainty set &‘. Some of our results on the uniqueness of the solutions do not depend on the properties of the uncertainty set; but the techniques of demonstration used here are simpler than those one can find in the literature of optimal control theory. Before doing that we need the following:

Lemma 1. If H is non-negative definite, T and G are such that

BJH+(B'HB)T=~, G=TA, then

(A+BG)'H(A+BG)s(A+BF)'H(A+BF),

(I+BT)'H(I+BT)~(I+Bu)'H(I+Bu),

for every (p x n) matrix F and every (p X n) matrix U.

C. Le Van, Sralionary uncerlainp fronliers 103

Proof. We have the following identities:

(A + BG)‘H(A + BG) = (A + BF)‘H(A + BF)

-(F- G)‘(B’HB)(F- G),

(z+BT)‘H(z+BT)=(z+BU’)H(z+BU)

-(T- U)‘(B’HB)(T- u),

where G and T are defined as previously. 0

Proposition 4. The following statements are equivalent:

(i) The matrix Riccati equation, associated with a non-negative matrix K, admits a solution yielding a stationary feedback control.

(ii) Minimising (K, 2) on d has a solution.

Proof. It is obvious that (i) is a necessary condition for problem (ii). Conversely, let H be a solution of the matrix Riccati equation:

H=K+(A+BG)‘H(A+BG), G=TA, B’H+(B’HB)T=O,

such that A + BG is stable. Then there exists a non-negative matrix 2 verifying

2= (A + BG)z(A + BG)‘+ (I+ BT)S(Z+ ST)‘.

2 belongs to 6” by the definition of 8. Let 7 be another element of 8,

T= (A + BF)T(A + BF)‘+ (I+ BU)S(Z+ BU)‘,

with F = UA, A + BF stable. By Lemma 1, one has

H=K+(A+BG)‘H(A+BG)<K+(A+BF)‘H(A+BF).

Let us define

C=A+BG, D=A+BF, N=Z+BT, M=Z+BU.

104 C. L.e Van, Stationary uncertain~ frontiers

Then we have

H= fj (C”KC’) s g (D’iKD’), i-0 i-0

x = f (ciNsN’c’i), i-0

7 = E ( D~M~M/D’~), i-0

and, using Lemma 1,

(K, T) = trace f K( D’MSM’D”) = trace f (MSM’)( D”KD’) i-0 i-0

r trace( MSM’H) = trace( SM’HM) 2 trace( SN’HN)

= C trace( NSN’C”KC’) i-0

= fj trace(C’NSN’C”K) = (K, 2). 0 i-0

Remark 4. From Proposition 4, it is obvious that the curve of the variance of two objectives stabilised by one instrument, with a family of quadratic welfare functions whose weight matrices are diagonal, is convex with respec to the origin. [See Taylor (1979), Deleau, Le Van and Malgrange (1984).]

We shall prove now the existence of solutions of matrix Riccati equations We proceed by two steps: first, using Proposition 4, we state the result wher the weighting matrix K is positive definite; then, we take a sequence o positive definite matrices {K,} converging to K, when this one is jus semi-definite; the result is obtained by moving to the limit.

Corollary 2. Under assumptions HI and H3 and if K is positive dejinite, ther the matrix Riccati equation has a unique solution and it yields a stationag feedback control.

Proof. One can always associate to matrices A and B an econometric mode verifying assumption H, and, consequently, an uncertainty set 6’.

C. Lu Van, Staiionary uncertainfy frontiers 105

We can observe that, since K is positive definite, every solution H of the matrix Riccati equation yields a stable feedback control. Indeed, let H be a solution:

H=K+(A+BG)‘H(A+BG),

and suppose that A + BG is unstable, i.e., that there exists x # 0, and A, with ]A] r 1, such that

(A + BG)x=Ax.

Then

x*Hx =x*Kx + 1X12x*Hx;

that is a contradiction. We prove now, that min BE #( K, Z) has a solution. Without loss of gener-

ality, one can assume that K is diagonal. Then

Since 7v is a closed convex set of RI and K is positive definite, this problem has always a solution. We prove now that this solution is unique. Indeed if there are two solutions Z, and Z,, then X, = (1 -X)X, + XX,, X ~]0,1[, is another solution. By Property 1, Z, is of rank greater than n -p, therefore, by Property 2, there exists another matrix r in d such that X, 2 7; it implies Zxii r rii, with at least a strict inequality, and we have

c Kii~ii < c KiiZlxii; i i

that is a contradiction. The unique solution X must belong to 9, by Property 2; its rank is equal to

n -p. By Proposition 2, the feedback control is then uniquely determined from X; this implies the uniqueness of the solution of the matrix Riccati equation. 0

Corollary 3. Under assumption HS and if K (non-negative de$nite) is such that B’KB is invertible, then the matrix Riccati equation,

H=K+(A+BG)‘H(A+BG), G=TA,B’H+(B’HB)T=O,

admits at least a non-negative definite solution.

Proof. Let { K, } be a sequence of positive definite matrices converging to K, and {H,,,} the sequence of solutions of matrix Riccati equations associated

106 C. L.e Van, Stationary uncertainty frontiers

with {K,,}. Under assumption H,, there exists a matrix F such that A + BF is stable. Then by Lemma 2, one has

H,,I f (A + BF)“K,,,(A + BF)‘. i-0

Thus, the matrices H,,, are bounded for m sufficiently large. One can suppose that H,,, converges to H 2 0. As H,,, 2 K,, for every m, then H 2 K. Since B’KB is invertible, (B’HB)-’ exists and G,,, converges to G = -(B’HB)-‘B’HA. Obviously H= K+(A + BG)‘H(A + BG). 0

Remark 5. If every (p xp) submatrix of B is invertible, then B’KB is invertible iff the rank of K is greater than, or equal to, p [Garbade (1976)].

In the following Propositions 5 and 6, which are standard [e.g., Preston (1979)], one notices that the strong condition of positive definite weighting matrix K, in Corollary 1, will be replaced by other conditions (detectability, observability) which ensure the uniqueness of stable non-negative solutions. But the existence conditions will be the same as in Corollary 1 (stabilisability and invertibility of B’KB) and seem to be the ‘best ones’ to obtain the existence results. 0

The following corollary reveals the nature of the solution yielded by the usual algorithm:

Corollary 4. Under assumption H3 and if K is such that (B’KB)-’ exists, then there exists a solution H, which is the least non-negative definite one in the set 01 the non-negative solutions of the matrix Riccati equation associated with K. The sequence { Hi }, defined by

H,=K,

Hi+l = K+ (A + BGi)‘Hi(A + BG,), Vi21,

Gi = - ( B’H,B)-‘B~H,A,

converges to H,.

Proof. Let H be a solution, which exists by Corollary 3. Observe first that Hi I H and H, s H2. By Lemma 1, one has, for i 2 2,

(A + BGi-,)‘Hi-,(A + BGi-1) I, (A + BGi)‘Hi-,(A + BGi),

(A + BGi-,)/Hi-,(A + BGi-1) I (A + BG)‘Hi-,(A + BG).

C. L.e Van, Stationary uncertainty frontiers 107

Then

Hi-Hi+ls (A +BGi)‘(Hiel-Hi)(A +BG,),

Hi-Hs(A+BG)‘(H,-,-H)(A+BG),

which implies

Hil Hi+l and Hi- H<O, Vi21.

The sequence {Hi} being increasing and bounded by H converges to a matrix H, r 0 [Kantorovich and Akilov (1964)] which is a solution of the matrix Riccati equation, and H, I H. 0

When the matrix K is only non-negative definite, the matrix Riccati equation associated with a cost function on the objective variables can admit an unstable solution [e.g., Holbrook (1972) and Tumovsky (1974)]. For that reason it is usual to introduce a cost function on the objectives and on the instruments. In the case of the deterministic optimal control with finite time horizon, the problem is to minimise

; (x:Kx, + d,‘Qd,), K20, Q20, r-0

under the constraints

x,=Ax,-,+m ,’

If we factorise K and Q in

K = c”ld, Q= D’D,

with

rankc=rankK, rankD=rankQ,

the equivalent problem is to minimise CTpOy:y,, under the constraints

x,=Ax,-,+Bd,, h=[jA]Xt-l+ [iB]d,.

The associated stochastic optimal control is the following:

min 2 traceE(y,‘y,), r-o

108 C. L.e Van, Stationary uncertainty frontiers

under the constraints

where sir, szt, ss, are random vectors of dimension n, p, n. The constraints can be rewritten as follows:

Z,=Az,~l+Af,+s,,

with

The associated matrix Riccati equation is

with 0: 0 K= y-l,,; . -[ 1

It is easily verified that

ii’I?ii = B’KB + Q.

If (B’KB + Q) is invertible, then, by Corollary 3, there exists a solution H of the matrix Riccati equation, H and the feedback control having the following forms:

Hl 0 H= 0 I,+, ’ [ 1 G = [G,,O], G,=(pXn)-matrix.

Hl and G, solve the following system:

H,=H,-K,

H2 = (A + BG,)‘H,(A + BG,) + GiQG, + K,

G, = - ( B’H,B + Q)-‘B;HA.

The two last relations define the matrix Riccati equation ‘of second type’.

C. Le Van, Stationary uncertainty frontiers 109

There exists, if B’KB + Q > 0, a one-to-one correspondence between the solutions of

H=K+(~+~?G)‘H(A+~~G),

G = - ( &H@-‘&HA”,

and Hz = (A + BG,)‘H,(A + BG,) + G;QG, + K,

G,= -(B’H,B+Q)-‘B’H,A.

The results given below concern the existence and uniqueness of the solutions of the matrix Riccati equations ‘of second type’. They need some additional assumptions. But before doing that, we recall the following definitions which are well-known in optimal control theory:

Dejnition 1. (c, A) is detectable if, for evev x0,

d;A”x,, + 0 =a A”x,-,O,

n+oo n+oo.

Definition 2. (6, A) is observable if

is of rank n.

A useful characterisation of detectability and observability was provided by Ha&us (1969):

(C”,A)isdetectable e 2x:, ,h,rl j , x = 0,

(c, A) is observable = cTx=o Ax=Xx * x = 0.

Proposition 5. The matrix Riccati equation ‘of second type ‘,

Hz = (A + BG,)‘H,( A + BG,) + GiQG, + K,

with

G,= -(B~H,B+Q)-‘B/H,A, K=&&o, ~>0,

110 C. Le Van, Stationary uncertainty frontiers

admits a unique solution which is positive semi-definite, and the matrix A + BG is stable if and only if (A, B) is stabilisable and (C, A) is detectable.

Proof. (i) Since Q is positive definite, there exists always a solution if (A, B) is stab&able, after the remarks given above and Corollary 3.

It will be shown that if (c”, A) is detectable, then A + BG is stable, and consequently the solution is unique.

Suppose that there are x # 0, X (1X1 2 l), such that

(A + BG,)x=Xx.

Then

x *H2x = IX12x *H2x + x *GiQGlx + x *Kx.

It implies G,x = 0, cx=o.

Then (6, A) is not detectable, in contradiction with the assumption. (ii) Conversely, suppose that there is a unique solution H,, which is positive

semi-definite, and the matrix A + BG, is stable. Suppose also that (C?, A) is not detectable, i.e., there exist x0 # 0 and A, such that

Ax,=Xx,, 1X121, dx, = 0.

Since the solution H2 is unique, the sequence { Li} defined by

L1=K=d’C”,

L,+,=(A+BF,)‘L,(A+BF,)+F,‘Q&+K for i>l,

where

4= -(B’L,B+ Q)-lB’L,A

converges to H2, the sequence {I;;} converges to G,, by Corollary 4. Since

6x0 = 0, LiXO = 0, Vi21, one has

H2x,, = 0, G,x, = 0,

and consequently

Ax, = (A + BG,)x,, = Ax,, IhI 2 1,

and we have a contradiction. •I

C. Le Van, Slalionaty uncerlain!v frontiers 111

One can easily prove the following:

Proposition 5’. The matrix Riccati equation ‘of second type’,

with

H2 = (A + BG,)‘H,( A + BG,) + G;QG, + K,

G, = -(B’H,B+ Q)-‘B’H,A,

K= C’&O,

Q> 0,

admits a unique, positive definite, solution such that the matrix A + BG, is stable if and only if (A, B) is stabilisable and (c, A) is observable.

But in the practice, the matrix Q is only non-negative definite, even null. So, we need the following proposition:

Proposition 6. The matrix Riccati equation ‘of second type’,

H2 = (A + BG,)‘H( A + BG,) + G;QGl + K,

with

G, = - ( B’H,B + Q)-‘B;H,A,

K= C&O, Q = D’D 2 0,

B’KB + Q > 0,

admits a unique solution positive (semi)definitc with a stable matrix A + BG, if and only if (A, B) is stabilisable a;d (C $ DG, A + BG) is (detectable) observable for evety G; the matrices C and D are of dimension (n + p) X p and defined as follows:

Proof. Observe that the Riccati equation can be rewritten as

H,=(A+BG,)‘H,(A+BG,)+(C+IjG,)‘(C+DG,).

Apply the same techniques as in Proposition 5. Cl

112 C. L4 Van, Staiionafy uncertainty yfrontiers

Remark 6. The algorithm the most usually utilised for solving the matrix Riccati equations are the one proposed in Corollary 4. This algorithm gives only the least positive semi-de&rite solution which can yield an unstable closed-loop system, i.e., the associated covariance matrix may not exist [Holbrook (1972) and Tumovsky (1974)]. The weakness of this algorithm is emphasised by another practice [e.g., Taylor (1979), Deleau, Le Van and Malgrange (1984)] consisting of controlling only the convergence of the variances of objective variables under interest. In order to apply this algorithm, one has to verify if the system verifies the necessary and sufficient conditions for the existence and uniqueness of the solutions or to verify numerically that the matrix A + BG is stable. If it is not the case, use the algorithm of Corollary 3: take a sequence of positive definite matrices K,, converging to K, the moduli of the eigenvalues of the limiting matrix A + BG are less than, or equal to, one, since those of the matrices A + BG,,, are less than one.

Appendix: Proofs of the Properties l-4

We need the following lemma whose proof is given in Albert (1972):

Lemma A.I. Consider a partition of a symmetric (n X n) matrix U:

where U, and U, are square matrices with dimensions p and n - p.

(i) If U is positive de$nite, then

U, - U,U; ‘U,l is positive definite.

(ii) If U is positive semi-definite and U3 is invertible, then

U, - U,U; ‘U. is positive semi-definite.

(iii) If U, is positive de$nite, then U is of rank n - p if and onb if

U 1 = UJJ3-‘U’ 2.

(iv) If U, is positive definite and

u, 2 u&J3-‘uz’,

then U is positive and semi-dejnite.

Let us consider model (1):

x,=Ax,-l+Bd,+s,.

C. Le Van, Stationary uncertainry frontiers 113

Since B is assumed to be of rank p, one can make a change of basis defined by the matrix

I 0 ‘= B j I+ * I 1

Then we have model (1’):

Y, = EY,-, + Fd, + u,, with

y,= P-lx I, u, = p-b,.

E = P-‘AP, F= !: . [ 1 0

The matrix P defines a one-to-one correspondence between the systems (1) and (l’), and, consequently, to the relation (3) in Proposition 1 corresponds, for the system (l’), relation (3’)

QEQ’ = QEBE’Q’ + QUQ’,

where 2 is the covariance matrix of y,, U is the one of u,, and Q is such that QF=O.

Obviously, one can choose

Q= [o / h-p].

Throughout this appendix, we shall consider the following partition of any square matrix S:

Sl s2 s = s, s, ’ i 1

where S, and S, are square matrices of dimensions p and n -p. If S is symmetric, then S, = S;, Si = S,, S; = S,.

Relation (3’) implies

2, = E&E, -I- E&E; + E&E; + E&E; + U,,

i.e., X, 2 Us. We shall now exhibit a (n x n) matrix N verifying

B=N(EXE’+ U)N’,

QN=Q.

(*I

(* *I (***I

114 C. Le Van, Stafionary uncertainty frontiers

Relation (* * *) implies that N must have the form

Since U is of full rank, 2 is of rank n -p, if and only if N is of rank n -p, i.e., N * = 0.

Relations ( * ) and (* * ) imply

N** =z,x;‘.

Hence, if 2 is of rank n -p, N is uniquely defned. We shall prove now Properties l-4 in system (1’).

Proof of Property 1. Define

z,=xz+(l-h)7, xE]O,l[, 2,7E&.

Suppose that 1, is of rank n -p. Its null space E, is of rank p and it implies that 2 and r are of rank n -p; in this case 2 and r must have the same null space which is E,.

If we use the partition of matrices defined above, by Lemma A.l, we have

The equation of the null space of 2 and 7 has two following equivalent expressions:

xz = -Z;‘z;x,, x2 = -r3-lr;x1,

where

(x1, x2) E RP x R”-P.

Hence

q’q= -1 r3 ri.

The matrices N associated with Z and r are the same. Since

Z = NEZE’N’ + NUN’, r = NErE’N’ + NUN’,

and since NE is stable (Proposition l), we have

Z=r. 0

C. Le Van, Stationaty uncertainty frontiers 115

Proof of Property 2. Let 2 be an element of 8’ with rank greater than n -p.

Then there exists a (n X n) matrix N of rank greater than n - p, such that

2 = NEZElN’+ NUN’, with

QN=Q.

Define V= NUN’; V is of rank greater than n -p. Partition V as previously. By Lemma A.l, one has

v 1 - vv-‘V’>O 23 2-*

Define

w 1 = v,v;‘v 29 w,= v,, w3= v3= u,.

By Lemma A.l, W is positive semi-definite of rank n - p and V - W 2 0. Since NE is stable, the equation

r = NErE’N’+ W

admits a unique solution r, positive semi-definite. .X - r verifies the equation

Z-r=NE(I-r)E’N’+ V- W.

Since NE is stable and V - W is positive and semi-definite, we have

it?--70, rE&. Cl

Proof of Property 3. Let 1 and r be in 8’ and of rank n -p. Suppose that Z 2 r. Then 2 and r have the same null space. By the same argument as in the proof of Property 1, we have

t:=r. Cl

Proof of Property 4. If L‘ is symmetric, of rank _n -p and such that 2, is positive definite, then by Lemma A.l, parts (iii) and (iv), 2 is positive semi-definite. 0

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Documents

Stationary uncertainty frontiers in macroeconometric models and existence and uniqueness of solutions to matrix Riccati equations