Journal of Economic Dynamics and Control 10 (1986) 225-229. North-Holland

STATIONARY UNCERTAINTY FRONTIERS IN MACROECONOMETRIC MODELS

An Approach for Solving Matrix Riccati Equations

Cuong LE VAN

CEPREMA P, 75013 Paris, France

1. Introduction

Deleau and Malgrange (1979) used the concept of contrastochastic policies in the macroeconometric models which are feedback controls, linear with respect to past values of endogenous and random variables, and introduced the concept of the stationary uncertainty frontier which is the set of least-positive semi-definite covariance matrices of the endogenous variables stabilised by these contrastochastic policies. In order to characterise this uncertainty fron- tier, they implicitly assumed that it coincided with the set of covariance matrices of objects stabilised by various quadratic welfare functions. Hence they used results on existence and unicity of solutions of matrix Riccati equations.

One analysis proceeds in the opposite direction. We characterise the sta- tionary uncertainty frontier independently of the quadratic cost function and of results on m.atrix 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 defini- tion 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 of matrix Riccati equations proved in the early 1970's by Wonhnarn (1968), Kucera (1972), Payne and Silverman (1973), and also results on the stability properties of optimal dynamic control [e.g., Chow (1970), Rappaport and Silverman (1971), Chow (1972), Aoki (1973), Turnovsky (1974)].

0165-1889/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

226 C. Le Van, Stationary uncertaint.v frontiers

2. Uncertainty frontier in maeroeeonometric models

2.1. Definitions and characterisation

Let us consider the following econometric model:

X t ~ " Ax,_ 1 + Bd, + s t , (1)

d, is a p-vector

and

H. I : B is of rank p,

E(s,) = 0 , Vt,

E(sqslt2 ) = 0, V/1 q: t2,

H.2: E(sts l t ) = S ,

rank S = n.

By definition, contrastochastic policies are the following feedback controls:

d, = T,(Ax,_~ + s,),

where T, is a ( p x n) matrix. If 17, is the covariance matrice of the objective variables and if d t is a

contrastochastic policy, then one has the relations

17, = ( A q- BG,)17,_l( A -1- BGt)'-[- ( I + BTt )S( I + BTt)',

where

G,-- TtA.

A contrastochastic policy is said to be stationary if the matrices {T t) converge to a matrix T when t tends to infinity and such that A + BTA is stable. In this case the matrices {17,} converge to a matrix I7 verifying

17= ( A + B G ) Z ( A + BG) ' + ( I + B T ) S ( I + B T ) ' ,

with G = TA.

where x t is an n-vector of objective variables at time t, ( p < n) of instruments, and s t is a random n-vector.

Model (1) can be viewed as a linearisation of a non-linear model around its steady state (reference trajectory).

The following assumptions are usually made:

C. Le Van, Stationary uncertainty frontiers 227

Let us denote by 8 the set of matrices ~ corresponding to the stationary contrastochastic policies, by ~ the set of 'least-positive semi-definite' matrices of o ~. ~ is called uncertainty frontier of model (1). Heuristically, ~ is the set of the 'incompressible' covariance matrices of the objective variables of model (1) when one uses stationary contrastochastic policies.

The following assumption ensures that ~ is non-empty:

H.3: There exists a matrix G such that A + BG is stable.

In this case (A, B) will be called stabilisable. First, we have the following results:

Proposition 1 [Le Van (1985)]. Under assumptions H.1-H.3 , 8 is a non- empty, conoex, closed set of R 2". ~ is the set of symmetric matrices ~ verifying

RY~R' = RA~,A'R' + RSR' ,

where R is a ( n - p ) × n matrix of rank n - p, such.that RB = O.

Proposition 2 [Le Van (1985)]. ~ is the set of'symmetric matrices Y~ of rank n - p oerifying

RY~R' = RAY, A'R' + RSR' ,

where R is defined as in Proposition 1.

2.2. Stationary uncertainty frontier and optimal control

Let us consider the minimisation problem, in finite time horizon,

T T

min E E(x~Kx t )= mill E t r (K~,) , K > 0, t = l t = l

under the constraints

2,, = ( A + BTtA)Y. t_I (A + BTtA )' + ( I + B T t ) S ( I + BTt)' ,

where N, is the covariance matrix, at time t, of the objective variables x t associated with contrastochastic policies d t = Tt( A x t . ! + st).

The Lagrangian expression, with H denoting the symmetric matrices of Lagrange multipliers; is

T T

L = E t r (K~,) - • t r [ H , ( Z , - (A + B T t a ) Z t _ l ( a + BTtA )' t = l t = l

- ( I + s r , ) s ( i + st,)')].

228 C. Le Van, Stationa~. uncertaint.v frontiers

Writing OL/OY., = a L / a T t = 0, and keeping in mind the constraints, one is lead to the following system:

H r = K ,

14, = K + (A + BGt)'Ht+I(A + BG,),

G, = TtA,

( ~ ' / L B ) r , + B'H, = 0,

~,, = ( A + BTtA)~, t_I( A + BTtA) ' + ( I + BT , )S ( I + BTt) ' ,

Z 0 = O.

The steady state of this system is characterised by

H = K + (A + BG) 'H(A + BG),

( B ' H B ) T + B'H=O,

Y. = ( A + BG),~( A + BG)' + ( I + B T ) S ( I + BT) ' ,

G = T A .

One can easily verify that, if a steady state exists, then the matrix A + BTA is stable. We say that the stationary optimal policy, T, is nonsingular if the associated matrix B'HB is invertible.

Then we have:

Proposition 3 [Le Van (1985)]. Under assumptions H.1-H.3, o °~ is the set of cooariance matrices associated with the non-singular stationary optimal policies.

3. Existence and unicity of solutions of matrix Riccati equations

By the previous propositions, if we want to find an element of ~-, we have to solve the matrix Riccati equation

H = K + ( A + BG) 'H( A + BG),

G-- - ( B ' H B ) - I B ' H A ,

such that A + BG is stable. The following proposition is the central result of this section.

c. Le Van, Stationary uncertainty frontiers 229

Proposition 4 [Le Van (1985)]. Under assumptions H.1-H.3, the following statements are equivalent:

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

(ii) Minimising the linear form Ei, jKi j2 , j has a solution on 8.

The wel l -known results on the existence and unici ty of solutions of matrix Riccati equa t ions yielding a stable feedback control can be deduced easily

f rom Proposi t ion 4 [Le Van (1985)].

References

Aoki, M., 1970, On the sufficient conditions for optimal stabilisation policies, Review of Economic Studies 40, 131-138.

Chow, G., 1970, Optimal stochastic control of linear economic system, Journal of Money, Credit and Banking 3,291-302.

Chow, G., 1972, Optimal control of linear econometric systems with finite time horizon, Interna- tional Economic Review 13, 16-25.

Deleau, M. and P. Malgrange, 1979, Efficient stabilisation of economic systems: Some global analytical results for the linear quadratic case, European Economic Review 12, 17-51.

Kucera, V., 1972, A contribution to matrix quadratic equations, IEEE Transactions on Automatic Control AC-17, 344-347.

Le Van, C., 1985, Stationary uncertainty frontiers in macroeconometric models, and Existence: Unicity of solutions of matrix Riccati equations, Working paper no. 8511 (CEPREMAP, Paris).

Payne, H. and L. Silverman, 1973, On the discrete time algebraic Riccati equation, IEEE Transactions on Automatic Control AC-18, 226-234.

Rappaport, D. and L. Silverman, 1971, Structure and stability of discrete-time optimal systems, IEEE Transactions on Automatic Control AC-17, 227-233.

Turnovsky, S.J., 1979, The stability properties of optimal economic policies, American Economic Review 64, 136-148.

Wonhna~, W.M., 1968, On a matrix Riccati equation of stochastic control, SIAM Journal on Control 6, 681-697.