Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue3.Mar2000:

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24 (2000) 325}346

Solution of

"

nite-horizon multivariate linear

rational expectations models and sparse

linear systems

Michael Binder

!

, M. Hashem Pesaran

"

,

*

!University of Maryland, USA

"Faculty of Economics and Politics, University of Cambridge, Sidgwick Avenue, Cambridge, CB3 9DD, UK, and University of Southern California, USA

Abstract

This paper presents e$cient methods for the solution ofxnite-horizonmultivariate linear rational expectations (MLRE) models, linking the solution of such models to the problem of solving sparse linear equation systems with a block-tridiagonal coe$cient matrix structure. Two numerical schemes for the solution of this type of equation systems are discussed, and it is shown how these procedures can be adapted to e$ciently solve

"nite-horizon MLRE models. As the two numerical schemes are fully recursive and only

involve elementary matrix operations, they are also straightforward to implement. The numerical schemes are illustrated by applying them to a"nite-horizon adjustment cost problem of expenditure shares under adding-up constraints, and to a "nite-horizon linear-quadratic optimal control problem. ( 2000 Elsevier Science B.V. All rights reserved.

JEL classixcation: C32; C63

Keywords: Multivariate linear rational expectations models; Sparse linear systems

*Corresponding author. Tel.:#44-1223-335216; fax:#44-1223-335471.

E-mail address:[email protected] (M.H. Pesaran)

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1. Introduction

In this paper we consider the solution of "nite-horizon multivariate linear rational expectations (MLRE) models. We establish a conceptual link between the solution of these models and the problem of solving sparse linear equation systems with a block-tridiagonal coe$cient matrix structure, and show how methods for the solution of such sparse linear equation systems can be adapted to e$ciently solve"nite-horizon MLRE models.

Sparse linear equation systems with a block-tridiagonal coe$cient matrix structure arise for a wide variety of scienti"c problems, including the numerical solution of certain classes of partial di!erential equations, linear}quadratic optimal control problems, and Gaussian optimal"ltering problems. Not sur-prisingly, then, the solution of linear equation systems with such a structure has been extensively studied in the numerical analysis literature. Drawing on recent work in this literature, we describe two numerical schemes for the e$cient solution of sparse linear equation systems with a block-tridiagonal coe$cient matrix structure, and provide analytical conditions under which these schemes can be successfully applied. Adapting the schemes to the solution of " nite-horizon MLRE models yields numerical algorithms that are e$cient as well as straightforward to implement.1One of the schemes we discuss is applicable also to problems involving coe$cient matrices with a high degree of singularity.

The solution of MLRE models has in the past few years received substan-tial attention in the literature. See, for example, Binder and Pesaran (1995, 1997), Broze et al. (1995), Anderson et al. (1996), Sims (1996), Anderson (1997), King and Watson (1997, 1998), Klein (1997), Uhlig (1997), and Zadrozny (1998). The primary focus of this research has been twofold: the deriva-tion of readily applicable methods for the determinaderiva-tion of the dimension of the solution set (Binder and Pesaran, 1995, 1997; Broze et al., 1995), and establishment of e$cient algorithms for the numerical solution of MLRE models with a unique solution. In either case, the concern of this literature has been with in"nite-horizon MLRE models having time-invariant solutions.2

1In the Appendix, we discuss application of one of these two schemes to the solution of a"nite-horizon linear}quadratic optimal control problem. MLRE models that arise from" nite-horizon linear}quadratic optimal control problems have been analyzed, for example, by Chow (1975), Kendrick (1981), and Aoki (1989). If the planning horizon is"xed, the numerical scheme discussed in the Appendix is likely to be signi"cantly more e$cient than the standard solution approach discussed in this literature, namely matrix Riccati equation based recursions.

2An exception is the work of Gilli and Pauletto (1997, 1998) that we became aware of after a"rst version of this paper had been completed. Gilli and Pauletto consider the solution of"nite-horizon MLRE models as a step in the solution of large-scale nonlinear rational expectations models. While Gilli and Pauletto also discuss the link between the certainty-equivalent solution of"nite-horizon MLRE models and the problem of solving sparse linear equation systems, the methods for the solution of such systems discussed in Gilli and Pauletto di!er from those discussed in this paper. We comment on these di!erences in more detail in Section 5 below.

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In this paper we consider "nite-horizon MLRE models, which, as is well known, generally do not have time-invariant solutions, and therefore re-quire a di!erent solution approach. There is a large number of problems of substantial economic interest that give rise to "nite-horizon MLRE models, such as the (log-linearized) optimality conditions of households' "nite-horizon intertemporal optimization problems.3It is therefore important to have e$cient methods available for the solution of "nite-horizon MLRE models.

The remainder of this paper is organized as follows: Section 2 introduces the class of"nite-horizon MLRE models with general lag- and lead-structure, and discusses in particular the various coe$cient matrix singularities that can arise and that are all accommodated by our approach, at most requir-ing a simple generalized inverse based transformation of the original model. Section 3 describes a solution method based on backward recurs-ions (the fully recursive method for the solution of MLRE models recently advanced in Binder and Pesaran, 1997). Section 4 shows how one may alterna-tively solve these models by rewriting them as sparse linear equation systems with a block-tridiagonal coe$cient matrix structure. Section 5 considers two numerical schemes for the e$cient solution of sparse linear equation systems with a block-tridiagonal coe$cient matrix structure, and shows how they can be adapted for the e$cient solution of "nite-horizon MLRE models. These schemes are illustrated by applying them to a lin-ear}quadratic adjustment cost problem involving expenditure shares in Section 6. As the expenditure shares naturally are subject to an adding-up constraint, this model is an example of an MLRE model with redundancies, that is, with a singularity in the canonical form coe$cient matrix premulti-plying the vector of current-period dependent variables. It is shown that one of the two numerical schemes of Section 5 is immediately applicable to this problem, and that the other is applicable upon using the generalized inverse based transformation introduced in Section 2. The application of one of the numerical schemes to a "nite-horizon linear}quadratic optimal control problem is discussed in an Appendix. Some concluding remarks are o!ered in Section 7.

3For an analysis of a"nite-horizonnonlinearrational expectations model that arises from the optimality conditions of a life-cycle consumption model possibly involving liquidity constraints, see Binder et al. (1999). Prucha and Nadiri (1984, 1991) as well as Steigerwald and Stuart (1997) argue that"rms may also operate with a"nite (but"xed) planning horizon, and have static expectations beyond their planning horizon.

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2. The model

A general formulation of the MLRE model with a "nite horizon can be written as4

n1 + i/0

n2 + j/0

ij E(zt`q`j~iDXt`q~i)!ft`q"0rC1, q"0, 1,2,¹!t, (2.1) where E(z

t`q`j~iDXt`q~i) is given for t#q#j!i'¹, i"0, 1,2,n1, and

j"0, 1,₂,n

2. In (2.1), zt`q denotes an r]1 dimensional vector of decision

variables, f

t`q represents a vector of forcing variables of the same dimension, M

ij (i"0, 1,2,n1, j"0, 1,2,n2) is anr]rdimensional matrix of"xed coe$ -cients, E(_'DX

t`q) is the conditional mathematical expectations operator, and

t`q represents a non-decreasing information set at time t#q, containing

(at least) current and lagged values of z

t`q and ft`q: Xt`q"Mzt`q,zt`q~1,2; f

t`q,ft`q~1,2N. For the state variables among the decision variables inzt`q, we

assume that the relevant initial conditions are given. We also assume that the forcing variablesMf

t`qNare adapted toMXt`qN.5

Throughout this paper, we will base our analysis of (2.1) on the following second-order canonical form:

t`q"Axt`q~1#BE(xt`q`1DXt`q)#wt`q, q"0, 1,2,¹!t, (2.2)

where x

t`q"(q@t`q,q@t`q~1,2,q@t`q~n1`1)@, q@

t`q"(z@t`q, E(z@t`q`1DXt`q),2,E(z@t`q`n2DXt`q))@, andw

t`qis anm]1 dimensional vector that consists of linear combinations of

the elements off

t`q, withm"n1(n2#1)r. See, for example, Binder and Pesaran (1995, 1997) for a de"nition of them]mdimensional matricesAandBin terms of theM

ij's (i"0, 1,2,n1, j"0, 1,2,n2), as well as a precise de"nition ofwt`q.

In (2.2), we have normalized the coe$cient matrix associated withx

t`qto be

the identity matrix. This is done without any loss of generality.6To see this, it

4While we formulate our model as one with a"nite and shifting planning horizon (so that the terminal period is"xed), the solution method we suggest is also readily applicable to the case where the planning horizon is "nite and"xed at all current and future dates. See Section 5 and the Appendix for a further discussion of the case of a"xed planning horizon.

5Note that this allows for many forms of nonstationarities and/or nonlinearities in the stochastic processes generating the forcing variables.

6Note that no rank restrictions have been imposed on theM

ij's (i"0, 1,2,n1,j"0, 1,2,n2),

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is simpler to go back for a moment to (2.1), as it may be readily veri"ed from the de"nition ofAandBin terms of theM

ij's (i"0, 1,2,n1, j"0, 1,2,n2) that the dimension of the null space of the coe$cient matrix premultiplyingx

t`qin (2.2)

is equal to the dimension of the null space ofM

00in (2.1). If, as assumed above,M

00is a square matrix, thenM00must be nonsingular for the MLRE model (2.1) to be complete in the decision variables z

t`q, q"0, 1,₂,¹!t: IfM

00is singular, then it is not possible to uniquely solve for the elements of z

t`q as functions of the elements of the lagged zt`q's, of

the current and future expectations of the elements of z

t`q, and of the

forcing variables in f

t`q. If one allows for the possibility that M00 is not a square matrix, but rather is a matrix of dimension, say, q]r, q'r, then for the model equations to be consistent, it must be the case that q!r rows of (2.1) are linear combinations of the remainingrrows of (2.1), and therefore are redundant.7If such redundancies are present, the rank ofM

00must be equal to

rfor (2.1) to be complete. Premultiplying (2.1) by the generalized inverse ofM 00,

00"(M@00M00)~1M@00, (2.3) one then obtains

n1 + i/0

n2 + j/0

ijE(zt`q`j~iDXt`q~i)!M~00ft`q"0rC1, q"0, 1,2,¹!t, (2.4) withMI

ij"M~00Mij, and thusMI 00"Ir, whereIrdenotes the identity matrix of

order r. Premultiplying (2.1) by the generalized inverse of M

00 removes any redundancies in the model.8(Of course, in the case whereq"r, the generalized inverse of M

00coincides with the inverse ofM00.) Therefore, even in the case where a model contains redundancies, working with the canonical form (2.2) does not involve any loss of generality, and (2.2) can always be obtained from (2.1) without any model-speci"c transformations.9

7Note that the presence of an identity by itself would not causeM

00to be singular. An identity

causes the variance}covariance matrix off

t`qto be singular, but does not present any new di$culty

as far as the solution of (2.1) is concerned.

8For an example see Section 6.

9An alternative to (2.2) would be to use the"rst-order canonical form of Blanchard and Kahn (1980), also employed, for example, in the work of Sims (1996) and King and Watson (1997, 1998). While the Blanchard and Kahn canonical form without coe$cient matrix singularity restrictions has the same level of generality as our second-order canonical form (2.2), reducing (2.1) to Blanchard and Kahn's"rst-order canonical form generally requires introducing new predetermined variables. If the second-order canonical form (2.2) is used, no predetermined variables need to be introduced, and the solution is directly obtained in terms of current-period decision variables.

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3. A backward recursive solution

As (2.2) is a "nite-horizon MLRE model which generally does not have a time-invariant solution, the standard methods available in the literature for the solution of in"nite-horizon MLRE models are not applicable to it. One approach to the solution of (2.2) would be to use backward recursions starting from time ¹. At time ¹, the solution for x

T, given xT~1 and the terminal conditionE(x

T`1DXT), is given by (2.2) forq"¹!t: x

T"AxT~1#BE(xT`1DXT)#wT. (3.1)

Proceeding recursively backward, we can obtainx

T~1as a function of xT~2, the terminal conditionE(x

T`1DXT), and of E(wTDXT~1) and wT~1. Combining (2.2) forq"¹!t!1 with (3.1), one readily obtains

T~1"(Im!BA)~1[AxT~2#B2E(xT`1DXT)#BE(wTDXT~1)#wT~1]. (3.2) Proceeding to period ¹!2, combining (2.2) for q"¹!t!2 with (3.2), the solution forx

T~2is given by x

T~2"[Im!B(Im!BA)~1A]~1 ]

C

T~3#B(Im!BA)~1B2E(xT`1DXT) #B(I

m!BA)~1BE(wTDXT~2) #B(I

m!BA)~1E(wT~1DXT~2)#wT~2

D

The pattern of these backward recursions should be apparent. We thus have the following proposition, which extends Proposition 4.1 in Binder and Pesaran (1997) to "nite-horizon MLRE models, and provides the solution for x

t`q,q"0, 1,2,¹!t.10

Proposition 3.1 (Backward recursive solution). Consider thexnite-horizon MLRE model(2.2).Let

T~t"Im, UT~t~i"Im!BUT~1~t~i`1A, i"1, 2,2,¹!t, (3.3)

10Binder and Pesaran (1997) also discuss how this backward recursive method may be used to compute the unique stable solution (if it exists) of in"nite-horizon MLRE models. Non-recursive methods may often be faster for the solution of such models than recursive solution methods (particularly for in"nite-horizon MLRE models for which the solution attcould be sensitive to the value ofE(x

T~t`1DXt) for large values of¹). On the other hand, the fully recursive method we suggest in this paper does not involve any matrix similarity transformations, but only requires elementary matrix operations (addition, multiplication, inversion). It may therefore be easier to grasp and implement than those methods in the literature for the solution of in"nite-horizon MLRE models that are based on matrix similarity transformations. Furthermore, it is often interesting to know for what values of

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and

T"BE(xT`1DXT)#wT,

T~i"BU~1T~t~i`1WT~i`1#wT~i, i"1, 2,2,¹!t. (3.4)

Suppose the matrices U

T~t~i are nonsingular for i"1, 2,2,¹!t. Then the

solution forx

t`qto(2.2)is given by x

t`q"U~1q Axt`q~1#U~1q E(Wt`qDXt`q), q"0, 1,2,¹!t. (3.5)

Note that the solution in all periods is a linear combination of the initial and terminal values, and the conditional expectations of the forcing variablesMf

t`qN.

As the forcing variables were assumed to be adapted to the information sets MX

t`qN, then so will be the solutionMxt`qN.

4. A block-tridiagonal system representation

While Proposition 3.1 provides the solution to (2.2) as long as the matrices

T~t~i,i"1, 2,2,¹!t, are nonsingular, signi"cant further insights into the

solution of (2.2) can be obtained by relating the solution of (2.2) to the solution of sparse linear equation systems, rather than applying the backward recursive approach underlying Proposition 3.1. Accordingly, we stack the canonical form (2.2) for periods¹,¹!1,2,t, and take conditional expectations with respect toX

tto obtain

A

m !A 0m 0m 2 0m 0m 0m !B I

m !A 0m 2 0m 0m 0m

m !B Im !A 2 0m 0m 0m

F F F F } F F F

m 0m 0m 0m 2 !B Im !A

m 0m 0m 0m 2 0m !B Im

BA

E(x

TDXt) E(x

T~1DXt) E(x

T~2DXt) F E(x

t`1DXt) x

B

A

E(w

TDXt) E(w

T~1DXt) E(w

T~2DXt) F E(w

t`1DXt) Ax

t~1#wt

B

A

BE(x T`1DXt)

mC1 0

mC1 F 0

mC1 0

mC1

B

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or, more simply,

t"ht, (4.2)

where the matrixC is of dimensionp]p, andu

tand htare p]1 dimensional

vectors, withp"m(¹!t#1). To solve forx

tfrom (4.2), one needs to invert the

matrix C. (As we will show below, one of our two numerical schemes for the inversion ofCin fact also yields the solutions for x

t`q, q"1, 2,2,¹!t.)

Considering the block-tridiagonal nature of the coe$cient matrixC, it is clear that (4.2) represents a sparse linear equation system. The sparsity ofCmay be yet more pronounced due to the canonical form matrices A and/or B being sparse also. The matricesAand/orBmay be sparse due to the inherent dynamic structure of the model under consideration and/or because the process of constructing the canonical form (2.2) may introduce a large number of zero entries inAand/orB. In the next section, we will show that by approaching the solution of (2.2) through the sparse linear equation system (4.2), one may obtain analytical conditions under which the recursions in Proposition 3.1 are well de"ned, and one may also achieve further gains in computational e$ciency relative to carrying out the recursions of Proposition 3.1 if certain nonsingular-ity restrictions are satis"ed.

5. Solution of block-tridiagonal linear equation systems

E$cient inversion of block-tridiagonal matrices has been studied in the recent numerical analysis literature. We provide here}in the context of (4.2)}a brief discussion of two methods to accomplish such inversions. One of these methods leads to the same recursions as in Proposition 3.1, but readily allows to establish analytical conditions under which these recursions are well de"ned. The second method, when applicable and when the planning horizon is"xed, is likely to lead to further gains in computational e$ciency, compared to carrying out the recursions of Proposition 3.1. Like the recursions of Proposition 3.1, both methods are straightforward to implement, as they involve only elementary matrix operations. Also the"rst method is applicable regardless of whether the block-subdiagonal matrixB is singular. The second method applies, however, only if B is nonsingular.11 Before proceeding, we might also note that our

11In the special case where m"1, techniques speci"cally geared towards the inversion of tridiagonal matrices are also available. One of these techniques has been used by Prucha and Nadiri (1991) to solve"nite-horizon univariate linear factor demand models. Some alternative schemes for the inversion of block-tridiagonal matrices to the ones outlined here are discussed in Gilli and Pauletto (1997, 1998). The methods discussed here have the advantage that analytical conditions are known under which they are operational. The nonstationary iterative methods presented in Gilli and Pauletto may have smaller storage requirements, however, and could therefore be attractive for large-scale models of the type discussed in Gilli and Pauletto.

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discussion is not meant to be a comprehensive guide to the inversion of block-tridiagonal matrices, but rather simply intends to convey how " nite-horizon MLRE models may be e$ciently solved using numerical schemes from the literature on the inversion of block-tridiagonal matrices.

Case1:Bpossibly singular.

Consider"rst the case whereBis possibly singular. The standard textbook approach to solving linear equation systems of the formCu

t"ht, de"ned by

(4.2), is by means of Gaussian elimination, or equivalently, LU-factorization; namely factorization ofCinto the product of a block lower triangular matrix LH and a block upper triangular matrix UH, and then solving the resultant block-triangular equation systems

LHl

t"ht, (5.1)

(by forward substitution), and UHu

t"lt, (5.2)

(by backward substitution). Such an algorithm does not exploit the sparse nature of C, however, and in general can be quite ine$cient. An alternative numerical procedure that utilizes the block-tridiagonal structure of C is the LDU-factorization, discussed, for example, in Axelsson (1994). DecomposeCas

C"Dc#Lc#Uc, (5.3)

where Dc, Lc, and Uc are the diagonal, subdiagonal, and

block-superdiagonal entries inC, and consider the factorization

C"LD~1U, (5.4)

where

L"D#Lc, (5.5)

and

U"D#Uc. (5.6)

Noting that

C"(D#Lc)D~1(D#Uc)"D#Uc#Lc#LcD~1Uc, (5.7)

it is easily seen that D"Dc!LcD~1Uc, and thus the matrix D satis"es the

recursions D

1"C11, Di"Cii!Ci,i~1D~1i~1Ci~1,i, i"2, 3,2,¹!t#1, (5.8)

whereC

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As discussed in Axelsson (1994), su$cient conditions for the recursions in (5.8) to be well de"ned are: (i)C is symmetric positive de"nite, or (ii)Cis a block H-matrix.12

Having factorized C as in (5.4), it is then a simple step to solve for u t by

splittingCu

t"htinto Lg_t"h

t or gt,1"D~11 ht,1, gt,i"D~1i (ht,i!Ci,i~1gt,i~1),

i"2, 3,₂,¹!t#1, (5.9) and

D~1Uu

t"gt or ut,T~t`1"gt,T~t`1, ut,i"gt,i!D~1i Ci,i`1ut,i`1,

i"¹!t,¹!t!1,₂,1. (5.10) The solution x

t to the MLRE model (2.2) is given by the last m entries inut.

We therefore have the following proposition:

Proposition 5.1 (Solution based on LDU-factorization). Consider thex nite-hor-izon MLRE model (2.2). Let

i"Im!BH~1i~1A, i"2, 3,2,¹!t#1, (5.11)

i"H~1i (BCi~1#wT`1~i), i"2, 3,2,¹!t#1, (5.12)

with the initial conditions H

1"Im, and C1"BE(xT`1DXT)#wT. Suppose the

matricesH

iare nonsingular fori"2, 3,2,¹!t#1.Then the solution forxtto

(2.2)is given by

t"H~1T~t`1Axt~1#E(CT~t`1DXt). (5.13)

It is easily veri"ed that the recursions in (5.11) and (5.12) match the recursions in Proposition 3.1. Theith recursion matricesH

iandCiin

Proposi-tion 5.1 are related to the ith recursion matrices U

T~t`1~i and WT`1~i

in Proposition 3.1 as follows:H

i"UT~t`1~i, i"1, 2,2,¹!t#1, andCi"

U_~1

T~t`1~iWT`1~i, i"1,2,2,¹!t#1. Notice that this equivalence also

im-plies that inversion ofCdoes not only yield the solution forx

t, but also yields the

solution forMx

t`qNTq/1~t. We have x

t`q"H~1T~t`1~qAxt`q~1#E(CT~t`1~qDXt`q), q"1, 2,2,¹!t. (5.14)

Proposition 5.1 provides a link between the recursions of Proposition 3.1 and theLDU-factorization, and establishes conditions under which these recursions

12See Axelsson (1994, Chapters 6 and 7) for a de"nition of block H-matrices. The class of blockH-matrices encompasses, but is not restricted to, matrices that are (generalized) diagonally dominant.

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are well de"ned. Notice that for these recursions to be well de"ned, it is by no means necessary that the coe$cient matricesAandBare nonsingular.13

While the recursions in (5.8)}(5.10) exploit the block-tridiagonal structure of C, theD

imatrices in general are full, even if the superdiagonal, diagonal and

subdiagonal blocks of C are sparse, as the inverse of a sparse matrix is, in general, full. A fully e$cient solution scheme to Cu

t"ht in the case where Aand/orBare sparse will therefore also incorporate sparse approximations of the inverses D~1

i , i"1, 2,2,¹!t#1, and sparse approximations of the

matrix product terms in the recursions in (5.8), C

i,i~1D~1i~1Ci~1,i. A variety of

numerical schemes accomplishing this by allowing the user to control the sparse blocks are discussed in Axelsson (1994, Chapter 8).

Case2:Bnonsingular.

In the case where the block-subdiagonal matrix Bis nonsingular, one may also solve the "nite-horizon MLRE model (2.2) by adapting the recursions suggested in Bowden (1989) for the inversion of block-tridiagonal matrices:

Proposition 5.2 (Solution based on Bowden's procedure). Consider the x nite-horizon MLRE model (2.2) with the coezcient matrix B nonsingular. Then the solution forx

tto(2.2)is given by x

t"BT~t`1Axt~1#

T₊~t i/0

B_T_~_t`_1~_iE(w

t`iDXt)#B1BE(xT`1DXt), (5.15)

where

B_i"F~1

T~t`2Fi, i"1, 2,2,¹!t#1, (5.16) F

1"Im, F2"B~1, Fi`1"(Fi!Fi~1Ai~1)B~1, i"2, 3,2,¹!t, (5.17)

and F

T~t`2"(FT~t`1!FT~tA). (5.18)

A proof of Proposition 5.2 can be constructed following the arguments in Bowden (1989).

Note that Bowden's procedure merely requires the inversion of two m]m

dimensional matrices. It is clearly an e!ective and straightforward method for the solution of the MLRE model, (2.2). To compute Mx

t`qNTq/1~t, however, the

13While we have not encountered in any application that we have considered near-singularity or singularity of any of the H

imatrices, if such (near-) singularities did arise, the recursions in Proposition 5.1 could be stabilized using techniques developed in the recent numerical analysis literature on (near-) rank-de"cient problems. See, for example, Hansen (1998), for an up-to-date survey.

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analog of (4.1) needs to be constructed for periodst#1,t#2,₂,¹!1, before the solution technique of Proposition 5.2 can be applied. Therefore, Bowden's procedure will typically be less e$cient for the computation of Mx

t`qNTq/1~t than the LDU-factorization based procedure of Proposition 5.1. However, in the case of MLRE models with a "nite and "xed planning horizon at all current and future dates, the structure of the matrix C in (4.2) remains unchanged in all periods, and Bowden's procedure is likely to be computationally more e$cient than the LDU-factorization based procedure.

6. An illustration:A consumer's optimal expenditure shares

In this section, we illustrate Propositions 5.1 and 5.2 by applying them to the solution of a model of a consumer's optimal expenditure shares if share adjust-ment to the target level is costly both in terms of the level of adjustadjust-ment and in terms of the speed of adjustment.14

Consider the following"nite-horizon adjustment cost problem:

min

Ms_t`_qNT_q/0~t

G

T+~t q/0

C

(st`q!sHt`q)@H(st`q!sHt`q)#*s@t`qG*st`q #*2s@

t`qK*2st`q

D

H

(6.1)

for given initial and terminal conditionss

t~1, st~2, E(sT`1DXT~1), E(sT`1DXT),

andE(s

T`2DXT), and subject to

i@s

t`q"1, q"0, 1,2,¹!t. (6.2)

In (6.1) and (6.2),s

t`qis anr]1 dimensional vector of the consumer's

expendi-ture shares, H, G, and K are r]r dimensional symmetric matrices of "xed coe$cients,b3(0, 1) is a constant discount factor,iis anr]1 dimensional vector of ones, andsH

t`qis a vector of desired (target) expenditure shares, derived, for

example, from the Almost Ideal Demand System of Deaton and Muellbauer (1980),

t`q"a#Clnpt`q#dln

A

t`q p

t`q

B

, q"0, 1,2,¹!t, (6.3)

14See Pesaran (1991) for a detailed discussion of adjustment costs both for the level and the speed of adjustment. See also Price (1992) and Binder and Pesaran (1995).

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where

A

a₁ a₂ F a_r

B

, C"

A

c₁₁ c₁₂ 2 c₁ r c₂₁ c₂₂ 2 c₂_r

F F F

c_r₁ c_r₂ 2 c_rr

B

A

d₁ d₂ F d_r

B

, and lnp t`q"

A

lnp 1,t`q lnp 2,t`q F lnp

r,t`q

B

In (6.3),p

i,t`qis the price de#ator of commodity groupi,yt`qis the consumer's

expenditure on all the commodities, andp

t`qis the general price index,

approxi-mated using the Stone formula lnp

t`q"x@0lnpt`q, (6.4)

wherex₀"(u₁₀,u₂₀,2,u_r₀)@, andu_i₀is the budget share of theith commodity group in the base year. Consumer theory imposes the following restrictions on the parameters of the target share equations:

(a) adding-up restrictions:

r + i/1

a i"1,

r + i/1

ij"0, and r + i/1

i"0, (6.5)

(b) homogeneity restrictions:

r + j/1

ij"0, (6.6)

and

ij"cji, jOi. (6.7)

We also assume that for given observations onp

t`qandyt`q, the parameters in

a,C, anddare such that the desired sharessH

i,t`qin (6.3) lie in the range [0, 1] for i"1, 2,₂,r, andq"0, 1,₂,¹!t. To computeE(sH

t`qDXt), we also assume that m

t"(lnp1t, lnp2t,2, lnprt, lnyt)@ follows the vector autoregressive process of

orders: m t"a# s + i/1 R

imt~i#*t, *t&i.i.d.N(0rC1,R*), (6.8)

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Forming the Lagrangian for (6.1)}(6.8), the Euler equations at timet#qcan after some algebraic manipulation be written as15

00st`q"N

C

10st`q~1#M20st`q~2#M01E(st`q`1DXt`q) #M

02E(st`q`2DXt`q)#HsHt`q

D

#hi, (6.9) q"0, 1,₂,¹!t, where M

00"H#(1#b)G#(1#4b#b2)K, M10"G# 2(1#b)K, M

20"!K, M01"bM10, M02"b2M20, N"Ir!hii@M~100, and h"1/(i@M~1₀₀i)_>0.16Noting thati@M~1₀₀N"0

r, it is easily seen thati@st`q"1, for

all q, as it should. The Euler equations (6.9) constitute a special case of the "nite-horizon MLRE model (2.1), and can be e$ciently solved using Proposi-tion 5.1, or, in the case where the adjustment costs are of"rst-order only,K"0

using Proposition 5.2.

Case1:KO0

To apply Proposition 5.1 (the solution method based on the LDU-factoriz-ation), we just need to rewrite (6.9) so that it"ts the canonical form (2.2). Let x

t`q"(s@t`q,s@t`q~1,E(s@t`q`1DXt`q))@. Then x

t`q"Axt`q~1#BE(xt`q`1DXt`q)#wt`q, (6.10)

with w

t`q"([M~100NHsHt`q#hM~100i]@, 0@rC1,0@rC1)@, A"

A

M~1

00NM10 M~100NM20 0r I

r 0r 0r

B

and

A

M~1₀₀NM

01 0r M~100NM02 0

r 0r 0r I

r 0r 0r

B

15We do not explicitly impose the conditions 04s

i,t`q41,i"1, 2,2,r, andq"0, 1,2,¹!t

in (6.9). We thus implicitly assume that the deviations of the shares from their target levels are su$ciently small. If any of the inequality constraints 04s

i,t`q41, i"1,2,2,r,

and q"0, 1,₂,¹!t, was violated in the absence of these constraints, the model would become nonlinear, necessitating application of the type of techniques discussed in Binder et al. (1999).

16Notice thatM

00must be a positive de"nite matrix if the second-order conditions for the global

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Proposition 5.1 can now be readily applied for the solution of (6.10).17Notice again that Proposition 5.1 does not require A and/or B to be nonsingular, and that E(w

t`q`hDXt`q) can be obtained noting that E(sHt`q`hDXt`q)"

a#(C_!_dx_@

0,d)E(mt`q`hDXt`q), and using (6.8).

Case2:K"0

If the adjustment costs have a "rst-order structure only, then the Euler equations (6.9) immediately become a special case of the canonical form (2.2):

t`q"Ast`q~1#BE(st`q`1DXt`q)#wt`q, (6.11)

with A"M~1

00NM10, B"M~100NM01"bA, and wt`q"M~100NHsHt`q#hM~100i. AsM~1

00Nis singular (recall thati@M~100N"0r), so will beB. In order to apply

Proposition 5.2 (the solution method based on Bowden's procedure), we there-fore"rst need to transform (6.11) so as to remove this singularity ofBthat is due to the constrainti@s

t`q"1. Observing this constraint, note that we can

decom-pose the vector of expenditure sharess t`qas s

t`q"e#Ps8t`q, (6.12)

wheres8

t`q"(s1,t`q, s2,t`q,2,sr~1,t`q)@,

A

0 0 F 0 1

B

, and P"

A

1 0 2 0 0

0 1 2 0 0

F F F F

0 0 2 1 0

0 0 2 0 1

!1 !1 2 !1 !1

B

withebeing of dimensionr]1, andPbeing of dimensionr](r!1). It is clear thatP is of rankr!1, and thatP@Pis nonsingular. Replacings

t`qfrom (6.12)

in (6.11), we now have

Ps8_t`q"APs8_t`q_~1#BPE(s8_t`q`₁DX

t`q)#wHt`q, (6.13)

where

wH_t`q"w

t`q#(A#B!Ir)e.

Equation system (6.13) is a special case of the type of MLRE model with redundancies discussed in Section 2: The last row of (6.13) is a linear

17GAUSS- and MATLAB-programs illustrating implementation of our solution methods for the model of this section may be downloaded from the following URL:

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combination of the "rstr!1 rows of (6.13). To remove this redundancy, we premultiply (6.13) by the generalized inverse ofPto obtain

s8_t`q"AIs8_t`q_~1#BIE(s8_t`q`₁DX

t`q)#w8Ht`q, (6.14)

where A_I"(P@P)~1P@AP, B_I"(P@P)~1P@BP"bA_I, and w8H

t`q"(P@P)~1P@wHt`q.

Proposition 5.2 can be readily applied to obtain the numerical solution of (6.14) ifBI, or, equivalently, ifP@BPis nonsingular.18

But since (forK"0

B"M~1

00NM01"bM~100(Ir!hii@M~100)G, we have

P@BP"bPI@(I

r!i8(i8 @i8)~1i8 @)GIPI,

where

PI"M~1@2₀₀ P, i8"M~1@2₀₀ i, and GI"M~1@2₀₀ GM1@2₀₀. It is now easily seen thatrank(PI)"rank(I

r!i8(i8 @i8)~1i8 @)"r!1. Therefore, for P@BP to be nonsingular,GI orGmust also have at least rankr!1.

OnceMs8_t`qNT_q_/0~tis computed, one may obtain therth expenditure share as Ms

r,t`qNTq/0~t"

G

r~1 + i/1

s i,t`q

H

T~t q/0 .

7. Conclusion

In this paper, we have shown that the numerical solution of "nite-horizon MLRE models can be reduced to the problem of solving sparse linear equation systems with a block-tridiagonal coe$cient matrix structure. The latter problem has been discussed in the recent numerical analysis literature, and there are e$cient algorithms available for this purpose. We have described two such numerical schemes in this paper. Their application to "nite-horizon MLRE models yields a procedure for the solution of these models that is e$cient, straightforward to implement, and allows investigation of a rich array of speci"cations for the forcing variables of the model, as the latter are not restricted to be generated by linear and/or covariance-stationary processes. Furthermore, in the case of one of the numerical schemes we have described, the

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procedure is applicable also to models where the coe$cient matrices in their companion canonical form involve a high degree of singularity.

Acknowledgements

We are grateful to Robert King, Ingmar Prucha, and particularly an associate editor and an anonymous referee for helpful comments and suggestions, and to Keith Bowden for sending us excerpts of his Ph.D. thesis.

Appendix. Solution of a5nite-horizon linear}quadratic optimal

control problem

In this Appendix, we consider the solution of the following "nite-horizon linear}quadratic optimal control problem:19

min

Mu t`qNTq/0~t~1

C

y@TZyT#

T~t~1 + q/0

t`qQ yt`q#u@t`qRut`qDXt

D

(A.1)

for giveny

t, and subject to y

t`q"Vyt`q~1#Wut`q~1#Net`q, et`q&**$ N(0gC1,Re),

q"1, 2,₂,¹!t, (A.2)

where y

t`q is the d]1 dimensional state vector, ut`q the f]1 dimensional

control vector,Z,Q, andVared]ddimensional matrices of"xed coe$cients, and the coe$cient matricesR,W, andNare of dimensionsf]f,d]f, andd]g, respectively. As is usually assumed, we take the coe$cient matricesZ, Q, and R as being symmetric and nonsingular. Terminal conditions in practice are typically imposed by choosing su$ciently large values for the appropriate elements of Z.20 While the objective function (A.1) does not involve cross-products of components of the state vector and the control vector, this is without loss of generality, as "nite-horizon linear}quadratic optimal control problems involving such cross-product terms can always be transformed to the form of (A.1)}(A.2).21 The information set X

t is speci"ed as

19For recent surveys of discrete time optimal control problems as relating to economics see, for example, Amman (1996) and Anderson et al. (1996).

20See, for example, Kendrick (1981) and Amman (1996).

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follows: X

t"Myt,yt~1,2;ut,ut~1,2;et,et~1,2N. Invoking certainty equiva-lence, we evaluate all random components,e

t`q,q"1, 2,2,¹!t, at their mean

values and solve the certainty-equivalent counterpart of Eqs. (A.1) and (A.2). The standard approach to the solution of the optimal control problem (A.1)}(A.2) is the sweep method of Bryson and Ho (see, for example, Lewis and Syrmos, 1995), which yields the optimal control law

uH_t`q"!K

t`qyt`q, (A.3)

where K

t`q"(W@St`q`1W#R)~1W@St`q`1V, (A.4) and

t`q"(V!WKt`q)@St`q`1(V!WKt`q)#K@t`qRKt`q#Q, (A.5) q"0, 1,₂,¹!t!1, with S

T"Z. Equation (A.5) is the (Joseph stabilized

version of the) matrix Riccati equation. The feedback gain matrices K t`q are

typically computed by backward recursions on (A.4) and (A.5). Note that these recursions involve computing ¹!t inverse matrices, (W@S

t`q`1W#R)~1, q"0, 1,₂,¹!t!1.22

In this Appendix, we show that the solution of the optimal control problem (A.1)}(A.2) can also be obtained by means of rewriting (A.1) and (A.2) in stacked form, and solving the resultant optimality conditions using Bowden's procedure for the inversion of a block-tridiagonal coe$cient matrix. Application of Bowd-en's procedure can signi"cantly reduce the computational e!ort needed to carry out the matrix Riccati equation recursions if the planning horizon is "xed. Alternatively, one could write the "rst-order conditions of (A.1)}(A.2) as a special case of the canonical form (2.2), and then use the solution techniques described in Section 5. However, it is easily veri"ed that the resulting canonical form does not contain lagged values, and only future expectations appear on the right-hand side of the equation system. The solution method developed in the remainder of this Appendix takes this special feature of the optimal control problem (A.1)}(A.2) into account.

Following Bowden (1983), we rewrite (A.1) and (A.2) after invoking certainty equivalence as

min

2My8 @QIy8#u8 @RIu8 N (A.6)

22In the in"nite-horizon case,S

t`qand thusKt`qunder certain conditions are time-invariant. In

this case the solution of the resulting algebraic matrix Riccati equation can also be computed non-recursively. E$cient methods for this purpose are reviewed, for example, in Lewis and Syrmos (1995) and in Anderson et al. (1996). See also Amman and Neudecker (1997).

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subject to VI y8"y8

t#WI u8, (A.7)

where

QI"

A

Q 0

d 0d 2 0d 0d 0d

d Q 0d 2 0d 0d 0d F F F } F F F 0

d 0d 0d 2 0d Q 0d

d 0d 0d 2 0d 0d Z

B

RI"

A

R 0

f 0f 2 0f 0f 0f

f R 0f 2 0f 0f 0f F F F } F F F 0

f 0f 0f 2 0f R 0f

f 0f 0f 2 0f 0f R

B

VI"

A

d 0d 0d 2 0d 0d 0d !V I

d 0d 2 0d 0d 0d

F F F } F F F

d 0d 0d 2 !V Id 0d

d 0d 0d 2 0d !V Id

B

WI "

A

W 0

dCf 0dCf 2 0dCf 0dCf 0dCf

dCf W 0dCf 2 0dCf 0dCf 0dCf

F F F } F F F

dCf 0dCf 0dCf 2 0dCf W 0dCf

dCf 0dCf 0dCf 2 0dCf 0dCf W

B

y8"(y@

t`1,y@t`2,2,y@T)@, u8"(u@t,u@t`1,2,u@T~1)@, and

t"((Vyt)@,0@dC1,2,0@dC1)@.

Substituting the constraints in (A.7) back into (A.6), we obtain the unconstrained optimization problem

min

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where

LI"VIQI~1VI @. (A.9) Taking derivatives of (A.8) with respect to u8 @, it is readily veri"ed that the optimal control vectoru8His given by23

u8H"!(RI~1!RI~1WI @(WI RI~1WI @#LI)~1WI RI~1)(WI @LI~1y8

t). (A.11)

Inspecting (A.11) and recalling the de"nitions ofRI andLI, the main computa-tional burden in solving foru8His easily seen to be given by the inversion of the block-tridiagonal matrixJI"WI RI~1WI @#LI,

JI"

A

WR~1W@#Q~1 !Q~1V@ 0

d 2 0d 0d 0d

!VQ~1 T !Q~1V@ 2 0

d 0d 0d

F F F } F F F

d 0d 0d 2 !VQ~1 T !Q~1V@

d 0d 0d 2 0d !VQ~1 F

B

(A.12) where

T"WR~1W@#VQ~1V@#Q~1, (A.13)

and

F"WR~1W@#VQ~1V@#Z~1. (A.14) Using Bowden's Procedure to invertJI results in the following recursions: Let

T~t"Id, NT~t~1"(VQ~1)~1F, N

T~t~i"(VQ~1)~1(TNT~t~i`1!Q~1V@NT~t~i`2),

i"2, 3,₂,¹!t!1, and

1"Id, P2"(WR~1W@#Q~1) (VQ~1)~1, P

i"(Pi~1T!Pi~2Q~1V@) (VQ~1)~1, i"3, 4,2,¹!t, P

T~t`1"(PT~tF!PT~t~1Q~1V@).

23In deriving (A.11), we have used the result (A~1₁₁#A

12A22A21)~1"A11!A11A12(A21A11A12#A~122)~1A21A11, (A.10)

which holds for any set of real valued matricesA

11,A12,A21, andA22for which the left- and

(21)

Then the (ij)th block ofJI~1is given by JI~1

ij "P@i(P@T~t`1)~1N@j for j*i, (A.15)

and J_I~1

ij "NiP~1T~t`1Pj fori'j, (A.16) i,j"1, 2,2,¹!t.

Note that computation ofJI~1involves the computation of only"ve inverses, R~1,Q~1,Z~1, (VQ~1)~1, and P~1_T_~_t`₁. Only the solution foru

tin (A.11) is the

solution for the stochastic optimal control problem (A.1)}(A.2), however, as the solutions forMu

t`qNTq/1~t~1do not re#ect the stochastic innovations in periods

t#1,t#2,₂,¹!t!1. To compute the solution forMu

t`qNTq/1~t~1, the analog of (A.8)}(A.9) needs to be constructed for the optimal control problems in periodst#1,t#2,₂,¹!t!1. The structure of these problems is the same as that att, (A.8)}(A.9), however, if the planning horizon is"xed at all current and future dates. Given that Bowden's procedure only involves computation of "ve inverses, it is likely to be signi"cantly more e$cient than the standard matrix Riccati equation recursions whenever the planning horizon is"xed at all current and future dates.

It is well known that the mathematical problems associated with the solution of the linear}quadratic optimal control problem and of the Gaussian optimal "ltering problem are dual. Therefore, the Gaussian optimal "ltering problem can also be reduced to the solution of a linear equation system with a block-tridiagonal matrix coe$cient structure. We do not go into the details here.

References

Amman, H., 1996. Numerical methods for linear-quadratic models. In: Amman, H.M., Kendrick, D.A., Rust, J. (Eds.), Handbook of Computational Economics, vol. 1. North-Holland, Amster-dam, pp. 587}618.

Amman, H., Neudecker, H., 1997. Numerical solutions of the algebraic matrix Riccati equation. Journal of Economic Dynamics and Control 21, 363}369.

Anderson, E.W., Hansen, L.P., McGrattan, E.R., Sargent, T.J., 1996. On the mechanics of forming and estimating dynamic linear economies. In: Amman, H.M., Kendrick, D.A., Rust, J. (Eds.), Handbook of Computational Economics, vol. 1. North-Holland, Amsterdam, pp. 171}252. Anderson, G.S., 1997. A reliable and computationally e$cient algorithm for imposing the saddle

point property in dynamic models. Mimeo, Board of Governors of the Federal Reserve System. Aoki, M., 1989. Optimization of Stochastic Systems, 2nd ed. Academic Press, San Diego. Axelsson, O., 1994. Iterative solution methods. Cambridge University Press, Cambridge. Binder, M., Pesaran, M.H., 1995. Multivariate rational expectations models and macroeconometric

modelling: a review and some new results. In: Pesaran, M.H., Wickens, M. (Eds.), Handbook of Applied Econometrics: Macroeconomics. Basil Blackwell, Oxford, pp. 139}187.

Binder, M., Pesaran, M.H., 1997. Multivariate linear rational expectations models: characterization of the nature of the solutions and their fully recursive computation. Econometric Theory 13, 877}888.

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Binder, M., Pesaran, M.H., Samiei, S.H., 1999. Solution of nonlinear rational expectations models with applications to"nite-horizon life-cycle models of consumption. Computational Economics, in press.

Blanchard, O.J., Kahn, C.M., 1980. The solution of linear di!erence models under rational expecta-tions. Econometrica 48, 1305}1311.

Bowden, K., 1983. Homological structure of optimal systems, Ph.D. Thesis, University of She$eld. Bowden, K., 1989. A direct solution to the block tridiagonal matrix inversion problem. International

Journal of General Systems 15, 185}198.

Broze, L., GourieHroux, C., Szafarz, A., 1995. Solutions of multivariate rational expectations models. Econometric Theory 11, 229}257.

Chow, G.C., 1975. Analysis and Control of Dynamic Economic Systems. Wiley, New York. Deaton, A.S., Muellbauer, J., 1980. Economics and Consumer Behavior. Cambridge University

Press, Cambridge.

Gilli, M., Pauletto, G., 1997. Sparse direct methods for model simulation. Journal of Economic Dynamics and Control 21, 1093}1111.

Gilli, M., Pauletto, G., 1998. Krylov methods for solving models with forward-looking variables. Journal of Economic Dynamics and Control 22, 1275}1289.

Hansen, P.C., 1998. Rank-de"cient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, Philadelphia.

Kendrick, D.A., 1981. Stochastic Control for Economic Models. McGraw-Hill, New York. King, R.G., Watson, M.R., 1997. System reduction and solution algorithms for singular linear

di!erence systems under rational expectations. Mimeo, University of Virginia and Princeton University.

King, R.G., Watson, M.R., 1998. The solution of singular linear di!erence systems under rational expectations. International Economic Review, in press.

Klein, P., 1997. Using the generalized Schur form to solve a system of linear expectational di!erence equations. Mimeo, Stockholm University.

Lewis, F.L., Syrmos, V.L., 1995. Optimal Control. 2nd ed. Wiley, New York.

Pesaran, M.H., 1991. Costly adjustment under rational expectations: a generalization. Review of Economics and Statistics 73, 353}358.

Price, S., 1992. Forward looking price setting in UK manufacturing. Economic Journal 102, 497}505.

Prucha, I.R., Nadiri, M.I., 1984. Formulation and estimation of dynamic factor demand equations under non-static expectations: a"nite horizon model. Technical Working Paper no. 26, NBER. Prucha, I.R., Nadiri, M.I., 1991. On the speci"cation of accelerator coe$cients in dynamic factor

demand models. Economics Letters 35, 123}129.

Sims, C.A., 1996. Solving linear rational expectations models. Mimeo, Yale University.

Steigerwald, D.G., Stuart, C., 1997. Econometric estimation of foresight: Tax policy and investment in the United States. Review of Economics and Statistics 79, 32}40.

Uhlig, H., 1997. A toolkit for analyzing nonlinear dynamic stochastic models easily. Mimeo, University of Tilburg.

Zadrozny, P.A., 1998. An eigenvalue method of undetermined coe$cients for solving linear rational expectations models. Journal of Economic Dynamics and Control 22, 1353}1373.

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procedure is applicable also to models where the coe

$

cient matrices in their

companion canonical form involve a high degree of singularity.

Acknowledgements

We are grateful to Robert King, Ingmar Prucha, and particularly an associate

editor and an anonymous referee for helpful comments and suggestions, and to

Keith Bowden for sending us excerpts of his Ph.D. thesis.

Appendix. Solution of a

5 nite-horizon linear

}

quadratic optimal

control problem

In this Appendix, we consider the solution of the following

"

nite-horizon

linear

}

quadratic optimal control problem:

19 min

Mu_t`qNT~t~1_q/0

1

2 E

C

y

@

T

Zy

T

#

T~t~1

₊

q

/0

y

@

t`

q

Q y

t`

q

#

u

@

t`

q

Ru

t`

q

D

X

t

D

(A.1)

for given

y

t

, and subject to

y

t`

q

"

Vy

t`

q

~1

#

Wu

t`

q

~1

#

N

e

t`

q

,

e

t`

q

&

**$

N

(0

g

C

1 ,

R

e

),

q"

1, 2,

₂

,

¹!t

,

(A.2)

where

y

t`

q

is the

d

]1 dimensional state vector,

u

t`

q

the

f

]1 dimensional

control vector,

Z,

Q, and

V

are

d

]

d

dimensional matrices of

"

xed coe

$

cients,

and the coe

$

cient matrices

R,

W, and

N

are of dimensions

f

]

f

,

d

]

f

, and

d

]

g

,

respectively. As is usually assumed, we take the coe

$

cient matrices

Z,

Q, and

R

as being symmetric and nonsingular. Terminal conditions in practice are

typically imposed by choosing su

$

ciently large values for the appropriate

elements of

Z.

20 While the objective function (A.1) does not involve

cross-products of components of the state vector and the control vector, this is

without loss of generality, as

"

nite-horizon linear

}

quadratic optimal control

problems involving such cross-product terms can always be transformed to

the

form

of

(A.1)

}

(A.2).

21 The

information set

X

t

is

speci

"

ed

as

19For recent surveys of discrete time optimal control problems as relating to economics see, for example, Amman (1996) and Anderson et al. (1996).

20See, for example, Kendrick (1981) and Amman (1996).

(2)

follows:

X

t

"M

y

t

,

y

t~1

,

2 ;

u

t

,

u

t~1

,

2 ;

e

t

,

e

t~1

,

2 N

. Invoking certainty

equiva-lence, we evaluate all random components,

e

t`

q

,

q"

1, 2,

2 ,

¹!t

, at their mean

values and solve the certainty-equivalent counterpart of Eqs. (A.1) and (A.2).

The standard approach to the solution of the optimal control problem

(A.1)

}

(A.2) is the sweep method of Bryson and Ho (see, for example, Lewis and

Syrmos, 1995), which yields the optimal control law

u

H

_t`

_q

"!

K

t`

q

y

t`

q

,

(A.3)

where

K

t`

q

"

(W

@

S

t`

q

`1

W

#

R)

~1

W

@

S

t`

q

`1

V,

(A.4)

and

S

t`

q

"

(V

!

WK

t`

q

)

@

S

t`

q

`1

(V

!

WK

t`

q

)

#

K

@

t`

q

RK

t`

q

#

Q,

(A.5)

q

"

0, 1,

₂

,

¹!t!

1, with

S

T

"

Z. Equation (A.5) is the (Joseph stabilized

version of the) matrix Riccati equation. The feedback gain matrices

K

t`

q

are

typically computed by backward recursions on (A.4) and (A.5). Note that these

recursions involve computing

¹!t

inverse matrices, (W

@

S

t`

q

`1

W

#

R)

~1

,

q"

0, 1,

₂

,

¹!t!

1.

22 In this Appendix, we show that the solution of the optimal control problem

(A.1)

}

(A.2) can also be obtained by means of rewriting (A.1) and (A.2) in stacked

form, and solving the resultant optimality conditions using Bowden

'

s procedure

for the inversion of a block-tridiagonal coe

$

cient matrix. Application of

Bowd-en

'

s procedure can signi

"

cantly reduce the computational e

!

ort needed to carry

out the matrix Riccati equation recursions if the planning horizon is

"

xed.

Alternatively, one could write the

"

rst-order conditions of (A.1)

}

(A.2) as

a special case of the canonical form (2.2), and then use the solution techniques

described in Section 5. However, it is easily veri

"

ed that the resulting canonical

form does not contain lagged values, and only future expectations appear on the

right-hand side of the equation system. The solution method developed in the

remainder of this Appendix takes this special feature of the optimal control

problem (A.1)

}

(A.2) into account.

Following Bowden (1983), we rewrite (A.1) and (A.2) after invoking certainty

equivalence as

min

1

2 M

y

8 @

Q

I

y

8 #

u

8 @

R

I

u

8 N

(A.6)

22In the in"nite-horizon case,S

t`qand thusKt`qunder certain conditions are time-invariant. In this case the solution of the resulting algebraic matrix Riccati equation can also be computed non-recursively. E$cient methods for this purpose are reviewed, for example, in Lewis and Syrmos (1995) and in Anderson et al. (1996). See also Amman and Neudecker (1997).

(3)

subject to

V

I

y

8 "

y

8 t

#

W

I

u

8 ,

(A.7)

where

Q

I

"

A

Q

0 d

2

0 d

Q

0 d

2

0 d

F

}

F

0 d

2

0 d

Q

0 d

2

0 d

Z

B

,

R

I

"

A

R

0 f

2

0 f

R

0 f

2

0 f

F

}

F

0 f

2

0 f

R

0 f

2

0 f

R

B

,

V

I

"

A

I

d

0 d

2

0 d

!

V

I

d

0 d

2

0 d

F

}

F

0 d

2 !

V

I

d

0 d

2

0 d

!

V

I

d

B

,

W

I

"

A

W

0 d

C

f

0 d

C

f

2

0 d

C

f

0 d

C

f

0 d

C

f

0 d

C

f

W

0 d

C

f

2

0 d

C

f

0 d

C

f

0 d

C

f

F

}

F

0 d

C

f

0 d

C

f

0 d

C

f

2

0 d

C

f

W

0 d

C

f

0 d

C

f

0 d

C

f

0 d

C

f

2

0 d

C

f

0 d

C

f

W

B

,

y

8 "

(

y

@

t`1

,

y

@

t`2

,

2 ,

y

@

T

)

@

,

u

8 "

(u

@

t

,

u

@

t`1

,

2 ,

u

@

T~1

)

@

,

and

y

8 t

"

((Vy

t

)

@

,

0 @

d

C

1 ,

2 ,

0 @

d

C

1 )

@

.

Substituting the constraints in (A.7) back into (A.6), we obtain the unconstrained

optimization problem

min

1

(4)

where

L

I

"

V

I

Q

I

~1

V

I

@

.

(A.9)

Taking derivatives of (A.8) with respect to

u

8 @

, it is readily veri

"

ed that the

optimal control vector

u

8 H

is given by

23 u

8 H

"!

(R

I

~1!

R

I

~1

W

I

@

(W

I

R

I

~1

W

I

@#

L

I

)

~1

W

I

R

I

~1

)(W

I

@

L

I

~1

y

8 t

).

(A.11)

Inspecting (A.11) and recalling the de

"

nitions of

R

I

and

L

I

, the main

computa-tional burden in solving for

u

8 H

is easily seen to be given by the inversion of the

block-tridiagonal matrix

J

I

"

W

I

R

I

~1

W

I

@#

L

I

,

JI"

A

WR~1W@#Q~1 !Q~1V@ 0

d 2 0d 0d 0d

!VQ~1 T !Q~1V@ 2 0

d 0d 0d

F F F } F F F

d 0d 0d 2 !VQ~1 T !Q~1V@

d 0d 0d 2 0d !VQ~1 F

B

,

(A.12)

where

T

"

WR

~1

W

@#

VQ

~1

V

@#

Q

~1

,

(A.13)

and

F"

WR

~1

W

@

#

VQ

~1

V

@

#

Z

~1

.

(A.14)

Using Bowden

'

s Procedure to invert

J

I

results in the following recursions: Let

N

T~t

"

I

d

,

N

T~t~1

"

(VQ

~1

)

~1F

,

N

T~t~i

"

(VQ

~1

)

~1

(TN

T~t~i`1

!

Q

~1

V

@

N

T~t~i`2

),

i"

2, 3,

₂

,

¹!t!

1,

and

P

1 "

I

d

,

P

2 "

(WR

~1

W

@#

Q

~1

) (VQ

~1

)

~1

,

P

i

"

(P

i~1

T

!

P

i~2

Q

~1

V

@

) (VQ

~1

)

~1

,

i"

3, 4,

2 ,

¹!t

,

P

T~t`1

"

(P

T~t

F!

P

T~t~1

Q

~1

V

@

).

23In deriving (A.11), we have used the result (A~1₁₁#A

12A22A21)~1"A11!A11A12(A21A11A12#A~122)~1A21A11, (A.10) which holds for any set of real valued matricesA

11,A12,A21, andA22for which the left- and right-hand sides of (A.10) are well de"ned.

(5)

Then the (

ij

)th block of

J

I

~1

is given by

J

I

~1

ij

"

P

@

i

(P

@

T~t`1

)

~1

N

@

j

for

j*i

,

(A.15)

and

J

_I

~1

ij

"

N

i

P

~1

T~t`1

P

j

for

i'j

,

(A.16)

i

,

j"

1, 2,

2 ,

¹!t

.

Note that computation of

J

I

~1

involves the computation of only

"

ve inverses,

R

~1

,

Q

~1

,

Z

~1

, (VQ

~1

)

~1

, and

P

~1

_T~t`1

. Only the solution for

u

t

in (A.11) is the

solution for the stochastic optimal control problem (A.1)

}

(A.2), however, as the

solutions for

M

u

t`

q

NT~t~1

q

/1

do not re

#

ect the stochastic innovations in periods

t#

1,

t#

2,

₂

,

¹!t!

1. To compute the solution for

M

u

t`

q

NT~t~1

q

/1

, the analog

of (A.8)

}

(A.9) needs to be constructed for the optimal control problems in

periods

t#

1,

t#

2,

₂

,

¹!t!

1. The structure of these problems is the same

as that at

t

, (A.8)

}

(A.9), however, if the planning horizon is

"

xed at all current

and future dates. Given that Bowden

'

s procedure only involves computation of

"

ve inverses, it is likely to be signi

"

cantly more e

$

cient than the standard

matrix Riccati equation recursions whenever the planning horizon is

"

xed at all

current and future dates.

It is well known that the mathematical problems associated with the solution

of the linear

}

quadratic optimal control problem and of the Gaussian optimal

"

ltering problem are dual. Therefore, the Gaussian optimal

"

ltering problem

can also be reduced to the solution of a linear equation system with a

block-tridiagonal matrix coe

$

cient structure. We do not go into the details here.