an alternative methodology for solving nonlinear forward-looking models

JOURNALOF

Ekonomic Dy namics

ELSEVIER Journal of Economic Dynamics and Control

19 (1995) 711-734 8~ Chtd

An alternative methodology for solving nonlinear forward-looking models

Raouf Boucekkine

Departamento de Economia. Universidad Carlos III de Madrid, 28903 Getafe, Spain CREST and CEPREMAP, Paris, France

(Received January 1993; final version received April 1994)

Abstract

This paper presents a new methodology for solving nonlinear deterministic forward- looking models. Based on a relaxation algorithm described by Laffargue (1990), the methodology is theoretically founded on a general multivariate linear model. Then, a complete experimental scheme is provided in the nonlinear case, including solution time horizon selection and saddlepoint path testing strategies. The proposed experiment is mathematically robust and it does not require any expert knowledge in numerical analysis. It is especially adapted to the simulation exercises conducted on medium scale economic models.

Key words: Expectations; Computational techniques JEL classi’cation: D84; C63

1. Introduction

Since the last decade, the solution of forward-looking models has been one of the major topics in computational economics. Indeed, given the numerous deficiencies of the adaptive expectations schemes (first emphasized by Lucas, 1976), there is no way to model intertemporal economic dynamics without some

This paper is part of a collective work involving Jean-Pierre Laffargue, Pierre Malgrange, Michel Juillard, and myself at CEPREMAP, Paris. I would like to thank Cuong Le Van, Jeff Fagnart, two

anonymous referees, and the editor for many stimulating suggestions. Of course, I am solely

responsible of any remaining error.

0165-1889/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved

SSDI 016518899400800 W

112 R. Boucekkine / Journal of Economic Dynamics and Control 19 (I 995) 71 I-734

forward expectations specifications. However, when integrating such specifications, the required solution techniques become nontrivial as forward expectations do not allow for dynamic recursivity. To get an immediate idea of this feature, let us consider the following infinite time univariate model:

x,+ 1 = a-x, + b.x,_ 1, t2 1, x0 given,

where a and b are parameters. The infinite time horizon form and the considered deterministic framework correspond to the specifications adopted in most forward-looking models used in economic policy design, the deterministic setting fitting the consistent expectations characteristic of these models. For such a model type, the initial condition x0 is no longer sufficient to compute the solution path {x,, r 2 l}; that is, x1’s value depends on the unknown expectational term x2. In practice, this indeterminacy problem is surmonted by imposing a second boundary value or constraint (terminal condition) at a chosen period T > 1 (solution horizon). The second boundary value could use either a model’s property (especially its long-run structure) or an off model informa- tion (through the construction of baseline solutions; see, for example, Masson et al., 1990). The infinite time model is then replaced by a finite time one with two boundary values. If we choose a terminal condition of the form xT+ 1 = x0, x0 and T fixed, the corresponding finite time system takes the following form:

S(T) x0 given, x,+~ = a-x, + b*x,_I, l<tlT, and x~+~=x’.

Nonetheless, truncating the infinite time structural models raises three important problems:

(i) Resolution problem - Even when imposing terminal constraints as second boundary values, the dynamic nonrecursivity is still effective and so, simul- taneous systems solution methods are required.

(ii) Approximation problem - As we ultimately solve finite time approximations of the structural models, we must ensure that the choices of the solution horizon and/or the terminal constraints allow for a good approximation of the original models.

(iii) Saddlepoint problem - The truncation approach raises a third problem related to a fundamental property of forward-looking models; that is they could admit an infinity of stable solutions. As one can see, the terminal condition instrument is basically a solution path selection procedure, and in this sense it only allows to locate a particular solution, possibly among an infinity of equally correct ones. From the point of view of economic forecasting, it is an embarrassing outcome and some authors exploited it to question the use of forward expectations in economics (see, for example,

R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 7/l-734 713

Gourieroux et al., 1982). Beginning with Begg (1982), a view emerges as solving forward-looking models makes sense as long as they admit a unique stable solution, namely a saddlepoint solution. Our third issue consists in identifying the numerical instruments allowing to test for the saddlepoint solutions.

Actually, most of the contributions in this area are devoted to the resolution problem (i). A natural method in the context of two boundary values problems is the well-known shooting technique. Applied on our introductive example S(T), it consists in writing x1 as a function of the second boundary value x0 and then computing x1’s solution value using a Newton-Raphson updating scheme (see Lipton et al., 1982, for details). Unfortunately, shooting methods suffer from important numerical instability problems (in particular, a huge sensitivity to xi’s initial choice), which make them problematic in practice.’

Further research was devoted to set more tractable devices. A major outline of this approach is the introduction of first-order iterative schemes (FOI) - see Fair and Taylor (1983), Hall (1985) and the ESRC Macroeconomic Modelling Bureau (Wallis et al., 1986; Fisher, 1990, 1992). Applied on our example S(T), the main idea is to solve simultaneously for {xi, x2, . . , xT} using an adequate FOI on a stacked matrix form of S(T). The latter statement describes exactly Hall’s method, whereas Fair and Taylor’s algorithm includes two main FOI loops: an inner loop solving for fixed expectations and an outer loop updating the expectations. Until now, Fair-Taylor’s technique is the most popular among the practitioners. It is a multiple-loop scheme allowing a better numerical control relatively to Hall’s algorithm, through a larger set of experimental parameters. However, recent contributions clearly showed its weaknesses with respect to the three problems mentioned above. First, the specific outer loop of this algorithm involves a high numerical cost in practice, as most macroeconometric models include an important number of expectational terms (see Fisher, 1992, Ch. 2, for details). This feature motivates the HollyyZarrop approach (1983), which consists in replacing the mechanical expectations updating scheme of Fair-Taylor’s algorithm with an optimization setting in the tradition of optimal control theory.’ On the other hand. the solution horizon fixation procedure in Fair-Taylor’s algorithm is particularly questionable.

’ Although the shooting could be decomposed into intermediate steps allowing for a better control of

the instability problems (multiple shooting), it ultimately necessitates a burdensome work.

*Although this approach is likely to lower the number of iterations in the outer loops, it is not sure

that it dominates Fair-Taylor’s algorithm in terms of numerical cost, precisely in the presence of

a high number of expectational terms.

714 R. Boucekkine / Journal of Economic Dynamics and Control 19 (I 995) 71 l-734

It involves a third loop on the solution horizon, whose convergence is achieved once the solution path on a given time interval of interest is stabilized. For a certain class of models (especially, near unit roots models as argued Laffargue, 1990), such a criterium could lead to select inadequate solution horizons with respect to the approximation problem. Finally, there is no guarantee that the solutions located by Fair-Taylor’s technique are saddlepoint (see some numerical evidence in Fisher, 1992, Ch. 3).

The latter arguments point out the necessity of building more global methodologies taking into account all the problems, and not only the resolution step. Along this line, a major contribution is due to the ESRC Macroeconomic Modelling Bureau. The proposed methodology is based on the FOI as a solution technique and on various sensitivity studies involving mainly the solution horizon and the terminal condition formulation. In our introductive example S(T), the latter sensitivity study consists in altering the condition xr+ 1 = x0, for example, replacing it by a constant level one, xT+ 1 = xT, or a constant growth one, xr+1/xr = xr/xr_i. If clearly more rigorous than the previous contributions, this methodology lacks robustness; following Fisher (1990, p. 127) himself: ‘ . . . Ultimately the choice of terminal conditions is dependent on the model structure . . . Testing is vital both to establish the stability properties of a model and to validate the terminal condition time-horizon employed . . . ’ Moreover, this approach requires a possibly important prior work in order to identify the adequate FOI, through the implementation of some grid search procedures. Therefore, the whole experiment could be hard to conduct.

In this paper, a simpler methodology is proposed. The methodology is proved to be efficient and robust on various models of about some tens of equations, say about 40 equations per period. It uses a Newton-Raphson relaxation algorithm described by Laffargue (1990) and some alternative procedures for solving both the approximation and the saddlepoint problems. At each linearization, the models are solved using a specific triangulation-backsubstitution (denoted TB hereafter) procedure, perfectly adapted to the distributions of zeros of the involved Jacobian matrices. Our theoretical arguments are presented on the latter procedure and then extrapolated to the general nonlinear case according to a local stability setting.

The paper is organized as follows: Section 2 provides some theoretical foundations for the TB procedure on a generalized Blanchard-Kahn multivariate model. In Section 3, we present some theoretical bases for solving the saddlepoint problem in line with the traditional numerical concept of well- conditioned problems. The result is used in Section 4 when defining the global alternative methodology in the nonlinear case. The specified methodological scheme is supported by some numerical evidence. We conclude with some computational aspects of the solution method and some limitations of our framework are examined.

R. Boucekkine J Journal of Economic Qvnamics and Control 19 (1995) 71 i-734 715

2. A theoretical analysis of the TB procedure

As explained in the introductive section, the triangulation-backsubstitution (TB) procedure covers the resolution technique in the linear case. So, we analyze it directly on a linear model of a generalized Blanchard-Kahn form:

UW Bx,- I + Aly, + A$x, + cy,. 1 = Dw,, t2 1, .x0 given,

where y, is the (p x 1) vector of the anticipated endogenous variables, x, is the (4 x 1) vector of predetermined endogenous variables, and w, is the (r x 1) vector of the exogenous variables. We set n = p + q, the number of endogenous variables and equations of the model (M). B(n x q), A 1 (n x p), Az(n x q), C(n x p), and D(n x r) are matrices of constants. We denote by A the horizontal concatenation of AI and A,: A = [A, AX]. A is a square (n x n) matrix, and it plays a central role in our theoretical developments. The considered model (M) is a generalized form of the deterministic version of Blanchard and Kahn’s model (1980) in the sense that we do not constrain it to be of a first difference structure. However, the lengths of the temporal leads and lags are required to be equal to one. Actually, this is a very weak assumption as it is almost always possible to transform the models with longer leads and lags into the generic form (M) using some very elementary algebraic operations (see, for example, Broze et al., 1989, Ch. 5).

For convenience, we set x0 = 0, r = 1, and we assume that the (n x 1) vector W, = Dw, is zero except for t = 1: W, = Dw; = dw # 0. This obviously corre- sponds to an impulse shock occurring at the first period. Once assumed that there is no unit root,3 we can conclude that the model has a unique stationary equilibrium, namely y* = 0 and x * = 0. Thus, we can consider the following finite time approximation system:

I x0 = 0,

S(T) &-I + Aly, + Azxt + CY,+~ = W,, 1 I t < T,

yT+1 =y* =o.

The consequences of considering a different terminal conditions are briefly examined in the concluding section. In this one, we are concerned with thk

3The assumption is made for convenience; even in the presence of unit roots or say hysteresis

variables, we can as well utilize the theoretical framework developed in this section and particularly

the boundary value yT+ 1 = 0, for example, by conveniently scaling certain variables as explained in

a recent contribution by Don and Van Stratum (1994).

716 R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 71 l-734

theoretical characterization of the TB procedure as described by Laffargue (1990). First, we analyze the triangulation step of the procedure for a fixed solution horizon.

2. I. A theoretical analysis of the triangulation step

Let us write the system S(T) in a stacked matrix form:

E(T)

I(q)

B Al A2 C

B Al A2 C

. . . . . .

. . . . . .

B AI A2 C

I(P)

x0

Yl

Xl

Y2

x2

YT

XT

YT+ 1

where I(k) is the identity matrix of dimension k, k > 0. We denote by J(T) the (n + 1) T square matrix affecting the vector [x0, y,, x1, . . . , yT, XT, yT+ 1]’ in the previous equation. Observe that matrix J(T) has a very special sparse form. To each solution period t, 1 I t I T, is associated a block, say BL,, and all the blocks share the same possibly nonzero elements, that is the concatenation [B A, A2 C]. Any block BL,, 2 I t I T, is obtained from the anterior block BL,_ 1 by translating matrix [B Al A2 C] to the right, such that the submatrices B and Al of BL, are exactly below the submatrices A2 and C of BL,_l, respectively. Given the special form of matrix J( T ), solution methods involving Gaussian elimination are natural, as argued by Stewart (1973, p. 131): ‘Gaussian elimination leads itself naturally to the reduction of matrices with special distributions of zeros . . . ’ More precisely, as the jacobian matrices are in block

R. Eoucekkine / Journal of Economic D_vnamics and Control 19 (1995) 71 I- 734 717

echelon form (see, for example, Baker and Porteous, 1990, p. 20), rows operations are sufficient to triangulate them, and whatever the order of these operations, the resulting reduced forms will be the same. Of course, it remains to see why the procedure described by Laffargue (1990) should work at least for well-specified models, as Gaussian elimination could fail (see Golub and Van Loan, 1989, p. 102). To this end, some key features of the triangulation procedure need to be isolated.

The main characteristic of the latter procedure is its block-by-block working, following the chronological order. It involves two operations:

(01)

(02)

Zeroing the submatrices B of each block BL,, 1 I t I T, using elementary Gaussian transformations.

Transforming the diagonal blocks of matrix J(T) into identity matrices, using Gaussian transformations and partial pivoting.

Concretely, operations (0,) are applied on the square blocks of type A = [A1 A21 and transform them into the identity matrix Z(n). Submatrices B being the unique possibly nonzero elements below the diagonal, we get a triangular form with unitary diagonal elements, once operations (0,) are performed. Partial pivoting, i.e., rows permutations, is the device allowing to control Gaussian elimination failures along the triangulation step. As argued Golub and Van Loan (1989, p. 115): ‘ . . . Partial pivoting could be used with confidence.. ’ - although its associated round-off error is theoretically nonzero, it provides very good performances in practice.

Let us study now the working of the TB procedure. As the required operations (0,) and (0,) are identically reproduced beginning with the block BL2, it is sufficient to analyze if for T = 2. So, we focuse on equation E(2):

I I(q)

B A, A2 C

B AI A2 C

I(P)

(i) Consider the first block, BLI = [B A, A2 C 0 . . . 01. To zero its submatrix B(n x q), we obviously use the identity matrix Z(q) of the initial conditions block. The corresponding Gaussian elimination is equivalent to multiplying equation

718 R. Boucekkine 1 Journal of Economic Dynamics and Control I9 (1995) 71 l-734

E(2) by the elementary Gaussian transformation matrix MB: I(q) - B Z(n) MB=

I 1 Z(n)

Z(P)

The diagonal block of J(2), included in BLr, is exactly the concatenation A = [A, A,] and so we have to transform matrix A into the identity matrix Z(n). The operation uses a product of permutation matrices, say PA(l), corresponding to the partial pivoting device, only affecting the rows of the block BL1 and allowing to overcome the occurrence of near-zero diagonal elements in A. The reduction also uses a product of Gaussian transformations, say MA(l), to zero the off-diagonal elements of A. Ultimately, these operations alter submatrix C of BL1 and the corresponding rows of the right side of E(2) (here dw). We denote by Q(1) = [Qr(l) Q*(l)]’ the altered form of C, Qr(1) (resp. Qz(l)) being the first p (resp. last q) rows of Q(1). We also denote by R(1) = [R,(l) R,(l)]’ the modified block of the right side of E(2), with analogous definitions for RI (1) and R,(l). Writing BL1 in its transformed form yields the equation:

Z(q) 0 Z(P) 0 QIU 0 0 z(q) QzU

B AI

1

1 A2 C

Z(P)

x0

Yl

Xl

Y2

x2

_ y3 1 = 0

RI(~)

k(l) 0

0 1. (ii) Given the previous matrix form, it is easy to see why submatrix B of BL2 is

zeroed exactly as in the anterior block BL1; that is, in the two cases, the submatrices B are just below the identity matrix Z(q). Nonetheless, the involved Gaussian transformation has different consequences in the case of BL2. As it consists in multiplying the equation by the matrix

MB=

-w Z(P)

Z(q) - B Z(n)

Z(P)

R. Boucekkine /Journal of Economic Dynamics and Control 19 (1995) 7/l-734 719

both submatrix AI of BL2 and the corresponding rows of the right side of the equation are modified. Precisely, Al is transformed into Al - BQ,(l), and a term - BR,(l) appears on the right side of the equation, which becomes [0, R,(l) R,(l) - BR,(I) 01’. The major consequence of this feature is that the diagonal block of the transformed J(2), included in the altered second solution period block, is the concatenation [A, - BQ,(l) AZ], and not [A, AZ] as is BL1. Thus, operation (0,) has to be conducted on the former submatrix, involving as before a permutation matrix P,(2) and a Gaussian transformation matrix M,(2). Similarly, the operations transform submatrix C of the second period block into Q(2) = [Qr(2) Q2(2)]’ an,d the corresponding rows of the right side of the equation into R(2) = [R,(2) R,(2)]‘, with the same conventions as before.

The final form of the system is

- 44

0 I(P) 0 Q,(l)

0 0 I(q) Q2(1)

0 I(P) 0 Q1(2)

0 0 I(q) Q,(2)

I(P)

_ - x0

Yl

Xl

4’2

x2

Y3 _ _

=

0 -

R,(l)

R,(l)

R,(2)

R,(2)

0

It is then easily solved by backsubstitution. As one can see, the working of the procedure can be summarized by the specification of the two sequences Q(t) and R(t), 1 < t I T. Hence, for analyzing the procedure, we can as well study the latter sequences. A first interesting result follows directly from the previous description of step (ii): clearly, the block-by-block working implies that some elements of a transformed block BL,_ 1, 2 I t i T, are involved in the operations of the posterior block BL,. These elements are identified in the description of step (ii) as being Q2(t - 1) and R2(t - 1). Using the latter remark, we can introduce a kind of transition functions formalizing this dependence relation:

Proposition 1. Provided that the sequences Q(t) and R(t) exist, then:

(a) Q(t) and R(t) are dejined by two time-dependent nonlinear matrix functions S I,[ and S2,, such that Q(t) = Sl,kQ2(t - 1)) and R(t) = S2.JQ2(f - I), R,(t - l)), 2 I t I T, Q(1) and R(1) gioen.

(b) Thefunctions S, ,t and S,,, are time-independent if the reduction of the matrices K, = [A, - BQ,(t - 1) A21 requires the same permutation matrices,4 i.e., tf PA(t) = PA(t - l), 2 I t I T.

41n matrix computation language, if the matrices K,, 2 I f I T. have the same pivot structure.

720 R. Boucekkine 1 Journal of Economic Dynamics and Control I9 (I 995) 71 I-734

Proposition 1 is used in Section 3. Property (a) is a simple formalization of the working of the triangulation: Sr ,,(.) (resp. S,,,(e)) is an (n x p) (resp. n x 1) matrix whose elements depend on the elements of submatrix Q2(t - 1) (resp. submatrices Q2(t - 1) and R,(t - 1)). The dependence is nonlinear because the reduction of the matrices K, requires the normalization of their diagonal elements. Property (b) is also straightforward: as the terms Q2(t - 1) appear in the same manner in the matrices K,, 2 I t I T, and given that the matrices Q(t) are the alterations of the same matrix (i.e., C), the functions S,,,(s) must be the same if the rows of K, are permuted in the same way. The same argument holds on the functions

S,.,(*). By matrix similarity arguments, the triangulation scheme should work if

the Jacobian matrices J(T), for every T, are invertible. For T = 1, we get the elementary regularity condition: matrix A invertible. If we write the model as

Bx,-1 + ACY, x,1’ + CY,+, = Dw,, this condition means that we can solve uniquely for the contemporaneous variables given the past (i.e., x,_ r) and the future (i.e., y, + 1 ).

As one can see, Gaussian elimination-based methods differ clearly from FOI schemes, to the extent that they solve ‘explicitly’ the involved linear systems. Whereas the convergence of the FOI for every initial condition requires a spectral radius condition, the triangulation procedure always works if a unique solution exists. To obtain an analogous property for the FOI, say for the Gauss-Seidel technique, we usually refer to symmetric positive definite matrices J(T) (see also Golub and Van Loan, 1989, Theorem 10.1.2, p. 509) a feature particularly unexpected in the case of forward-looking models. That is why we argue that triangulation schemes are much more natural in the latter case: for well-specified models, no prior work on the initial conditions and no prelimi- nary grid search procedures are needed. Let us now complete the analysis of the TB procedure by examining its backsubstitution step and varying the solution horizon T.

2.2. TB procedure and stable solution paths

Hereafter, we assume that the finite time systems admit unique solutions for sufficiently high solution horizons, such as we can conduct the triangulation device. The backsubstitution step is then applied on the following relations induced by the triangular system:

(TrJ Y, + Ql(t)yt+ 1 = R,(t) for ljtjT,

W-2) xt + Q2(t)yt+l = R,(t) for 1 I t I T.

R. Boucekkine 1 Journal of Economic Dynamics and Control 19 (I 995) 71 l-734 721

Denoting by y:(T) and x:(T), 1 I t I T, the solution values, the backsubstitution yields

T-1

(Bl) y:(T)=Ri(t)+ x(-l)’ R,(t+i) for 1 <rlT, i=l

W) x:(T) = R20) - Q~(~)Y;C+I(T) for 1 I t 2 T.

The solution values of the finite time systems depend on the solution horizon, T.

It remains to study how x:( T ) and y$( T ), 1 I t I T, evolve when T increases. Given that the procedure TB solves ‘explicitly’ the finite time systems, we can already conclude for two cases, regarding to the stability properties of the structural infinite time model (M), and provided the terminal condition type:

(9

(ii)

If the model (M) is unstable, then the TB procedure will fail to detect a solution, for sufficiently high solution horizons, due to the terminal condition yT+ I = 0.

If the model (M) is stable ‘saddlepoint’, namely if the model admits a unique stable solution, then the main matrices J(T) should be invertible for sufficiently high solution horizons, and the TB procedure will detect a unique solution path.

The problem arises when the model (M) admits an infinity of stable solutions. That is, the use of the terminal condition y r + i = 0 could allow to select a unique solution path even for very high solution horizons T. We will establish this feature concretely in the next section.

Before, we need a formal stability setting with respect to the TB procedure working. First, observe that a solution path {y:( T ), x:( T ), 1 I t I T ), selected by the TB procedure, must satisfy two trivial admissibility conditions to be consistent with the stable solutions of the structural model (M):

(Ad,) For every t fixed, x:(T) and y:(T) must admit finite limits when T goes

to infinity.

(Ad,) The limits of xF( T) and y?(T) are zero when T goes to infinity.

Condition (Ad,) simply means that a stable solution path should not contain a point to such that x:(T) or y:(T) are explosive when T goes to infinity. Condition (Ad,) is the long-run state requirement: the selected solution paths must fit the long-run state of the structural model (M).

Conditions (Ad,) and (Ad,) could be expressed in terms of the TB procedure outcomes, according to the following proposition:

722 R. Boucekkine J Journal of Economic Dynamics and Control 19 (1995) 711-734

Proposition 2. (i) Conditions (Ad,) and (Adz) are satisfied if:

(C,) the sequence R(T), T 2 1, admits a null limit when T goes to in$nity.

(C,) the series CTiI’( - 1)’ {nj;k Qi (j + l)} RI (i + 1) has a finite limit when

T goes to injinity, and

(C,) the sequence Q(T) is bounded.

(ii) Ifconditions (Ad,) and (Ad,) are satisjed, then conditions (C,) and (C,) hold.

Condition (C,) allows to fulfill the admissibility requirement (Ad,). It is obtained directly from equations (B,) and (B2) with t = T and using the terminal condition yT+ I = 0; that is,

y;(T) = R,(T) and x;(T) = R,(T).

It follows that (Ad,) is satisfied if and only if the vector R(T) = [R,(T) R,(T)]’ admits a zero limit when T increases. Condition (C,) and the latter result allow to prove that, for every i > 0, both yF(T + i) and x$( T + i) go to zero, when T goes to infinity. Writing equations (Tr,) for successive solution horizons T + i, i > 0, and setting t = T yields

y*,(T + 4 + QI(T)Y?+I(T + 4 = Rl(T).

For i = 1, condition (C,) ensures that the limit of R,(T) and Y~+~(T + 1) is zero when T grows. As Q,(T) is bounded by condition (C,), then the limit of y ;( T + 1) is zero. For i = 2, we use the latter result: as yt + 1 (T + 2) goes to zero, y;(T + 2) goes also to zero. The complete ascending recurrence scheme is straightforward. The same argument is used to show the zero limit property for x;( T + i) on equations (Tr2) as Q2( T) and R2 (T ) have the same characteristics as Q 1 (T ) and RI (T ) respectively, by conditions (C,) and (C,).

We can now conclude that it exists To > 0 such that for every t 2 To, the limits of x:(T) and y:(T) are zero when T goes to infinity. For t < To we use also equations (Tr,) and (Tr,):

(Trl) Y:(T) + QlWy;"+l(T) = R,(t),

(Trz ) x:(T) + Q2(t)y;C+1(T) = R2(t) for 1 I t I T,, - 1.

Condition (C,) stipulates that y:(T) has a finite limit when T goes to infinity. Using equation (Tr, ) for t = 1, it follows that the limit of yf (T) is also finite.

R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 71 I-734 723

Repeating the same argument on the successive equations (Tr,) and exploiting the fact that the terms Qi(t) and R,(t), 2 I t I To - 1, are independent of T, straightforwardly allow to check that the limits of y:(T) are finite for every 1 I t < T,, - 1. Again, the same ascending recurrence scheme is applied on the successive equations (Tr2) to ensure that the limits of x:( T ), 1 I t 5 T, - 1, are finite. The nonexplosivity condition (Ad,) and the result (i) are consequently fulfilled.

The result (ii) is more immediate. If condition (Ad,) holds, the limit of y:( T) is finite and so condition (C,) is checked. On the other hand, as seen before, condition (Ad,) is equivalent to condition (C,).

Using Propositions 1 and 2, we are now able to study the outcomes of the TB procedure with respect to the saddlepoint problem. In the next section, using a simple bivariate model, we show that the solution procedure may select a unique admissible solution path, even if the structural model (M) admits an infinity of stable solution paths.

3. The saddlepoint problem: A simple illustration

Let us consider the following bivariate model (p = 4 = l), denoted (M,) hereafter:

bxt-I +~lY,+~2x,+cY,+l =o,

a$, + b’xml = w, for r 2 1.

We set c = a; = 1 for normalization, x0=0 and w,=O for t# 1, wi = Aw # 0, as in the previous section. We also assume that matrix

A = al a2 [ 1 0 1

is invertible, which implies a, # 0. The eigenvalues of the model (M,) are E., = - al and A2 = - b’. Of course, we assume that A2 is less than one in absolute value, otherwise the second equation is explosive. We do not allow for unit roots as in the previous section, and we use the TB procedure for solving the finite time approximations of the model (M,), obtained by the terminal conditions yr+ i = 0 for solution horizons T.

3.1. The stable solution paths attained by the TR procedure

Given the elementary structure of the model (M,), we can derive explicitely the transition functions S,,, and S2,,, 1 I t < T, mentioned in Proposition 1.

724 R. Boucekkine / Journal of Economic Dynamics and Control I9 (1995) 71 l-734

The triangulation step does not require partial pivoting in this case, so as the latter functions are time-independent, Sr,, = S1 and S,,, = SZ for every r 2 2. We use conditions (C,), (C,), and (C,) of Proposition 2 to check that the solution paths selected by the procedure are not explosive when T grows. These findings are summarized in the following proposition:

Proposition 3. (i) The transition functions S1 and S2 are given by

Q(t) = Sl(Q2(t - 1)) = [ + 0 1 ‘, 2 I t I T, with Q(1) = 1

[ G 1 0 I’ ,

W) = S2tQ20 - 11, R2(t - 1))

R,(r - 1) L2R2(t - 1) 1 ‘)

2stsT, with R(1) = [F dw Awl’.

(f’i ~o;diri~ (C,) and (C,) are always checked, Condition (C,) holds ifand only if

1 A2 <

Establishing the result (i) only requires very elementary rows operations on the matrices J(T) associated to the finite time approximations of the model (M,), along the lines of Section 2.1. The corresponding matrices Q(t) are found constant; in particular, submatrices Q2(t), defined by Q(t) =

[QI (t) Q2@1’, are always null. As a consequence, the diagonal blocks of the main matrices J(T), included in each solution period block, are all equal to matrix

A= al a2 . [ 1 0 1

Since a, is assumed nonzero, the reduction of A to the identity matrix I(2) does not require any pivoting.

The result (ii) is obtained as follows. Condition (C,) is directly checked. Utilizing the transition function S2, we get

R(T) = CR,(T) &V)I = R2(t - 1) (A2)T-1d~ I’ .

R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 71 l-734 125

A2 being strictly less than one by assumption, R,( T ) goes to zero when T grows. AS

R,(T) goes to zero. Consequently, R(T) has a null limit, and the stability condition (C,) is always checked. Condition (C,) stipulates that the series ~~=~l’( - 1)’ (nj:=&QI(j + 1)) Rl(i + 1) is of a finite limit when T goes to infinity. Using the transition functions S1 and S2, the series could be written as

b + a& T-1 12 i

1112

AwE x. i=l 0 1

(C,) is then fulfilled if and only if 1 A1 /A2 ( < I, which establishes the result (ii). In the saddlepoint case, the condition (C,) is always verified, and the solution

procedure always locates a unique stable solution path. When the structural model (MO) admits an infinity of stable solutions (i.e., 1 i1 1 < 1 and ( A2 1 < l), the TB procedure may locate a unique stable solution path (if I A2 I < I II I < 1) or may not (if Ii1 1 < /A21 < 1). This simple exercise points out the necessity of building some complementary procedures testing for the saddlepoint solutions. In the following subsection, we briefly present a new testing strategy proposed by Boucekkine and Le Van (1993). The test is illustrated on the model (MO), and supported by some numerical evidence in Section 4.

3.2. Testing for the saddlepoint solutions

The test is in the spirit of the traditional problem condition analysis in matrix computations. The finite time systems:

S(T) Rx,_1 + Ary, + A2x, + Cy,,, = W, for 1 I t I T,

.YTfl = 0,

are perturbed in the following way:

I x0 = 0,

Sp(T) Rx:-, + Aryr + A,xr + Cy:+l = Wp for 1 I t < T,

YT+l = 0,

126 R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 711-734

wtth x;p = xt + ar, y, ’ = y, + a2, 1 <t I T, with c1r and t12 two constants vectors, ~1~ # 0, and consequently, Wr = W, + (B + Al)al + (C + Az)c+ Of course, as x2 # 0, and W, = 0 for t > 1 in our setting, W p is generally nonzero for t > 1, and so the zero long-run state is no longer sustainable. As a consequence, the nonexplosivity condition (Ad,) of Section 2.2 should be modified to fit the problem S’(T) structure:

(Ad;) The limits of x; and y*,(T), solutions of SP(T), are finite when T goes to infinity.

Then, we can prove, independently of the resolution method, that the solution paths of the systems SP(T) check the nonexplositivity conditions (Ad,) and (Ad;) if and only if the structural infinite time model admits a unique stable solution (Boucekkine and Le Van, 1993).

The test could be significantly illustrated on our example (M,) solved by the TB procedure. First, we must modify conditions (C,) and (C,) of Proposition 3 to fit the new admissibility condition (Ad;), that is:

(C;) R(T) admits a finite limit when T goes to infinity.

(C;) The sequence Q(T) has a finite limit when T goes to infinity.

Let us set ~1~ = 0 and az = c( # 0. The corresponding perturbed finite time system S’(T) is

x0 = 0,

S’(T) bx,- r + alyP + uzx, + yp+ r = 6, for 1 I t I T,

‘

x, + b’x,_1 = w,,

YT+1 =o,

with 6, = 6 = (a1 + l)cr, 1 I t 5 T - 1, and 6r = 6’ = arcr. The working of the TB procedure on the system S’(T) can be summarized as

follows:

Proposition 4. (i) The transition function S, of Proposition 3 is unchanged. (ii) The transition function S2 of Proposition 3 is altered as follows:

R(t) = S,(QJt - 1). R,tr - 1))

=[(“+A:2’2) R2(t - 1) - ; A,R,(t - 1) 1 I‘, 2<t<T,

R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 71 I-734 721

and

Aw I’ .

(iii) Conditions (C;) and (C;) are always verijied. Condition (C,) is checked fand only if \A,\ > 1, i.e., if the saddlepoint conditions are checked.

As the main matrices J(T) are not altered by the perturbation, the outcomes related to J(T) are not modified; consequently matrices Q(t) and the transition function Sr are unchanged. On the contrary, the perturbation alters the right sides of the equalities, and consequently the vectors R(t). The perturbation only adds additive terms (i.e., 6,/L,, 1 I t I T) to the initial expression of R(t). It is easy to see that the assumption 1 ,I2 1 < 1 allows to check directly the condition (C;), i.e., R(T) goes to [ - 6’/1r 01’ when T goes to infinity. Condition (C;) is directly checked as before.

With the new transition function SZ, the series invoked in condition (C,) is

Expliciting the sums CrLr’ (L2/L,)i and xrit (l/ir)‘, we obtain the following

expression for the series once the constants are eliminated:

As 1 i2 ) < 1 and h/(1, - 1) - 6’ is nonzero, if follows that the series is conver- gent if and only if 1 I, 1 > 1, which establishes the result (iii).

Hence, perturbing the initial finite time systems allows to distinguish clearly the saddlepoint solutions from the other stable solutions. This powerful device is used in the definition of a complete methodology for nonlinear systems, as explained in the following section.

4. A complete experimental methodology for nonlinear models

In this section, we consider the exact nonlinear extension of the model (M):

(NM) .f(x,- l,Yr,xf,yt+ r,w,) = 0 for t 2 1, .x0 given.

728 R. Boucekkine J Journal of Economic Dynamics and Control I9 (1995) 71 I-734

with the same conventions as before. fis an (n x 1) vector function. We assume that a long-run equilibrium of the model is known5, that is [ys xJ’. According to the previous sections, we set the terminal condition yr+ r = y, for a solution horizon T. For convenience, we choose x0 = x,.

The corresponding finite time approximation NS(T) is

x0 = xs,

NW) f(~~-l,~,,~~,y~+l,w,)=O for 1 ~tl T,

yT+l = Ys,

4. I. The methodology

The chosen solution method is a relaxation Newton-Raphson scheme, integrating the TB procedure. For T fixed, the relaxation is initialized with the long-run values y, and x,; if we denote by L,(T) = {xp, yp, 1 I t I T } the trajectory base, the initialization required in our framework is given by xp = x, and yp = y, for 1 I t I T. The system NS(T) is then linearized around the base L,(T), and if it exists, a solution L,(T) = {x:(T), y:(T), 1 I t I T } is computed by the TB procedure. If the trajectory Lr (T) is sufficiently close to L,(T), i.e., if the distance between L1(T) and L,(T) is less than or equal to a chosen convergence tolerance level, say E, L,(T) is the selected solution path. Other- wise, the model is linearized around L,(T) and the same operations are conducted until the location of a fixed point path, L*(T ). We denote this algorithm NTB, hereafter.

The whole experiment, including the choice of the adequate solution horizon T and the treatment of the saddlepoint problem, is conducted in two -main steps:

Step 1 - Solution horizon choice and local stability study This step involves only theBrst linearization: (v) Initialize with the base Lo(T ), i.e., exclusively with the long-run values.

Solve the first linearized form with increasing solution horizons T, using the TB procedure. If the distance between the solution path L1 (T ) and Lo( T,) increases with T, stop. Otherwise, select a time horizon T* such that for every t, l<t<T*, IIx:(T* + 1)-x:(T*)II IE’, IIY:V* + I)-y:(T*)/I I&‘, 11 xi.(T *) - x, 11 I E’, and )I yi.( T *) - y, 11 I E’, with 11 * (I a vector norm and E’ < E a chosen real number. Go to (vv).

50f course, we do not assume the uniqueness of such an equilibrium. [y. x,]’ could also be a baseline solution for a residuals-augmented version of the models (as for MULTIMOD, see Masson et al., 1990).

R. Boucekkine / Journal of Economic Dynamics and Control 19 (I 995) 71 I-734 729

(vv) Initialize with the base Lc(T*) = {xp,yp, 1 I t I T*} with xp = x, + CI~ and yp = y, + CI~ for 1 I t I T *, with ~1, and CI~ two real numbers of small magnitude, cc2 # 0. Solve the first linearized form with increasing solution horizons using the TB procedure. If the distance between the solution path L 1 (T ) and L!( T ) increases, stop. Otherwise, go to step 2.

Step 2 ~ Resolution of the nonlinear model Set T = T *, initialize with the base L,( T ), and solve the nonlinear model

with the algorithm NTB.

Step 1 only uses the first linearized form. Its first part (v) is devoted to select the adequate solution horizon T *, according to the admissibility conditions (Ad,) and (Ad,) of Section 2.2. The second part (vv) uses the saddlepoint problem analysis of Section 3: in particular, the perturbation device is implemented through the trajectory base of the relaxation. Of course, the described methodology should work for local problems. Actually, this restriction is unim- portant: from a theoretical point of view, global stability analysis is generally intractable (see Stokey and Lucas, 1989, Ch. 6); in economic practice, almost all the exercises are local.6

The solution method employed in step 2 is in the spirit of Hall’s algorithm (1985), to the extent that it uses only one iteration loop. In comparison with multiple-loop schemes, as Fair-Taylor (1983) or Holly-Zarrop (1983) algorithms, the approach is obviously much more efficient in terms of iterations number. The only inconvenience of Hall’s technique is its iteration type, which does not guarantee convergence for every choice of the trajectory base. From this point of view, multiple-loop FOI schemes allow for better numerical control, as argued in the introductive section. Our solution method takes the one iteration loop advantage of Hall’s algorithm, with a solution procedure per relaxation step, ensuring the convergence for one particular initialization if the considered models are stable. The proposed experiment is consequently much more tractable. Let us illustrate it now on a formal nonlinear model of five equations.

4.2. Some numerical evidence

The considered nonlinear model contains two anticipated variables y, = [yi., Y~,~]‘, two predetermined variables x, = [x 1 ,* x2,*]‘, and one static variable z,, in order to signify that the solution procedure also works in the presence of variables of the latter group. The equations use rational polynomial forms, and the long-run structure of the model allows for the existence of a unique

61n practice, baseline solution paths are perturbed with small magnitude shocks.

730 R. Boucekkine 1 Journal of Economic Dynamics and Control I9 (1995) 71 l-734

equilibrium, CY, z, ~1’ = CY~,~ Y,,, s z xl,s xl,J’. The equations of the model are:

X 2.1 - 0.75. y + 1.25 = 0,

d-1 Y?.t+ 1 - C’ L y2

.I +1 YlJ = 09 (5)

where a, b, c, and d are free parameters and w, is the exogenous variable of the model. The long-run values [y, z, xJ’ are assumed to be consistent with a unitary value of the exogenous variable w,. At t = 0, x1,o = x~,~ and xzqo = x~,~. For t = 1,2,3, w, deviates from its unitary equilibrium value with a magnitude equal to 1%. We aim at computing the response of the model to this temporary shock, along the lines of the methodological scheme described before. By varying the parameters a, b, c, and d, we generate the three possible local spectral configurations and we analyze each of them. The convergence tolerance levels E and E’ are taken equal to lo-’ and 5.10-6, respectively. The distance between the base L:(T) = {yt,,, y!,,, zf, xt,,, x!,,, 1 I t I T} and the solution path of the first linearized form L,(T) = {y:,*, y:.,, z:, xi,,, xi,,, 1 I t I T }, denoted by AL1(T), is defined as follows:

(i) Locally explosive models - Such a case is obtained when setting a = - 3, b = 0.5, c = 2.5, and d = 1.95. The local spectrum of the model is Sp = (4.46, -1.4, 1.14, 0.46). Such models are rejected at the step l(v) of our methodological scheme. Setting Lb,(T) = L,( T ), i.e., initializing exclusively with the long-run values, and solving the first linearized form with the TB procedure, for increasing solution horizons T, the distance ALI should be explosive when T grows. The explosivity is clear in our example, as reported in Table 1.

(ii) Injinity of local stable solutions - This configuration is obtained with a= -3, b = 1.5, c = 2.5, and d = 0.5, the local spectrum being Sp = (2.03, -0.5,0.77,0.36). Such models are rejected at step l(vv) of our scheme. Indeed, step l(v) may be insufficient to distinguish between these models and the locally saddlepoint ones. In our case, setting Lb,(T) = L,(T), and varying T from 50 to 1000, no explosive effect is detected on the first linearized

R. Boucekkine 1 Journal of Economic Dynamics and Control 19 (1995) 71 l-734 731

Table 1

T 50 100 200 ALI(T 1 4.12 1894 8.10’

T = solution horizon, AL,(T) = distance between the base

and the solution horizon path of the first linearized form.

Table 2

T 50 100 200

ALI(T 1 0.001 5.96 3.10”

T = solution horizon, A LI (T) = distance between the base

and the solution horizon path of the first linearized form.

form. On the contrary, beginning with T * = 50, y :.i is stabilized at 2.4 and x :,i at 3.71. The long-run values are attained before T * = 50. Using the perturbed base L:(T), with ~1~ = x2 = [10e4 10-4]‘, the explosivity effects are clearly obtained as reported in Table 2.

(iii) Locally saddlepoint models - Setting a = -2. h = 1.5, c = 2.5, and d = 1.5, the local spectrum is Sp = { -2.12,3.42,0.92,0.61} and the saddlepoint conditions are checked. Only this class of models is accepted by our testing strategy. Neither step l(v) nor step l(vv) reveals any explosivity feature. In particular, even for CI~ = t12 = CO.1 O.l]‘, the perturbed first linearized form remains nonexplosive. Step l(v) selects T * = 50 and the complete Newton-- Raphson algorithm NTB succeeds in locating a fixed-point path after three linearizations: for example, y T, 1 = 0.002 and x :, 1 = 7.14.

The experiment has been showed robust on various medium scale models. The methodology is perfectly adapted to economic models of about some tens of equations, say 40 equations per period, as one can deal with solution horizons of some hundreds of periods without any feasibility problem. Whether the experiment could be implemented on large scale models (say models of some hundreds of equations per period) depends on feasibility considerations, exactly as for the other methodologies. It is worth pointing out that in our case, the limitation is due to a purely feasibility issue and not to a mathematical robustness problem. The concluding section sheds light on this problem and on some other computational aspects.

5. Concluding remarks

The paper presents a new methodology for simulating forward-looking models. Due to its theoretical bases, the methodology is mathematically robust.

732 R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 71 l-734

The experiment is very tractable: the relaxation base is given and no grid search procedure is necessary. Consequently, it is very likely to fit the numerical exploratory work of the economists-theorists, on general equilibrium models. For stochastic investigations, we could as well employ the Monte Carlo extension of the relaxation algorithm NTB, as explained in Boucekkine (1993).

The algorithm NTB is, by construction, more expensive in terms of storage cost and floating point operations per relaxation step than the FOI. However, in the general nonlinear case, no definitive conclusion could be got out in terms of computing time. For two reasons, at least:

0

00

Given the local weak nonlinearities of economic models, and the small magnitudes of the shocks involved in economic exercises, the convergence of the NTB algorithm is usually achieved after three or four linearizations whereas the optimal FOI require some hundreds iterations (see Fisher, 1992, Ch. 2). The advantage of the former algorithm is of course accentuated relatively to the multiple-loop FOI. The time computing comparison depends on the programming language. If we adopt a matrix language as GAUSS, we could as well replace the per element Gaussian elimination of the TB procedure by a block Gaussian elimination, as matrix inversion in such a case is very’cheap in terms of computing time.’ Then, on a standard PC 486, 66 MHz, the resolution of the model used in Section 4.2 is instantaneous for T * = 50.

Concerning the feasibility issue in the case of large-scale models, the problem is not specific to the adopted solution. method. As the TB procedure works block-by-block, we do not need storing the whole Jacobian matrices, but only some elements of each block. Therefore, the algorithm NTB is likely to be feasible for large-scale models, at least for short solution horizons, but this restriction holds also for the FOI users. Nonetheless, as the experiment required in step 1 is based on explosivity effects detection and as the latter depends on the local spectra, the first step may be misleading for large-scale models (i.e., the explosivity effects may not be revealed by short solution horizons). For a medium-scale model, the whole experimental scheme can be easily implemented on standard PC machines. For large-scale models, it remains to see how it could be amended. The point, here, is the nature of the model’s long-run structure and the terminal conditions specifications. In the ESRC Macroeconomic Modelling Bureau methodology, large-scale models are solved for short solution horizons and the terminal conditions play a central role in the validation work. The latter sensitivity study is not possible when using the NTB algorithm: as this algorithm is based on a block-by-block ‘explicit’ resolution procedure, an analogous

‘The software, developed by Michel Juillard at CEPREMAP, uses exactly this device.

R. Boucekkine / Journal of Economic Dynamics and Control 19 (1995) 71 l-734 733

sensitivity study does not make sense. A current investigation of us focuses on the possibility of utilizing steady state growth paths, in the tradition of Deleau et al. (1990), for rigorously solving short solution horizons large-scale models. A possible device is to use terminal conditions of the form yT+ 1 = y”( T + l), with y”(T + 1) the point of the extrapolated steady state growth path corresponding to the period T + 1. Of course, the approach requires the prior computation of the steady state growth paths (see the first successful attempt on a large-scale model in Loufir and Malgrange, 1994) but we can also question the scientific validity of the solution paths ‘computed through purely heuristic sensitivity studies, without a serious knowledge about the long-run structures of the models.

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an alternative methodology for solving nonlinear forward-looking models

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