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This paper was presented at the Second International Conference on Computing and Economics and Finance, Geneva Switzerland, 26-28 June 1996. We thank Raouf Boucekkine, Jean-Pierre Laffargue, Pierre Malgrange, and Steve Symansky for encouraging us to do work in this area and Peter Hollinger for incorporating the solution algorithm into portable TROLL. We also thank Helen Hwang and Susanna Mursula for their excellent technical assistance. Juillard is assistant professor at the University Paris 8, and a researcher at CEPREMAP, Paris. Laxton is an economist in the Economic Modeling and External Adjustment Division of the Research Department. The views expressed in this paper are those of the authors and do not necessarily reflect those of the International Monetary Fund.
The original prototype for the L-B-J algorithm was written in GAUSS for a PC (Juillard, 1996). It has recently been integrated in a much more efficient manner into Portable TROLL by Peter Hollinger at Intex Solutions.
In this paper, we ignore the issue about how to solve for the true terminal conditions and just impose artificial base-line estimates. There are two general approaches to this problem. The first, which was suggested by Fair and Taylor (1983), just adds an additional iterative scheme to the basic algorithm and this iterative method is then used to eventually buildup the true model-consistent terminal conditions. The alternative is to derive a steady-state analogue model and then use this model to compute the true terminal conditions—for an example of this methodology see Black, et al (1994).
As shown is in Section IV, this method can efficiently handle problems that are substantially larger than this. Indeed, one enormous advantage of this algorithm compared to some existing algorithms is that solution speed is approximately a linear function of the simulation horizon.
It is possible that because of linearization that a zero element may suddenly become nonzero. This problem is easily dealt with by algorithms that are designed to exploit sparse systems.
In this example, the convergency criterium is set so that max||f(zt)||≤ 10-5.
Much of the discussion in the literature of the relative merits of Newton-Raphson based methods versus Gauss-Seidel has ignored techniques to deal with the sparse and repetitive structure of the Jacobian—see, for example, Hughes Hallett and Fisher (1992). Obviously, for even small models, the matrix inversion problem for an Newton-Raphson based algorithm can become incredibly time consuming without some technique to exploit the sparse structure of the Jacobian. For a discussion of available sparse matrix techniques, see Press et al (1992).
Hughes Hallett, Ma, and Yin (1996) provide an example of a nonlinear model of resource extraction where Newton-Raphson fails to converge. Of course, this is always a possibility in highly nonlinear models but we have never encountered any of these situations yet in our work with models that have a well-defined balanced growth path. However, users of MULTIMOD have found many situations where Fair-Taylor iterations would not converge without damping or using a different ordering of the model. Armstrong et al (1995) reports similar results for development work done on the Bank of Canada’s Quarterly Projection Model.
For an exhaustive review of the properties of these types of models see Bryant, Hooper and Mann (1993).
The same problems have been experienced with other software—for example Pioro, McAdam and Laxton (1996) experienced these sorts of problems using SLIM.