Comparative studies of Gauss-Seidel and Newton-type algorithms for solving large sparse systems of equations are reported by Nkpomiastchy and Ravelli (1978), Gabay et a/. (1980) and Norman et a/. (1 983). Thc first two favour Newton's method, the third favours Gauss-Seidel. Apart from working on different test models, their setups differ in the implementation o f both schemes.
This paper studies the performance of both methods on ten different econometric models of varying size and complexity. First the choice of implementation (equation reordering, updating rules for Newton's Jacobian) is studied on a relatively small model. Qualitative and quantitative feedback criteria are considered, and an efficient reordering algorithm is discussed. On the ten models considered, the selected Newton method is almost uniformly cheaper, generally reducing the number of iterations by more than 30 percent. A final section draws attention to the possible extra gains of Newton's method in evaluating multipliers for policy analysis.
KEY WORDS Solution algorithms Fecdback Newton's method
Equation reordering Econometric models
Forecasting in economics often involves solving a large system of equations. Today's large computers can d o many iterations in a simple Gauss-Seidel solution scheme both cheaply and fast. Yet it is still useful to study more efficient solution methods because (1) the advent of powerful micro-computers that can indeed handle large econometric models has made response time an interesting criterion for model solution schemes, (2) models that contain rational expectations require solving several time periods simultaneously, and (3) in optimal policy problems with many instruments the calculation of the Jacobian still requires a considerable number of model solutions. We only consider models that are recursive in time. For an application of the ideas of this paper to a rational expectations model, see Jurriens (1985). For a practical problem with many instruments, see e.g. Sandee et al. (1984).
An important feature of most empirical forecasting models is their sparsity, i.e. the fact that each single equation involves only a few variables. The chain of relations between variables then usually can be written in an almost recursive order. A model representation with little feedback has been found useful for analysing model interdependencies, see e.g. Gallo and Gilli (forthcoming). We will discuss feedback structure in section 1 below, and use it in the design of model solution algorithms.
Two well known basic solution methods are proposed in the literature, the Gauss-Seidel