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 modelsForecasting 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
Summary A complete explicit formula for the expectation of the product of an arbitrary number of quadratic forms in normally distributed variables is derived, extending and confirming recent results of Magnus [4]. Incidentally, some results on traces of matrix products are presented.
What is now known in English as the Netherlands Bureau for Economic Policy Analysis (CPB) has been involved in econometric model building since its foundation in 1945. During the 60 years of developing and using the models reviewed in this article, CPB's model building has evolved significantly. Over this period, a shift of emphasis can be observed from econometrics and empiricism to economic theory. New questions from policymakers and new features in the national economy have guided research, while new developments in econometrics and economic theory were taken on board wherever they helped to improve the quality and scope of the analysis. Although considerable progress has been achieved in several spheres, even the most sophisticated and up-to-date models continue to be riddled with some longstanding limitations and weaknesses.
Frisch and Tinbergen founded the standard framework for finding the optimal economic policy by maximizing the welfare function under constraints supplied by the econometric model. Frisch worried about the reliability of the model and Tinbergen thought that it would be too difficult to specify the welfare function. Looking at current practice in Dutch policy making, both worries are relevant but the solutions proposed by the founders are not very helpful. Rather, the solution is found in applying an iterative trial-and-error procedure interfacing between the policy maker and the model-cum-expert system. The main contributions of the standard framework are its useful set of concepts, the famous order condition for a feasible solution, and the clear definition of role models for the two parties in the interaction.
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