COMPUTATIONAL ECONOMICS AND ECONOMETRICS Advanced Studies in Theoretical and Applied Econometrics Volume 22 Managing Editors: J.P. Ancot, Netherlands Economic Institute, Rotterdam, The Netherlands A.J. Hughes Hallet, University of Strathclyde, Glasgow, United Kingdom Editorial Board: F.G. Adams, University of Pennsylvania, Philadelphia, U.S.A. P. Balestra, University of Geneva, Switzerland M.G. Dagenais, University of Montreal, Canada D. Kendrick, University of Texas, Austin, U.S.A. J.H.P. Paelinck, Netherlands Economic Institute, Rotterdam, The Netherlands A.S. Pindyck, Sloane School of Management, M.I. T., U.S.A. H. Theil, University of Florida, Gainesville, U.S.A. W. Welfe, University of Lodz, Poland The titles published in this series are listed at the end of this volume. CCoommppuuttaattiioonnaall EEccoonnoommiiccss aanndd EEccoonnoommeettrriiccss EEddiitteedd bЬуy HHaannss M. Amman М. Аттan DDaavviidd A. BBeellsslleeyy А. LLoouuiiss F. Pau Р. Раи SSPPRRIINNGGEERR SSCCIIEENNCCEE++BBUUSSIINNEESSSS MМEEDDIIAA,, BВ..VУ.. Library of Congress CataIoging-in-Publication Data Computational econom ies and econometrics I edited by Hans M. A.man, David A. Belsley, Louis F. Pau. p. c •. -- (Advanced stud,es in theoretical and appllec econometrics ; v. 22) Includes index. ISBN 978-94-010-5394-5 ISBN 978-94-011-3162-9 (eBook) DOI 10.1007/978-94-011-3162-9 1. Econometrlcs--Congresses. 2. Computer simulation--Congr.sses. 1. Amman, Hans M. II. Belsley, DavId A. III. Pau, L.-F. (Louls -Fran~ols), 1948- IV. Serles. HB139.C645 1991 330' .01·5195--dc20 91-16876 ISBN 978-94-010-5394-5 Printed an acid-free pap er AlI Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 Softcover reprint ofthe hardcover lst edition1992 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS Preface vii PART ONE: ECONOMETRICS Likelihood evaluation for dynamic latent variables models David F. Hendry and Jean-Fran~ois Richard 3 Global optimization of statistical functions: Preliminary results William L. Goffe, Garry D. Ferrier and John Rogers 19 On efficient exact maximum likelihood estimation of high-order multivariate ARMA models Stefan Mittnik 33 Efficient computation of stochastic coefficients models /-Lok Chang, Charles Hallahan and P.A.V.B. Swamy 43 The degree of effective identification and a diagnostic measure for assessing it David A. Belsley 55 PART TWO: MODEL STIMULATION AND OPTIMIZATION A splitting equilibration algorithm for the computation of large-scale constrained matrix problems: Theoretical analysis and applications Anna Nagurney and Alexander Eydeland 65 Nonstationary model solution techniques and the USA algorithm P.G. Fisher and A.J. Hughes Hallett 107 Implementing no-derivative optimizing procedures for optimization of econometric models Gyorgy BaraMs 121 Information in a Stackelberg game between two players holding different theoretical views: Solution concepts and an illustration Henriette M. Prast 137 vi TABLE OF CONTENTS Exchange rate uncertainty in imperfect markets: A simulation approach Hans M. Amman and Lidwin M.T. van Velden 157 Subject index 165 Authors'index 169 PREFACE The field of Computational Economics is a fast growing area. Due to the limitations in analytical modeling, more and more researchers apply numerical methods as a means of problem solving. In tum these quantitative results can be used to make qualitative statements. This volume of the Advanced Series in Theoretical and Applied and Econometrics comprises a selected number of papers in the field of computational economics presented at the Annual Meeting of the Society Economic Dynamics and Control held in Minneapolis, June 1990. The volume covers ten papers dealing with computational issues in Econo metrics, Economics and Optimization. The first five papers in these proceedings are dedicated to numerical issues in econometric estimation. The following three papers are concerned with computational issues in model solving and optimization. The last two papers highlight some numerical techniques for solving micro models. We are sure that Computational Economics will become an important new trend in Economics in the coming decade. Hopefully this volume can be one of the first contributions highlighting this new trend. The Editors H.M. Amman et a1. (eds), Computational Economics and Econometrics, vii. © 1992 Kluwer Academic Publishers. PART ONE ECONOMETRICS LIKELIHOOD EVALUATION FOR DYNAMIC LATENT VARIABLES MODELS 1 DAVID F. HENDRY Nuffield College, Oxford, U.K. and JEAN-FRANc;mS RICHARD ISDS, Pittsburgh University, Pittsburgh, PA, U.S.A. Abstract. We propose a general Monte Carlo simulation technique for evaluating the likelihood function of dynamic latent variables models, based on artificial factorizations of the sequential joint density of the observables and latent variables. The feasibility of the proposed technique is demonstrated by means of a pilot application to a one-parameter disequilibrium model. Extensions to models with weakly exogenous variables and the use of acceleration methods are discussed. 1. Introduction Since the 1970s there has been a major resurgence of interest in the topic of Dynamic Latent Variables (DLV) models. For the purpose of the present discussion, latent variables are broadly defined as variables which enter the formulation of an econometric model and yet are not observable. See in particular the discussion in Aigner et al. (1983). Latent variables are widely recognized to be major components of the behavior of economic agents. They are inherently dynamic for a broad class of models such as intertemporal optimization, search or duration processes, error correction mechanisms, models of habit formation or persistence, and state dependence. Important examples are discrete choice models (Heckman, 1981) and disequilibrium models (Quandt, 1988; references in Quandt, 1989). Unfortunately, with the exception of linear Gaussian models, the likelihood functions of DLV models are often analytically intractable, largely because the elimination of the latent variables requires high-dimensional numerical integration. Many important techniques have been developed over recent years that address the specific issue of numerical tractability. For expository purposes we can usefully regroup these contributions into three broad categories (not mutually exclusive): 1. Simplifying the dynamic structure. A characteristic example of this approach is the use of autocorrelated error terms in the (static) supply and demand equations of disequilibrium models. In Laffont and Montfort (1979), the resulting likelihood function leads to direct numerical evaluation of the maximum likelihood estimators. 1 Financial support for this work has been provided by the Ford Foundation, the National Science Foundation (SES-90 12202) the Pew Charitable Trust, and by the UK Economic and Social Research Council under Grants BOO22012 and R231184. We are pleased to acknowledge useful comments from J. Danielsson and J. Geweke. The usual disclaimer applies. H.M. AnuMn et aI. (ells), Computational Economics and Econometrics, 3-17. © 1992 Kluwer Academic Publishers. 4 D.E HENDRY AND J.-F. RICHARD 2. Observable surrogates for latent variables. Typical examples are the use of price-adjustment equations in disequilibrium models (see Quandt (1982, 1988) or Maddala (1983» or the use of "fixed-effects" formulations as in Heckman and McCurdy (1980). 3. Simulation. Even though the likelihood functions associated with DLV mod els may be intractable, the models themselves are often amenable to joint simulation of the latent and observable processes. Hence the recently devel oped Method of Simulated Moments (MSM), as discussed e.g. by McFadden (1989) or Pakes and Pollard (1989), is applicable to DLV models and is dis cussed further below. Each line of attack has its merits. The first often results in more manageable formulations. The second offers the major advantage that the inclusion of sensible surrogates generates additional information. The simulation techniques may be computationally more demanding, but they are extremely general and therefore have considerable potential for preserving the full dynamic structure of the problem under consideration. In the present paper we aim at evaluating the likelihood function itselfby means of simulation techniques. The motivation for doing so is obvious: a broad range of statistical techniques (estimation, hypothesis testing, Bayesian methods) is likeli hood based. The difficulty to be addressed lies in the fact that the joint distribution of the latent variables conditional on the observables is generally not available for DLV models. On then has to select an approximating distribution, known as an importance function, in the form of a random number generator. The main object of our paper is to propose a general procedure for the automatic selection of importance functions and demonstrate its feasibility with a pilot application. This paper describes work in progress, and only briefly considers the design of "variance reduction" or "acceleration" techniques, and extensions to models with unmodelled variables. The reader can also usefully refer to the current literature on Monte Carlo integration (see e.g. Geweke (1989) for many technical details and for additional references). This paper is organized as follows: Section 2 considers the closely related literature on MSM; the automatic selection of importance functions is discussed in Section 3, and an application is proposed in Section 4. Acceleration techniques are noted in Section 5. The treatment of exogenous variables is examined in Section 6, and Section 7 concludes. 2. Method of Simulated Moments The objective of this section is to describe, and relate our own work to, current developments, in particular to the Method of Simulated Moments (MSM). For ease of presentation, we temporarily abstract from specifically discussing the sequential (d ynamic) structure of the class of models under consideration and adopt instead a