Table Of ContentRATIONAL EXPECTATIONS IN MACROECONOMIC MODELS
Advanced Studies in Theoretical and Applied Econometrics
Volume 26
Managing Editors:
A.J. Hughes Hallet, University of Strathclyde, Glasgow, United Kingdom
J. Marquez, The Federal Reserve System, Washington, D.C., US.A.
Editorial Board:
F.G. Adams, University of Pennsylvania, Philadelphia, US.A.
P. Balestra, University of Geneva, Switzerland
M.G. Dagenais, University of Montreal, Canada
D. Kendrick, University of Texas, Austin, US.A.
J.H.P. Paelinck, Netherlands Economic Institute, Rotterdam, The Netherlands
R.S. Pindyck, Sloane School of Management, M.I. T., US.A.
H. Theil, University of Florida, Gainesvi/le, US.A.
W. Welte, University of Lodz, Poland
The titles published in this series are listed at the end of this volume.
Rational Expectations
in Macroeconomic Models
by
Paul Fisher
Economics Division, Bank of England"
·The research reported in this book was undertaken whilst the author was with the ESRC Macro
economic Modelling Bureau, University of Warwick.
Springer-Science+Business Media, BV.
Library of Congress Cataloging-in-Publication Data
F1sher, Paul.
Ra t 1o na 1 expectat 10ns 1n lIacroeconom1 c m"ode 1s I Pau 1 F 1s her.
p. cm. -- (Advanced stud1es 1n theoret1cal & app11ed
econometr1cs ; vol. 26)
Includes b1b11ograph1cal references.
ISBN 978-90-481-4188-3
1. Macroeconom1cs--Econolletr1c lIodels. 2. Rat10nal expectatlons
(Econoiilc theory)--Econometr1c models. 1. Tltle. II. Ser1es.
HB172.5.F544 1992
339' .01·5195--dc20 92-22157
ISBN 978-90-481-4188-3 ISBN 978-94-015-8002-1 (eBook)
DOI 10.1007/978-94-015-8002-1
Printed an acid-traa papar
AII Rights Reserved
© 1992 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1992
Softcover reprint of the hardcover 1st edition 1992
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.
CONTENTS
Acknowledgements vii
Summary vii
Abbreviations viii
1. INTRODUCTION 1
2. LARGE SCALE MACROECONOMIC MODELS AND 5
FORWARD EXPECTATIONS
3. SOLUTION METHODS FOR NONLINEAR FORWARD 37
EXPECTATIONS MODELS
4. TERMINAL CONDITIONS, UNIQUENESS AND STABILITY 73
5. EXPERIMENTAL DESIGN AND STOCHASTIC SIMULATION 111
6. ALTERNATNE MODEL FORMS AND SOLUTION MODES: 147
mSTORICAL TRACKING
7. CONTROL AND POLICY ANALYSIS: EXPERIMENTAL DESIGN 167
8. CONCLUSIONS 197
9. BIBLIOGRAPHY 201
APPENDIX: MODEL VINTAGES 211
ACKNOWLEDGEMENTS
This book has been developed during the course of the research programme of
the ESRC Macroeconomic Modelling Bureau, partially in response to the needs of
that work. The majority of the content was included in a Phd Thesis, "Simulations
and control techniques for nonlinear rational expectations models", University of
Warwick, 1990. Indirect support by the ESRC is therefore acknowledged. Special
thanks are due to my supervisor and the director of the Bureau, Professor
K.F. Wallis and to the Bureau's principal research fellow Dr. J.D. Whitley. I am
also grateful for the comments and encouragement of former Bureau colleagues
M.J. Andrews, D.N.F. Bell, A.J. Longbottom, S.K. Tanna, and D.S. Turner.
Some of the material in this book was originally developed in joint research
projects with other members of the Bureau and Chapter 3 draws heavily on joint
research with A.J. Hughes Hallett and Chapter 6 with D. Turner. The present
author claims full responsibility for this presentation and any errors which it
contains.
Finally I should like to thank S. Hall, B. Henry, S. Holly, E. Karakitsos,
K. Matthews, A. Rastogi, B. Rustem, M. Salmon, R. Smith and R. Whittaker for
informal discussions on these issues and the models over a number of years.
SUMMARY
This book presents a comprehensive set of techniques for solving, simulating,
analysing and controlling large scale, nonlinear, econometric models that contain
rational expectations of future dated variables. These expectations are generally
treated as model consistent, whereby the expectation is set to the deterministic
projection of the model.
Solutions to such models are distinguished from those of conventional models by
the fact that they are not recursive in time. The outcome for the current period
depends on the expected outcome for future periods as well as past periods. This
property means that all of the basic numerical procedures need to be altered.
We consider the following topics: solution algorithms for the two-point
boundary value problem; terminal conditions, uniqueness and stability;
experimental design and stochastic simulation; the role of the exchange rate; control
methods including optimal control. We find that suitable procedures allow us to
undertake all of the experiments usually conducted with conventional models.
vii
viii
Each topic is illustrated by application to three large scale models of the United
Kingdom economy which contain rational expectations terms. Only one of these
models is constructed following the new-<:lassical paradigm and hence their
comparative properties revealed by our experiments are of some interest.
ABBREVIATIONS
BE Bank of England
EEC European Economic Community
FGS Fast Gauss-Seidel
FIML Full information maximum likelihood
FOI First order iterations
FT Fair-Taylor
FWW Fisher, Wallis and Whitley
GDP Gross domestic product
HMT Her Majesty's Treasury
III Incomplete inner iterations
IV Instrumental variables
JOR J aco bi-over-relaxation
LBS London Business School
LPL Liverpool Research Group in Macroeconomics
NIESR National Institute of Economic and Social Research
NPV Net present value
OLS Ordinary least squares
PSBR Public sector borrowing requirement
RMSE Root-mean-square error
SOR Successi ve-over-relaxation
UK United Kingdom
US United States of America
VAT Value added tax
2SLS Two stage least squares
CHAPTER 1
INTRODUCTION
Throughout the last two decades, expectations formation has been at the heart of
macroeconomic research. In particular, following the work of Muth (1961),
attention has been concentrated on the concept of rational expectations i.e. that
agents form expectations consistent with their knowledge of the underlying
processes of the economic system and taking into account all available information.
The rational expectations hypothesis has sufficiently penetrated all areas of macro
economic theory and applied work to have caused a "revolution in macroeconomics"
according to Begg (1982). Developments in macroeconomic theory are usually
incorporated, sooner or later, into large-scale macroeconomic models. In the United
Kingdom there are now three large, nonlinear, empirical, macroeconomic forecasting
models which incorporate rational expectation terms as part of their basic structure.
The Liverpool model (LPL) is an annual model which has been based on
rational expectations since its inception in 1979. It is new classical in structure with
a set of long-run equilibrium equations determined only by the supply side. The
London Business School (LBS) and National Institute of Economic and Social
Research (NIESR) models are quarterly and used rational expectation models for
their forecasts for the first time during 1985 (Economic Outlook, October; National
Institute Economic Review, November). The LBS model has three expectation
terms in its large financial sector. It is based around the income-expenditure
framework and has a tradition as an international monetarist model. A more
accurate description of recent versions would be a sluggish price adjustment model.
The NIESR model initially remained largely Keynesian despite the incorporation of
the expectations of eleven variables with leads of up to four periods. The model is
therefore a quantity adjustment model, being driven more by expenditures than
relative prices. In recent years the NIESR model has been developed so that, in the
long-run, its properties are more classical in nature (Wren-Lewis, 1988).
1
2
The actual processes by which agents form expectations are generally
unknown and their precise expectations concerning future macroeconomic quantities
are usually unobserved. The assumption of rational expectations in economic theory
therefore leads to complications both in estimating equations which contain explicit
expectation terms and in the numerical procedures which are designed to solve,
simulate and analyse large-scale models which contain such terms. The
econometric literature now suggests a range of statistical methods which allow the
estimation of equations containing rational expectation terms (e.g. see Begg, 1982,
Ch.5). This book concerns methods which allow the use of such equations in
large-scale models.
1.1 The Purpose of this Book
The primary objective of the research described in this book is to establish a set of
numerical methods for the simulation, analysis and optimal control of large-scale,
nonlinear, macroeconomic models that contain rational expectations of future-dated
variables. These methods allow us to undertake on rational expectations models all
of the experiments usually performed with conventionallarge-scale models such as
forecasting, policy analysis or stochastic simulation.
Numerical methods are often designed for specific problems or for specific
models. In this book we aim to establish a general set of techniques which can be
applied to a variety of models and problems. Starting from these general methods
we can then modify the procedures to suit the requirements of particular
experiments.
A secondary objective is to use our procedures to investigate the properties of
the three publicly available models of the U.K. economy which incorporate rational
expectations terms. As noted by Wallis (1988):
"In the United States the eventual adoption of the rational
expectations hypothesis was associated in addition with the new
classical equilibrium business cycle models and the policy ineffective
ness proposition."
In the U.K. this association has not been universally adopted and the properties of
those models with rational expectations may not coincide with the new-classical
model. The elucidation of the properties of these models is therefore of some
interest.
In the three different U.K. models containing rational expectations, the only
3
common expectations term is that of the Sterling effective exchange rate. In a
relatively small open economy, operating under a floating rate regime, fluctuations
in the exchange rate can be a particularly important transmission mechanism.
Expectations of future exchange rates changes appear to play an important
behavioural role in determining its current value (!sard, 1988; Fisher et al, 1990)
and these terms are therefore of fundamental importance to the properties of U.K.
models (e.g. see Wallis et al., 1987, pp.44-48). When explaining our results we pay
particular attention to the behaviour of the exchange rate and Chapter 6 deals
explicitly with the role of the exchange rate in the models.
1.2 Overview
In Chapter 2 we present a wide-ranging literature survey covering all aspects of
large-scale modelling but paying particular attention to the numerical methods
used. The role of expectations is discussed and we consider the problems caused by
including expectation terms of future-dated variables. We conclude that the issues
raised have only partially been resolved by the existing literature. Some problems,
such as terminal condition choice, do not appear to have been considered in depth
and some existing numerical methods, such as those for the optimal control of
nonlinear rational expectations models, do not appear to be well understood.
In the third chapter we consider the basic problem of solving a rational
expectations model such that the expectations terms are consistent with the model's
solution. We develop a family of first-order iterative techniques which encompasses
some specific solution algorithms suggested elsewhere in the literature. These
methods are tested on three large-scale models to derive the most efficient forms of
solution algorithm. We contrast and compare our methods with two alternative
approaches to solving nonlinear rational expectations models.
In Chapter 4 we address the issues of uniqueness and stability in the
solutions of rational expectations models. Having established the conditions for a
unique stable solution to exist, we are faced with the problem of locating that
solution. This comes down to the choice of terminal values for the expectation
terms in the final solution period. A number of possible terminal conditions are
proposed and evaluated on both a small demonstration model and the three large
scale models. Particular attention is paid to the implications of different choices
when there is not a unique stable solution.
In Chapter 5 we present a general discussion of alternative model forms
(structural form, reduced form, final form) and solution modes (single-€quation,
