A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends

A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends Over the last thirty years there has b...

Author: Albert Rex Bergstrom | Khalid Ben Nowman

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A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends Over the last thirty years there has been extensive use of continuous time econometric methods in macroeconomic modeling. This monograph presents the first continuous time macroeconometric model of the United Kingdom incorporating stochastic trends. Its development represents a major step forward in continuous time macroeconomic modeling. The book describes the new model in detail and, like earlier models, it is designed in such a way as to permit a rigorous mathematical analysis of its steady state and stability properties, thus providing a valuable check on the capacity of the model to generate plausible long-run behavior. The model is estimated using newly developed exact Gaussian estimation method for continuous time econometric models incorporating unobservable stochastic trends. The book also includes discussion of the application of the model to dynamic analysis and forecasting. The late ALBERT REX BERGSTROM was Emeritus Professor of Economics, a former Dean of the School of Social Studies, Chairman of the Department of Economics and Pro Vice Chancellor at the University of Essex and a Fellow of the Econometric Society. He was one of the world’s leading authorities on continuous time econometric modelling. Professor Bergstrom was formerly Professor of Econometrics at the University of Auckland and Reader at the London School of Economics. His professional papers appeared in leading journals such as Econometrica and Econometric Theory. Professor Bergstrom’s earlier books include The Construction and Use of Economic Models (1967), Continuous Time Econometric Modelling (1990), Statistical Inference in Continuous Time Economic Models (editor, 1976) and Stability and Inflation: Essays in Memory of A. W. Phillips (edited with A. J. L. Catt, M. H. Peston and B. D. J. Silverstone, 1978). KHALID BEN NOWMAN is Professor of Finance at the Westminster Business School, University of Westminster in London. He previously worked at City University Business School, London Business School, University of Essex, Durham University, University of Kent and in the banking sector at the Bank of England, First National Bank of Chicago and Barclays Bank. Professor Nowman’s papers have appeared in leading journals such as Econometric Theory, Journal of Finance, Journal of Financial and Quantitative Analysis, and the Journal of Economic Dynamics and Control, among others.

A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends

Albert Rex Bergstrom University of Essex

Khalid Ben Nowman University of Westminster

CAMBRIDGE UNIVERSITY PRESS

˜ Paulo Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521875493 Estate of Albert Rex Bergstrom and Khalid Ben Nowman 2007 c Cambridge University Press 2007

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2007 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Bergstrom, A. R. (Albert Rex) A continuous time econometric model of the United Kingdom with stochastic trends / Albert Rex Bergstrom, Khalid Ben Nowman. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-521-87549-3 (hardback) ISBN-10: 0-521-87549-8 (hardback) 1. Great Britain – Economic policy – Econometric models. 2. Finance – Great Britain – Econometric models. 3. Econometric models. 4. Stochastic processes. I. Nowman, Khalid Ben, 1962– II. Title. HC256.7.B47 2007 330.941 086 – dc22 2006037265 ISBN 978-0-521-87549-3 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

BN: To Miho

Contents

xi xiii xix

List of Figures and Tables

page

Foreword by Peter C. B. Phillips Preface

1

Introduction to Continuous Time Modelling

1

1.1 1.2 1.3

1 3

1.4 1.5 1.6 1.7 1.8

Introduction Why Model in Continuous Time Introduction to General Continuous Time Models Continuous Time Models in Finance Continuous Time Macroeconomic Modelling Policy Analysis in Continuous Time Macroeconomic Models Stochastic Trends in Econometric Models An Outline of Contents

vii

9 18 31 42 45 47

Contents

2

Continuous Time Econometrics with Stochastic Trends 2.1 2.2 2.3 2.4 2.5

3

Introduction The Continuous Time Model The Exact Discrete Model and Its VARMAX Representation Estimation and Forecasting Conclusion Appendix A: Formulae for the Coefficient Matrices of Exact Discrete Model Appendix B: Formulae for the Autocovariance Matrices

50 50 53 58 67 79 80 85

Model Specification

114

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

114 116 125 128 130 134 136 138 141 144 145 146 147 147 149 150

Introduction Equations and General Properties of the Model Private Consumption Residential Fixed Capital Employment Private Non-Residential Fixed Capital Output Price Level Wage Rate Interest Rate Imports Non-Oil Exports Transfers Abroad Real Profits Interest and Dividends from Abroad Cumulative Net Real Investment Abroad Exchange Rate

viii

Contents

4

5

3.17 Stocks 3.18 Conclusion Appendix A: Derivation of General Adjustment Equations Appendix B: Distributed Lag Relations

151 152

Steady State and Stability Analysis

173

4.1 4.2 4.3 4.4 4.5

173 175 180 192 197 197 203

Introduction The Steady State Stability Analysis Stability and Bifurcations Conclusion Appendix A: Steady State Level Parameters Appendix B: Transformed Model

152 164

Empirical Estimation of the Model and Derived Results

213

5.1 5.2 5.3 5.4 5.5 5.6

213 214 223 232 240 248

Introduction Estimation from UK Data Time Lag Distributions Steady State and Stability Properties Post-Sample Forecasting Performance Conclusion Appendix A: Linear Approximation about Sample Means Appendix B: Data

249 262

269 285 288

References Author Index Subject Index

ix


Figures 1.1

UK and US Interbank Interest Rates

1.2

Japan and Singapore Interbank Interest Rates

5.1

Lag Distribution Private Consumption

226

5.2

Lag Distribution Employment

227

5.3

Lag Distribution Residential Fixed Capital

227

5.4

Lag Distribution Private Non-Residential Fixed Capital

228

5.5

Lag Distribution Output

228

5.6

Lag Distribution Interest Rate

229

5.7

Lag Distribution Imports

229

5.8

Lag Distribution Non-Oil Exports

230

5.9

Lag Distribution Current Transfers Abroad

230

xi

page 22 23


5.10 Lag Distribution Profits Interest and Dividends from Abroad

231

5.11 Lag Distribution Investment Abroad

231

Tables 1.1

Single Factor Continuous Time Interest Rate Models

21

5.1

Parameter Estimates from United Kingdom Quarterly Data

219

5.2

Estimated Time Lag Parameters

225

5.3

Values of Exogenous Variable Time Path Parameters Assumed in Steady State Analysis

235

5.4

Derived Steady State Growth Rates

235

5.5

Derived Steady State Levels

236

5.6

Eigenvalues for Linear Approximation about Steady State

239

Root Mean Errors of Post-Sample Multi-Period Forecasts

242

Post-Sample Multi-Period Forecasts

243

5.7

5.8

xii

Foreword

The daunting complexity of aggregate economic behaviour has led recent generations of macroeconomists to pursue a rich diversity of approaches in modelling. Some have focused on the development of tightly specified small scale systems that embody rational expectations, real business cycle mechanisms and intertemporal optimization principles to strengthen the economic foundations of the models and furnish meaningful prior restrictions. Others have preferred the use of model formulations that are more convenient in coping with the time series properties of the observed series, using both parametric and semiparametric approaches, and allowing for mechanisms that can accommodate the regime changes that can occur in practice. Still others have begun to work with large dimensional panels and dynamic factor decompositions.

xiii

Foreword

One unifying force amongst the growing diversity of empirical modelling research in macroeconomics has been acknowledgement of the importance of long-run behaviour and the recognition that trending mechanisms in economics are stochastic. A second area of commonality lies in the use of nonlinear dynamics, both systems dynamics and volatility dynamics, the latter being especially important in financial market applications. The present volume reports the construction and implementation of a new macroeconomic model of the UK economy that embodies most of these themes. While the model differs from much of the mainstream of modern macroeconomics in terms of its genesis and form, it shares some commonality with mainstream work in its detailed attention to nonlinear dynamics, its concern for stochastically trending data and its use of dynamic optimization principles in the derivation of adjustment relations. The model developed here is capable of describing diverse patterns of cyclical behaviour in economic aggregates and long-run growth. It is formulated in continuous time as a system of mixedorder stochastic differential equations. It synthesizes real, monetary, financial, labour and production sectors of the economy. It allows for market disequilibria in a systematic way, with parameterized adjustment mechanisms that

xiv

Foreword

measure responsiveness to deviations from partial equilibria whose values are internally consistent and determined by economic theory. The model’s long-run properties are explored analytically, an explicit steady state growth path is obtained and plausible regions of the parameter space are determined a priori, thereby providing useful guidelines for empirical model evaluation. In many of these respects, the model is very different from prevailing traditions in empirical macroeconomic modelling. The model is the latest and most sophisticated in a succession of empirical econometric models of the United Kingdom that have been developed under the leadership of the senior author Rex Bergstrom. The present model therefore represents the culmination of a long research agenda in which the cardfile of models has steadily increased in complexity. The incorporation of internally embedded stochastic trends, the allowance for temporally aggregated flow data and the introduction of higher order and more complex lag responses distinguish the present contribution, giving the new model important elements of realism and features that are comparable to and in some ways exceed those of other empirical macro models. Indeed, the reduced form discrete time version of the present model is a higher order vector autoregressive (VAR) moving average model with

xv

Foreword

exogenous variables that involves 18 endogenous variables, 27 speed of adjustment parameters, 33 long-run coefficients and 3 trend parameters. In this respect, it has time series dynamics that are considerably more complex than those of most structural and unrestricted VAR specifications, many other time series models and dynamic structural equation macroeconomic models. Bergstrom and Wymer (BW, 1976) developed an early model in this general class for the UK economy. The BW model is widely recognized as the first aggregate model of economic activity that was formulated in continuous time as a set of interdependent stochastic differential equations. That model was highly successful as a parsimonious representation of economic activity in a modern industrialized economy and it has since served as a prototype for similar models in many other countries. The present volume continues the BW tradition of continuous time macroeconometrics; it integrates modelling enhancements that were introduced in the subsequent higher order model reported in Bergstrom, Nowman and Wymer (BNW, 1992), and it breaks new ground in empirical continuous time modelling by embedding stochastic trend elements into the system and growth paths.

xvi

Foreword

This book carries the indelible signature of Bergstrom’s superb scholarship. The theoretical model is developed with great attention to underlying economic ideas, the econometric methodology is systematically built on the extraction of an exact discrete model and an algorithm that constructs the Gaussian likelihood, the empirical implementation is painstakingly conducted, the size of the system and its complex transcendental matrix nonlinearities push to the limits of present computational capacity and the empirics involve specification testing and prediction evaluation against a highly competitive VAR system with exogenous inputs. Ragnar Frisch, one of the founders of the Econometric Society and a co-recipient of the first Nobel Prize for economics, believed that the ultimate strength of econometrics as a scientific tool for understanding economic activity comes through the combined use of three elements – mathematics, statistical analysis and economic theory. This Frisch mantra is easy to cite as a guiding principle in quantitative economic research, yet it is far more difficult to implement in practice. The present work makes a substantive contribution to econometric knowledge but is of wider methodological interest because it succeeds in meeting Frisch’s criteria

xvii

Foreword

and aspirations for productive research in econometrics. In so doing, it lays out a paradigm for others to follow. This volume sadly concludes Rex Bergstrom’s distinguished work on continuous time econometric modelling, the manuscript being completed just a few months before his unexpected death on 1 May 2005 and subsequently revised by his former student and co-author, Ben Nowman. We hope that the research agenda will continue into the future and attract further generations of researchers as well as those who were themselves trained by Bergstrom or his students. The present volume will remain a distinguished magnum opus that concludes Bergstrom’s own work, a testimony to his sustained accomplishments in founding this field, a resource for present and future researchers and a basis for further work as the frontiers of macroeconometric modelling and continuous time econometrics continue to unfold. Peter C. B. Phillips Yale University May 2006

xviii

Preface

Over the last 30 years there has been a growing use of continuous time econometric methods in macroeconomic modelling. Economy-wide continuous time macroeconometric models have been developed for most of the leading industrial countries of the world. This monograph presents the first continuous time macroeconometric model of the United Kingdom incorporating stochastic trends. Its development represents a major step forward in continuous time macroeconomic modelling. The book describes the new model in detail and like earlier models it is designed in such a way as to permit a rigorous mathematical analysis of its steady state and stability properties, thus providing a valuable check on the capacity of the model to generate plausible long-run behaviour. The model is estimated using exact Gaussian estimation methods for continuous time econometric models

xix

Preface

incorporating unobservable stochastic trends. The book also includes discussion on the application of the model to dynamic analysis and forecasting. An earlier version of this work was presented at the Cowles Foundation Conference on Recent Developments in Time Series Econometrics, Yale University, 1999; the 3rd Japan Economic Policy Association International Conference, Tokyo, 2004; Westminster Business School, 2004; and the 3rd Nordic Econometric Meeting, Helsinki, 2005. We are grateful for comments from these presentations. Ben Nowman would like to give special thanks to Peter Phillips for his extensive comments and support in helping him to revise the book. I also thank Peter Phillips for kindly sending me the photo of Rex Bergstrom and for permission to use it. He also thanks his colleagues at Westminster Business School for their support and especially Len Shackleton. I also thank William Barnett and Yijun He for sending me research papers. We are very grateful to Marcus Chambers for his comments and constant support over many years in our work. We would like to thank Scott Parris, the economics editor of Cambridge University Press, for his encouragement and support in this project and two anonymous referees for valuable comments. Lastly we both give our warmest thanks to Miho Takashima for her constant support during the

xx

Preface

writing of this book. Royalties from this volume will be used to contribute to the A. R. Bergstrom Prize in Econometrics established in 1993 for research excellence in econometrics. A. R. B. and K. B. N. January 2005

xxi

CHAPTER ONE

Introduction to Continuous Time Modelling 1.1 Introduction

T

he world’s economies and many of its financial markets operate 24 hours a day and the formulation of realistic

models to represent the economy and these markets is one of the major challenges facing economists and finance specialists today. Given that the economy is made up of millions of agents making decisions continuously the use of a continuous time model that allows for these interactions to be incorporated will be a more realistic description of the underlying phenomena we are trying to model. In response to this challenge, the econometric estimation of continuous time models in economics and finance has been a major ongoing development over the last 30 years. The major problem faced in such econometric modelling is that we do not have a

1

Continuous Time Econometric Model of UK with Stochastic Trends

continuous time record of observations and most economic data are available only at discrete time periods (annually, monthly, quarterly). In finance, data are more frequently available at a higher frequency, for example, daily, hourly and more recently tick-by-tick transactions data. The problem facing an econometric investigator, therefore, is how best to utilize such data in the estimation of an underlying continuous time system so that the fitted model can then be used in economic forecasting, policy analysis and derivative pricing. In financial markets continuous time models have found widespread applications in financial asset and option pricing since the seminal work of Black and Scholes [1973] and Merton [1973a] and in the general area of continuous time term structure modelling and bond valuation over the last 20 years. These developments have opened up an explosion of research activity developing new continuous time models in finance and associated econometric estimation techniques for fitting continuous systems with the available discrete data. The history of the earliest work on continuous time stochastic processes (for example, Bachelier [1900], Einstein [1906], Wiener [1923] and Ito [1946, 1951]) and continuous time econometric modelling (Bartlett [1946], Koopmans

2


[1950], Phillips [1959], Durbin [1961]) has been well documented in survey papers by Bergstrom [1984a, 1988, 1990, Ch. 1, 1996]. Our aims in this chapter are to give a brief introduction to continuous time modelling and econometrics in areas that are important to economists and finance specialists working in academia, government and financial markets. This chapter is organized as follows. Section 1.2 explains the advantages and problems of continuous time modelling. Section 1.3 introduces the general idea of a continuous time model and the estimation of its parameters with available discrete time data. Section 1.4 discusses the application of continuous time models to term structure models in finance and Section 1.5 introduces continuous time macroeconomic modelling. Section 1.6 discusses policy analysis and Section 1.7 introduces the idea of stochastic trends in econometrics. Lastly, Section 1.8 outlines the contents of the book.

1.2 Why Model in Continuous Time In economic and financial modelling, researchers face a fundamental choice between modelling in continuous and discrete time. This choice raises the important question of why models should be formulated in continuous time rather than

3


discrete time. There are a number of advantages of modelling in continuous time that we briefly summarize here (see Bergstrom [1996] for an extensive discussion). First, since the economy and financial markets are continuously operating and the underlying decision processes we are trying to model involves millions of decisions by economic agents within the recorded data observation interval, realistic models will depend on the continuous passage of time. A continuous system will therefore be sympathetic to the underlying interactions being modelled and captured in the data, whereas traditional discrete time models are inherently less flexible because they restrict the underlying decisionmaking lag structures to match the observation interval in the data exactly (daily, monthly, quarterly), so that at best an economy’s equilibrium and disequilibrium characteristics are being captured at successive points of time. The potential for misspecification therefore seems greater in a discrete time system when the underlying phenomenon is continuous. Secondly, economic and financial models typically comprise two types of variables. Stock variables observed at points in time (for example, the money stock, inventories, fixed capital) and flow variables (for example, consumption, exports, output and imports) observed as aggregations over

4


the unit observation period. Continuous time econometric methods, as applied in this monograph, allow for the correct treatment of these different types of variables in the direct and exact estimation of the models. This solves the temporal aggregation bias that commonly occurs in using discrete time models with flow variables, where no distinction is given to stock and flow variables generally. Thirdly, when economists in government and central banks develop econometric models for short- and long-term economic forecasting, the construction of a quarterly or monthly model is commonplace due to the availability of national accounts data of that frequency. This dependence of the formulation of the model on the observed data frequency is a major disadvantage of discrete time models. For example, the construction of a model on monthly data will be different from one based on daily data. In contrast, the specification of a continuous time model in economics and finance is independent of the available data frequency. Related to the previous point, the fourth advantage of continuous time models is in economic and financial forecasting. The use of continuous time models allows one to obtain continuous time paths of the variables that can be used to make forecasts at shorter intervals to the available data used in the estimation of the model. For example, the continuous

5


time path forecasts of manufacturing production and gross domestic product in the economy would be useful to government Treasury departments and central banks in their policy setting. In the business sector, companies would find it useful to have the continuous time path of retail sales projections for setting future production levels for example. A discrete time model is less flexible where a quarterly macroeconomic model would generate forecasts of gross domestic product at quarterly intervals and if we wanted forecasts on a weekly or monthly interval would have to be interpolated. A fifth advantage of continuous time modelling is concerned with the modelling of dynamic adjustment mechanisms in the economy where continuous systems allow for more realistic specifications of the partial adjustment processes compared to a discrete time model. Typically, continuous systems in formulating partial adjustment (error correction) mechanisms allow the dependent variable to adjust continuously in response to deviations from its partial equilibrium level, which may be set to continuously depend on other variables in the model according to some underlying economic theory. These partial adjustment equations usually take the form of first- or second-order differential equations but could, of course, be much more general. The

6


main reason why economic variables adjust gradually, rather than instantaneously, to their partial equilibrium level is that there are adjustment costs, which depend on the rate of change and, possibly, the acceleration of the adjusting variable. The resulting mechanisms can be formally derived as part of the solution of a dynamic optimization problem, which takes account of the costs of adjustment, as is done in the present monograph. The sixth advantage of continuous time models is their natural application in the development of finance theory and derivative pricing models since the pathbreaking work of Merton [1969, 1971, 1973a,b] and Black and Scholes [1973]. Sundaresan [2000] provides a major review of these developments, continuous time models in finance and their many applications. There are a number of other advantages of continuous time modelling and the reader is referred to Bergstrom [1990, Ch. 1, 1996] and Gandolfo [1981, 1993] for further discussion and motivation. One of the disadvantages of continuous time modelling is that the estimation of these models has involved the development of complicated and sometimes specialized econometric methods that have in the past largely remained in the province of econometric theorists and finance specialists. But over the last 10 years there has been a substantial

7


increase in the number of Ph.D. educated graduates in econometrics, finance, physics and mathematics who are capable of and interested in using more advanced econometric techniques in financial and macroeconomic applications. These graduates have taken up employment in government departments and the banking and financial sectors, as well as academia, and are busy working on quantitative research, risk management, asset price modelling, hedge fund research and trading floor activities, where advanced econometric methods are now being more commonly used. Secondly, the specification of a macroeconomic model in continuous time typically involves a system of structural equations derived from economic theory and detailed mathematical analysis of its long-run properties and steady state analysis. This work is very time consuming mathematically and re-specification and analysis is seldom easy to do. This compares to discrete time macroeconomic modelling in which a typical model is constructed of individually specified equations and least squares or instrumental variable estimation methods are usually employed. Re-specification and updating of estimation results as new quarterly data come available for a quarterly forecasting round can be individually done and easily achieved. On the other hand, the

8


explosive development of continuous time modelling in economics, and more especially finance over the last 20 years, together with the enormous increases in computational capability has made the specification and estimation of continuous time econometric models a much more plausible practical enterprise.

1.3 Introduction to General Continuous Time Models We present in this section a review of the general form of continuous time models and the basic ideas of econometric estimation using discrete time data. The general model considered by Bergstrom [1966a] is a first-order continuous time system specified as d x(t) = {A(θ)x(t) + b(θ)}dt + ζ(dt) (t ≥ 0),

(1.1)

where x(t) = {x1 (t), . . . , xn (t)} is a n-dimensional continuous time random process, A(θ) is an n × n matrix whose elements are functions of a vector θ = [θ1 , . . . , θ p ] of unknown structural parameters ( p ≤ n(n + 1)) and b(θ) is a vector that is a function of θ . The error term ζ(dt) is assumed to be a vector of white noise innovations (see Bergstrom

9


[1984a] for a precise definition and interpretation of this system). We assume the continuous time model generates equispaced discrete data observed as the sequence {x(0), x(1), . . .} and our objective is to estimate the parameters of the continuous time model. Two approaches that have been used in the past have become known as the Discrete Approximation Method and the Exact Discrete Model approach. The first approach was originally developed by Bergstrom [1966a] (see also Houthakker and Taylor [1966], Sargan [1974, 1976], Wymer [1972]). The discrete approximation is obtained by using a trapezoidal approximation that gives an approximate simultaneous equations system of the form 1 A(θ){x(t) + x(t − 1)} + b(θ ) + ut 2 E(ut ) = 0, E ut ut = , E us ut = 0, s = t, s, t = 1, 2, . . . .

x(t) − x(t − 1) =

(1.2)

In using this model it was shown in Bergstrom [1966a] that the estimates have a small asymptotic bias. The continuous time model can be extended to include an m-dimensional vector of exogenous variables z(t) = {z 1 (t), . . . , z m (t)} with

10


an associated n × m coefficient matrix B(θ) so the model has the form: d x(t) = {A(θ)x(t) + B(θ)z(t)} dt + ζ(dt) (t ≥ 0).

(1.3)

The first element of the exogenous vector will typically represent the vector of constants in the system. The approximate simultaneous equations model is now given by 1 A(θ){x(t) + x(t − 1)} 2 1 + B(θ){z(t) + z(t − 1)} + ut 2 E(ut ) = 0, E ut ut = , E z s ut = 0, s, t = 1, 2 . . . . E us ut = 0, s = t,

x(t) − x(t − 1) =

(1.4)

The use of this approximate simultaneous equation model has been extensive in continuous time empirical work in many areas of economics, macroeconomic modelling, financial and commodity markets over the last 30 years. These applications are listed and discussed in Bergstrom [1988, 1996]. The second and more complicated approach of using the exact discrete model was suggested and explored in Phillips [1972]. The exact discrete model is obtained from the

11


solution of the continuous time model (see Bergstrom [1984a, Theorem 3, p. 1167]) and is given below for the simple closed model case: x(t) = F (θ)x(t − 1) + g(θ) + εt 1 1 F = e A(θ ) = I + A + A2 + A3 + · · · 2! 3! g = e A(θ ) − I A−1 (θ)b(θ) 1 E(εt ) = 0, E εt εt = e rA e rA dr = , 0 E εs εt = 0, s = t.

(1.5)

As can be seen from this formulation, the exact discrete model coefficients are complicated functions of the underlying parameters of the continuous time model, and such was the available computer power in the 1960s and 1970s that it was infeasible to use this approach in general empirical work. It was, however, shown in Phillips [1972] that this exact model can be used to obtain consistent and asymptotically efficient estimates of the parameter vector θ, and in an important Monte Carlo study Phillips [1972] found that these estimates have finite sample properties superior to those obtained by using the approximate simultaneous model (see Phillips [1972] for details). This has important implications for obtaining more accurate economic forecasts from the estimated model in practice.

12


More complicated dynamic adjustment mechanisms in the economy can be handled by using second or higher order continuous time models. There is, of course, no theoretical economic reason for adjustment processes in the economy to be of first order and the implied time lag distributions generated by higher order continuous time models are more complicated (see Chapter 3 for precise forms). A general second-order model was considered by Bergstrom [1983, 1985, 1986, 1990] and may be presented as d[D x(t)] = [A1 (θ)D x(t) + A2 (θ)x(t) + B(θ)z(t)] dt + ζ(dt), x(0) = y1 ,

D x(0) = y2 ,

(t ≥ 0) (1.6)

where {x(t), (t > 0)} is a n-dimensional vector of endogenous variables, z(t) is an m-dimensional vector of nonrandom functions (exogenous variables), A1 (θ) and A2 (θ) are n × n system matrices and B(θ ) is an n × m system matrix whose elements are known functions of θ, y1 and y2 are nonrandom n × 1 vectors, D is the mean square differential operator and ζ(dt) is a vector of white noise innovations. The first element of z(t) will normally be unity and so the first column of B will be a vector of constants in the system. Usually dynamic models include terms for the rate of change, D z(t) and acceleration, D 2 z(t) of the exogenous variables and this

13


can be done by using the interpolation formulae of Nowman [1991](see also Chambers [1991]). Bergstrom [1986] provides a precise interpretation of this system. Associated with these more complex dynamic structures during the last 20 years, there have been enormous developments in computing technology that have facilitated the estimation of such models and that use estimation methods, which take account of the exact restrictions on the distribution of the discrete data implied by the continuous time model. The most extensively used of these new methods is the exact Gaussian estimation method developed by Bergstrom [1983, 1985, 1986, 1990] (see also Agbeyegbe [1984, 1987, 1988], Chambers [1991, 1999], McCrorie [2000, 2001, 2002], McGarry [2003] and Nowman [1991]). These methods are applicable to models formulated as systems of higher order and mixed-order stochastic differential equations and under appropriate conditions, yields exact maximum likelihood estimates of their structural parameters from a sample of mixed stock and flow data. Typical economic models comprise both stock and flow variables and we may assume that the stock variables are observed at specific points in time such as the points 0, 1, 2, . . . , T and the flow variables are observed as integrals over the unit intervals [0, 1], [1, 2], . . . , [T − 1, T ]. We let the

14


elements of x(t), z(t), y1 and y2 be ordered (without loss of generality) so that

x(t) = y1 =

x s(t)

x f (t) s y1 f

y1

,

,

z(t) =

z s(t)

z f (t) s y2 y2 = f , y2

,

where x s(t) is a vector of endogenous stock variables, x f (t) is a vector of endogenous flow variables, z s(t) is a vector of exogenous stock variables, z f(t) is a vector of exogenous flow variables. We then define the observable vectors x¯ t and z¯ t (t = 1, . . . , T ) by x s(t) − x s(t − 1) x¯ t = t (t = 1, . . . , T ), f t−1 x (r )dr 1 s s {z (t) + z (t − 1)} (t = 1, . . . , T ). z¯ t = 2 t f t−1 z (r )dr

(1.7)

(1.8)

We also define the observable part of the initial state vector that comprises the level stock vector as y1s = x s (0). We define y = [y1 , y2 ] as the unobservable part of the initial state vecf

tor that has to be estimated along with the other parameters of the model. In addition to these parameters, we also have to estimate the parameters of the covariance matrix of the white noise errors E[ζ(dt)ζ (dt)] = dt(µ), where is a

15


matrix whose elements are known functions of µ. The total parameters to be estimated then comprise [θ, µ, y]. Our aim in empirical work is to estimate these underlying parameters of the second-order continuous time model. This is achieved by deriving the exact discrete model corresponding to this second-order system, under certain assumptions concerning the unobservable continuous time paths of the exogenous variables (see Phillips [1974a, 1976]). This was obtained in Bergstrom [1986] for the general case of stock and flow variables, and is given below: x¯ 1 = G11 y1 + G12 y2 + E 11 z¯ 1 + E 12 z¯ 2 + E 13 z¯ 3 + η1 , x¯ 2 = C11 x¯ 1 + G21 y1 + G22 y2 + E 21 z¯ 1 + E 22 z¯ 2 + E 23 z¯ 3 + η2 , x¯ t = F1 x¯ t−1 + F2 x¯ t−2 + E 0 z¯ t + E 1 z¯ t−1 + E 2 z¯ t−2 + ηt , (t = 3, . . . , T ). (1.9)

The coefficient matrices and the error term covariance matrix E(ηη ) = were shown to be complicated functions of the underlying parameters of the continuous time model (see Bergstrom [1986] for precise forms). Bergstrom [1986] then derived the exact Gaussian likelihood function to obtain the exact Gaussian estimates. Letting L(θ, µ, y) be minus twice the logarithm of the Gaussian likelihood and by using a Cholesky factorization = MM , where M is a real lower triangular matrix with positive elements along the

16


diagonal, Bergstrom [1983, 1985, 1986] was able to define the Gaussian likelihood function in a simpler form below: L=

nT

εi2 + 2 log mii

(1.10)

i=1

in which the term mii represents the ith diagonal element of M and ε = [ε1 , . . . , εnT ] is a vector whose elements can be evaluated recursively from Mε = η. The Gaussian estimates ˆ µ, [θ, ˆ yˆ ] of [θ, µ, y] are then obtained from the iterative procedure proposed by Bergstrom [1985, 1986]. It was shown in a Monte Carlo study in Nowman [1993] using a secondorder version of an extended trade cycle model based on the model of Phillips [1954] (see also Bergstrom [1967, Ch. 3] and Phillips [1972]) that the Gaussian estimation method performed well in finite samples. The model used is represented below: d[DK(t)] = {−γ5 DK(t) + γ6 [β2 Y(t) + β3 M(t) − K(t)]} dt + ζ1 (dt) d[DC(t)] = {−γ1 DC(t) + γ2 [β1 Y(t) − C(t)]} dt

(1.11)

+ ζ2 (dt) d[DY(t)] = {−γ3 DY(t) + γ4 [DK(t) + C(t) + G(t) − Y(t)]} dt + ζ3 (dt),

where Y(t) is real net national income or output at time t, C(t) is real private consumption, K(t) is the amount of fixed

17


capital, G(t) is real government consumption, M(t) is real money supply and γ1 , γ2 , γ3 , γ4 , γ5 , γ6 , β1 , β2 and β3 are the parameters of the model. Another general approach to estimating the parameters of these continuous time models has been to use the Kalman filter algorithm as discussed in Harvey and Stock [1985] (see also Zadrozny [1988]). Readers are referred to Bergstrom [1984a, 1996] for extensive reviews on the developments and estimation of continuous time models in econometrics. We now turn to a discussion of the application of continuous time models in finance and their application to the important area of interest rate modelling that is widely used in financial markets today.

1.4 Continuous Time Models in Finance The development and application of general continuous time models in finance has been a major ongoing development originating in the important work of Merton [1969, 1971, 1973a, b] and Black and Scholes [1973] (see Merton [1990] and Sundaresan [2000]). Over the last 30 years a significant area of application of continuous time models has been in the modelling of interest rates using linear and nonlinear models for use in fixed income financial markets.

18


Typically, these models are used in the valuation of different types of bonds and bond options in various bond markets (for example, government, corporate, mortgage-backed). The various features of interest rates we would like to capture include a drift element and also the idea of mean reversion in interest rates. When rates happen to be high, they tend over time to revert to normal levels, and vice versa. The other important element to incorporate is any possible link between the level of rates and the volatility of rates the so-called “level-effect”. One important interest rate model was developed by Chan, Karolyi, Longstaff and Sanders [1992, CKLS, hereafter], which had the advantage of nesting as special cases a range of different short rate models. These special cases included the models of Merton [1973a], Vasicek [1977], Cox, Ingersoll and Ross [1985, CIRSR], Dothan [1978], Brennan and Schwartz [1980, BRSC], Cox, Ingersoll and Ross [1980, CIRVR], Constant Elasticity of Variance (CEV) model of Cox [1975], Cox and Ross [1976] and the Geometric Brownian Motion (GBM) model (see Sundaresan [2000] for further models). All these models differ in their specification of the conditional mean and diffusion components. The correct specification is of importance to fixed income traders to obtain the correct implied pricing of fixed income

19


securities. The CKLS general continuous time interest rate model is presented below: dr (t) = {α + βr (t)} dt + σ r γ (t)ζ(dt) (t ≥ 0),

(1.12)

where {r (t), t > 0} is a short-term interest rate, α and β are the unknown drift and mean reversion structural parameters, σ is the volatility of the short-term rate, γ is the proportional volatility exponent and ζ(dt) is a white noise error term. As can be seen, the conditional mean specification is of the same form as Bergstrom [1966a] but the diffusion component is now time varying, which differs from the assumption of constant volatility originally formulated in Bergstrom [1966a]. The special cases of the CKLS model are given in Table 1.1. There are many approaches available to estimate the parameters of these models with discrete data. Melino [1994] provides an excellent early review of the estimation of continuous time models in finance. In CKLS they used the Generalized Method of Moments method of Hansen [1982] and an approximate discrete time model. Their important findings for the US Treasury market was that term structure models with volatilities more highly sensitive to the level of rates have a closer fit to the data. In particular, the magnitude

20


Table 1.1. Single Factor Continuous Time Interest Rate Models α

Model CKLS Merton Vasicek CIRSR Dothan GBM BRSC CIRVR CEV

β

σ2

γ

γ

dr (t) = {α + βr (t)}dt + σ r (t) ζ(dt) dr (t) = αdt + σ ζ(dt) dr (t) = {α + βr (t)}dt + σ ζ(dt) dr (t) = {α + βr (t)}dt + σ r 1/2 (t) ζ(dt) dr (t) = σ r (t) ζ(dt) dr (t) = βr (t) dt + σ r (t) ζ(dt) dr (t) = {α + βr (t)}dt + σ r (t) ζ(dt) dr (t) = σ r 3/2 (t) ζ(dt) dr (t) = βr (t) dt + σ r γ (t) ζ(dt)

0

0 0

0

0 0

0

0 0 1/ 2 1 1 1 1/ 2

of the relationship was shown to be γ = 1.5 and far higher than assumed by the other interest rate models. Since banks trade in a range of international currencies extensive empirical analysis has been carried out in other markets. Examples include Dahlquist [1996], who considered rates for Denmark, Sweden, Germany and the United Kingdom; Hiraki and Takezawa [1997] considered offshore rates in Japan; Tse [1995] considered money market rates for 11 countries and Adkins and Krehbiel [1999] looked at London Interbank Offer rates. In Figures 1.1 and 1.2 the relationship between the level of rates and the volatility of rates for the United Kingdom, United States Japan and Singapore are displayed using weekly 1 month interbank rates

21

16

UK

US

10

14 8 12 6

10 8

4

6 2 4 2

0 1992 1994 1996 1998 2000 2002 2004

1992

UK volatility 3.0 2.5

1994 1996 1998 2000 2002 2004

US volatility

2.0

1.6

2.0 1.2 1.5 0.8 1.0 0.4

0.5 0.0

0.0 1992 1994 1996 1998 2000 2002 2004

1992

1994 1996 1998 2000 2002 2004

Figure 1.1 UK and US Interbank Interest Rates 22

9

Japan

Singapore

12

8 10 7 6

8

5 6 4 3

4

2 2 1 0

0 1992 1994 1996 1998 2000 2002 2004

1.6

Japan volatility

1992

1994 1996 1998 2000 2002 2004

Singapore volatility

4

1.4 1.2

3

1.0 0.8

2

0.6 0.4

1

0.2 0.0

0 1992 1994 1996 1998 2000 2002 2004

1992

1994 1996 1998 2000 2002 2004

Figure 1.2 Japan and Singapore Interbank Interest Rates 23


obtained from DatastreamTM over the period from 1990 to 2005 where volatility is defined as |rt − rt−1 |. One of the general conclusions obtained in these empirical studies is the marked differences in the link between the level of rates and the volatility of rates in different financial markets. Another estimation approach is to use the general Gaussian estimation methods of Bergstrom [1983, 1985, 1986, 1990] and this was first demonstrated by Nowman [1997]. To apply these methods to the discrete model of Phillips [1972] (see also Bergstrom [1984a]), which assumed a constant conditional second moment that is not usually satisfied by financial data, it was necessary to modify the discrete model used in the estimation of the parameters of the continuous time interest rate models. Nowman [1997] assumed that as an approximation to the true underlying model given by Equation (1.12) that over the interval [0, T ], r (t) satisfied the stochastic differential equation dr (t) = {α + βr (t)} dt + σ {r (t − 1)}γ ζ(dt),

(1.13)

where t − 1 is the largest integer less than t. Nowman [1997] assumed that the volatility of the interest rate changes at the beginning of the unit observation period (day, week, month) and then remained constant. This allowed the modified discrete model of Phillips [1972] and Bergstrom

24


[1984a, Theorem 2] derived in Nowman [1997] to be used to obtain the Gaussian estimates below α r (t) = e β r (t − 1) + (e β − 1) + ηt (t = 1, 2, . . . , T ) β E(ηs ηt ) = 0 (s = t) E(ηt2 )

=

t

(1.14) (1.15)

2γ

e 2(t−τ )β σ 2 {r (t − 1)} dτ

t−1

σ 2 2β (1.16) = (e − 1){r (t − 1)}2γ = m2tt . 2β The Gaussian estimates are then obtained from the Gaussian

likelihood function where θ = [α, β, γ , σ 2 ] and  2  α β β r (t) − e r (t − 1) − (e − 1)  T  β   L(θ) = 2 log mtt + , 2   m tt t=1 (1.17)

which in Nowman [1997] can be defined as L(θ) =

T

2 log mtt + εt2 .

(1.18)

t=1

The transformed residuals ε1 , . . . , εT can be computed from mtt εt = ηt . The approximate discrete model had the advantage of reducing over the CKLS discrete approximation some of the temporal aggregation bias. The Nowman [1997] estimator is a quasi-maximum likelihood estimator

25


since the CKLS, CIRSR and BRSC models have a diffusion term that relates the level of rates to the volatility of rates and is non-Gaussian. It was also shown in empirical work on the US and UK financial markets in Nowman [1997] that the volatility of rates was not sensitive to the level of rates in the United Kingdom using monthly 1-month interbank rate with a level effect of γ = 0.2898, and using the same monthly 1-month US Treasury bill rate data used in CKLS a level effect of γ = 1.3610, the differences to the CKLS estimate being due to the different estimation methods used. Applications of this Gaussian estimation approach to international markets include Nowman [1998a, 2002] using monthly Eurocurrency data for the United States, Japan, France, Italy and Certificate of Deposit and Gensaki rates for Japan. An extensive study in Episcopos [2000] investigated the level effect for 10 countries using interbank rates again showing marked differences in the level effects. More recently, Yu and Phillips [2001] made an important contribution in deriving an exact likelihood function based on a new Gaussian discrete time form of the CKLS model to estimate the parameters. The method has the interesting property that it uses non-equispaced observations and a time change transformation in the exact discrete model to achieve Gaussianity. They used the idea of a time change in

26


the process to convert a continuous time martingale into a Brownian motion using the Dambis, Dubins-Schwarz theorem to achieve this (see Yu and Phillips [2001, Lemma 3.1] and Revuz and Yor [1999]). Yu and Phillips [2001] suggested an estimation algorithm that involved using the Nowman [1997] estimator and approximate discrete time model in an intermediate stage and then the Yu-Phillips transformed exact discrete model and exact Gaussian estimator. Monte Carlo evidence reported by them found that the Nowman estimator provided very good estimates of the diffusion term in finite samples but that the Yu and Phillips [2001] approach improves the estimation of the drift and mean reversion parameters in a range of frequencies (see also Kawai and Maekawa [2003] and Lo [2005]). Yu and Phillips [2001] also presented some empirical results, using the UK data used in Nowman [1997] and the US data used in CKLS, and found that the Yu-Phillips estimates were close to the Nowman estimates for the United Kingdom but not for the United States. A recent application of the Yu-Phillips procedure to ˜ ıguez Japanese financial markets is given by Nowman and N´ [2005a]. Due to the complicated nature of interest rate behaviour, continuous time models with more complex nonlinear drift and volatilities have been proposed for example by

27


A¨ıt-Sahalia [1996]. Also many other estimation methods have been put forward including A¨ıt-Sahalia [1996], Stanton [1997], Jiang and Knight [1997], Bandi and Phillips [2003] and Phillips and Yu [2005]. An important implication of these continuous time models for traders in fixed income financial markets is the implied derivative prices. CKLS provided a comparison of prices of bond options implied by the historically estimated CKLS specification and some special cases on US data and found significant variations in the call option values. Nowman and Sorwar [1999, 2005b] applied Gaussian estimation to the CKLS, Vasicek, CIRSR and BRSC models on monthly Eurocurrency rates for a number of countries and found that default free bond prices and contingent claim prices are sensitive to the underlying interest rate model used in a range of currencies. Phillips and Yu’s [2005] jackknifing method provided substantial improvements in bond options pricing over existing methods (see Phillips and Yu’s [2005] for further details). Continuous time multi-factor models have found widespread application in financial market derivative valuation. Examples of the factors include in addition to the short rate, the long rate, volatility of the short rate, time varying central tendency of the short rate, spreads and news

28


streams. An important two-factor continuous time model was developed by Brennan and Schwartz [1979] where the first factor is the short rate and the second factor the consol rate. To estimate their term structure model, they used the estimation procedure of Malinvaud [1966] and the exact discrete model (Phillips [1972]). Examples of other multi-factor models include Beaglehole and Tenney [1991], Longstaff and Schwartz [1992], Chen and Scott [1992], Hull and White [1994], Chen [1996], Duffie and Kan [1996], Andersen and Lund [1997] and Babbs and Nowman [1999]. The model of Babbs and Nowman [1999] was a general multi-factor Generalized Vasciek model and a subclass of the general Gaussian model of Langetieg [1980]. They dealt with the most general case of n-factors and derived the general pure discount bond price for the model. In Babbs and Nowman [1999] they assumed a possible description of the instantaneous spot interest rate, r, is r (t) = µ(t) −

J

X j (t),

(1.19)

j=1

where µ is a constant representing the long-run average rate and X1 (t), . . . , X J (t) are unobserved state variables representing the current effect of J streams of economic “news”. The “news” streams for example may represent interest rate

29


rumours from the Monetary Policy Committee of the Bank of England or the Bank of Japan’s Policy Board Meetings on monetary policy and long-term economic news representing monthly and quarterly economic statistics that all effect the yield curve. The arrival of each type of “news” is modelled by the innovations of Brownian motions, which may be correlated, while the impact of a piece of “news” dies away exponentially as the time since it was received increases. Babbs and Nowman [1999] assumed that X j (t) are generated by the processes below: d X j (t) = −β j X j (t) dt + c j ζ j (dt),

(1.20)

where the terms are defined in Babbs and Nowman [1999]. They used the Kalman filter (see Harvey [1989]) to estimate one-, two- and three- factor Generalized Vasicek models on US data and found that the two- and three-factor models were able to explain the US yield curve. In Bergstrom and Nowman [1999] they applied Gaussian estimation methods and derived the exact discrete model and the exact Gaussian likelihood function for the Babbs and Nowman [1999] model. More recently, Nowman [2001, 2003, 2006] considered the Gaussian estimation of multi-factor CKLS and CIRSR models and empirical applications on Japanese interest rate data indicated various level effects for the CKLS

30


ˇ [2003] model in the multi-factor cases. Recently, Saltoglu applied Gaussian estimation to the Brennan and Schwartz [1979] model. This area of application of continuous time models continues to be one of the most active research areas in finance. We now turn to a brief introduction to continuous time macroeconomic modelling.

1.5 Continuous Time Macroeconomic Modelling Continuous time modelling has also been extensively applied in macroeconomics since the development of the first continuous time macroeconomic model by Bergstrom and Wymer [1976]. Economy-wide continuous time macroeconometric models have been developed for most of the leading industrial countries of the world, including the United Kingdom (Bergstrom and Wymer [1976], Knight and Wymer [1978], Bergstrom, Nowman and Wymer [1992]), Australia (Jonson, Moses and Wymer [1977]), Canada (Knight and Mathieson [1979]), Italy (Gandolfo and Padoan, [1982, 1984, 1987a, 1990], Germany (Kirkpatrick [1987]), New Zealand (Bailey, Hall and Phillips [1987]), the United States (Donaghy [1993]), Sweden (Sjo¨ o¨ [1993]), Netherlands (Nieuwenhuis [1995]), Czech

31


Republic (Stavrev [2001]) and various other countries. A complete list is provided in Bergstrom [1996]. The Bergstrom and Wymer [1976] model was a closed system of 13 interdependent differential equations (10 structural equations and 3 identities). See also Bergstrom [1966b, 1967] for some earlier prototype models. An interesting feature of the model was that it contained only one exogenous variable a simple time trend. The trend terms were allowed for the unobservable factors: technical progress, the growth of labour supply and growth in demand for exports. The model was based on extensive economic theory and had 35 key parameters, which comprised 16 long-run elasticities and propensities β, 16 speed of adjustment parameters γ and 3 trend parameters λ. The model in its deterministic form is given below:

DC β1 Y = γ1 log (1.21) C C β2 e −λ1 t {Y −β4 − β3 K −β4 }−1/β4 DL = γ2 log (1.22) L L Dp Y 1+β4 −r + + β5 − k (1.23) Dk = γ3 γ4 β3 K p DY = γ5 {(1 − β6 )(C + DK + E) − Y} + γ6 {β7 (C + DK + E) − S}

32

(1.24)


DI = γ7 {β6 (C + DK + E) − I } + γ8 {β7 (C + DK + E) − S} DE β8 p −β9 e λ2 t = γ9 log E E  Y β4 −(1+β4 )/β4  −λ1 t 1 − β3 K β β we Dp  10 2  = γ10 log   p p

(1.25) (1.26)

(1.27)

Dw β2 e −λ1 t {Y −β4 − β3 K −β4 }−1/β4 = γ11 log (1.28) w β11 e λ3 t Dr β12 pY β13 r −β14 = γ12 log (1.29) r M E β16 e λ3 t Dm = γ13 log + γ14 log β15 I L E β16 e λ3 t + γ15 Dlog + γ16 Dlog (1.30) β15 I L DS = Y + I − C − DK − E DK =k K DM =m M

where C = real consumption Y = real net income or output K = amount of fixed capital

33

(1.31) (1.32) (1.33)


E = real exports I = real expenditure on imports S = stocks L = employment M = volume of money r = interest rate p = price level w = wage rate k = proportional rate of increase of fixed capital m = proportional rate of increase in volume of money t = time D = d/dt and 0 < β1 < 1, β4 > −1, 0 < β6 < 1, all other parameters positive (except possibly λ3 ). An important feature of nearly all existing continuous time macroeconometric models, starting with the model of Bergstrom and Wymer [1976], is that they are designed in such a way as to permit a rigorous mathematical analysis of their steady state and stability properties, thus providing a valuable check on the capacity of the model to generate plausible long-run behaviour. The general methodology used in this analysis was developed by Bergstrom [1966b, 1967], using smaller models. For the model of Bergstrom and Wymer [1976] and the many continuous time

34


macroeconometric models for which that model has served as a prototype including the model of Bergstrom, Nowman and Wymer [1992] below, it was shown that, when the exogenous variables satisfy certain conditions the model has a steady state solution in which all variables grow at constant exponential rates (possibly zero). Moreover, for each of these models, it was possible to obtain an explicit differential equation system in the logarithms of the ratios of the variables to their steady state paths, and so that the nonlin´ ear part satisfy the Poincare–Liapounov–Perron conditions (see Bellman [1953, p. 93] and Coddington and Levinson [1955, p. 314]). The asymptotic stability of the steady state solution can then be deduced from the eigenvalues of the linear part of this system. Most previous continuous time macroeconometric models have not been estimated by econometric methods that take into account the exact restrictions on the distribution of the discrete data implied by the continuous time model. They have used the discrete approximation method with a transformation developed by Phillips [1974b] and Bergstrom and Wymer [1976] for the temporal aggregation problem arising from flow variables with the full information maximum likelihood estimator. This leads to estimates that are not asymptotically efficient compared to the

35


exact discrete model approach. The Bergstrom and Wymer [1976] model was estimated on quarterly UK data over 1955–66 using the exact discrete model, but even in this case the exact restrictions were imposed only on the autoregressive coefficient matrices and therefore the final estimates were also not asymptotically efficient as the moving average component of the VARMA model was only approximately allowed for using their transformation for flow variables. The first macroeconomic application of the exact Gaussian method of Bergstrom [1983, 1986] was in the estimation of the second-order continuous time macroeconometric model of the United Kingdom developed by Bergstrom, Nowman and Wymer [1992]. The model based on the Bergstrom and Wymer [1976] model also made intensive use of economic theory to obtain a parsimonious parametrization and cross-equation restrictions. One of the innovations of the Bergstrom, Nowman and Wymer [1992] model in addition to being specified as a second-order system of differential equations is that it had a more elaborate financial sector. The model, in its deterministic form, is set out below. It is a system of 14 second-order differential equations with 63 structural parameters, including 27 long-run parameters or elasticities, 33 speed of adjustment parameters,

36


and 3 trend parameters representing the rate of technical progress, the rate of labour supply (the non-accelerating inflation level of employment) and the target rate of growth of the money supply. The variables and equations of the model, in its deterministic form, are as follows: D 2 logC = γ1 (λ1 + λ2 − DlogC ) β1 e −{β2 (r −Dlog p)+β3 Dlog p} (Q + P ) + γ2 log T1 C D 2 logL = γ3 (λ2 − DlogL) β4 e −λ1 t {Q −β6 − β5 K −β6 }−1/β6 + γ4 log L D 2 logK = γ5 (λ1 + λ2 − DlogK ) β5 (Q /K )[1+β6 ] + γ6 log r − β7 Dlog p + β8

(1.34)

(1.35)

(1.36)

D 2 logQ = γ7 (λ1 + λ2 − DlogQ ) {1 − β9 (qp/pi )β10 }(C + Gc + DK + E n + E o ) + γ8 log Q (1.37) D 2 log p = γ9 (Dlog(w/p) − λ1 ) β11 β4 T2 we −λ1 t {1 − β5 (Q /K )β6 }−(1+β6 )/β6 + γ10 log p (1.38)

37


D 2 logw = γ11 (λ1 − Dlog(w/p)) + γ12 Dlog( pi /qp) β4 e −λ1 t {Q −β6 − β5 K −β6 }−1/β6 + γ13 log β12 e λ2 t

(1.39)

D 2r = −γ14 Dr + γ15 [β13 + rf − β14 Dlogq + β15 { p(Q + P )/M} − r ]

(1.40)

D 2 logI = γ16 (λ1 + λ2 − Dlog( pi I /qp)) β9 (qp/pi )β10 (C + Gc + DK + E n + E o ) + γ17 log ( pi /qp)I (1.41) D 2 logE n = γ18 (λ1 + λ2 − DlogE n ) β β16 Yf 17 ( pf /q p)β18 + γ19 log En

D 2F = −γ20 DF + γ21 {β19 (Q + P ) − F }

(1.42)

(1.43)

D 2P = −γ22 DP + γ23 [{β20 + β21 (rf − Dlog pf )}K a − P ] (1.44) D 2K a = −γ24 DKa + γ25 [{β22 + β23 (rf − r ) − β24 Dlogq − β25 dx }(Q + P ) − K a ]

38

(1.45)


β26 eλ3 t D 2 logM = γ26 (λ3 − DlogM) + γ27 log M En + E o + P − F + γ28 Dlog ( pi /qp)I E n + E o + P − F − DKa + γ29 log ( pi /qp)I β27 pf D 2 logq = γ30 Dlog( pf /qp) + γ31 log qp En + E o + P − F + γ32 Dlog ( pi /qp)I E n + E o + P − F − DKa + γ33 log ( pi /qp)I

(1.46)

(1.47)

Endogenous variables: C = real private consumption E n = real non-oil exports F = real current transfers abroad I = volume of imports K = amount of fixed capital K a = cumulative net real investment abroad (excluding change in official reserves) L = employment M = money supply P = real profits, interest and dividends from abroad

39


p = price level Q = real net output q = exchange rate (price of sterling in foreign currency) r = interest rate w = wage rate Exogenous Variables: dx = dummy variable for exchange controls (dx = 1 for 1974–79, dx = 0 for 1980 onwards) E o = real oil exports Gc = real government consumption pf = price level in leading foreign industrial countries pi = price of imports (in foreign currency) rf = foreign interest rate T1 = total taxation policy variable ((Q + P )/T1 is real private disposable income) T2 = indirect taxation policy variable (Q /T2 is real output at factor cost) t = time Yf = real income of leading foreign industrial countries. The model was estimated using the exact Gaussian estimator for the general case of Bergstrom [1986] on UK quarterly data over the period 1974–84 with the predictive

40


performance tested over 1985–86. As judged by the plausibility of the parameter estimates, the post-sample forecasting performance and the analysis of its steady state properties, this model was very successful. One of the problems in estimating the parameters of continuous time models concerns the issue of the identification of the structural parameters. We have the aliasing problem where one cannot distinguish between structures generating cycles whose frequencies (per unit observation period) differ by integers. For example, the case of half a cycle per observation period and one and a half cycles per observation period. This phenomenon was originally discussed in a study by Phillips [1973], where it was shown how identification could be achieved through Cowles Commission type restrictions on the structural parameters. Hansen and Sargent [1983] also investigated the aliasing problem. They showed that there at most a finite number of structures that are indistinguishable (on the basis of the observations) from the true structure because of the aliasing phenomenon. One of the implications of their theorem is that the true structure will be identifiable provided that the unit observation period is sufficiently short. In Bergstrom, Nowman and Wymer [1992] they used an alternative approach to identification, suggested by the last result of Hansen and Sargent [1983],

41


but as pointed out by them not dependent on their theorem. This involved the use of bounds on the speed of adjustment parameters and they were able to show how to achieve for the complete vector of structural parameters of their model identification (see Bergstrom, Nowman and Wymer [1992]).

1.6 Policy Analysis in Continuous Time Macroeconomic Models Continuous time macroeconomic models have been extensively used in policy analysis (see, for example, Gandolfo and Padoan [1982, 1984], Gandolfo and Petit [1987b], Jonson and Trevor [1981], Jonson, McKibbin and Trevor [1982], Kirkpatrick [1987], Sassanpour and Sheen [1984], Stefansson [1981] and Tullio [1981]). The Bergstrom and Wymer [1976] model was used for policy analysis in Bergstrom [1978, 1984b], where an analysis of monetary and fiscal policy feedbacks was made. This analysis was based on feedbacks of the type introduced into economic models by Phillips [1954] and Bergstrom [1967, Ch. 6]. A more advanced approach is to use optimal control theory. In Bergstrom, Nowman and Wandasiewicz [1994], they investigated monetary and fiscal policy feedbacks, using the Bergstrom, Nowman and Wymer [1992] model. As the

42


model’s steady state computed from the estimated parameters was slightly unstable (although, statistically they could not reject the hypothesis that it is stable), they investigated the use of simple fiscal policy feedbacks and the use of control theory considered in Bergstrom [1987] that could reduce or eliminate the instability. Optimal feedbacks were derived by minimizing an infinite horizon quadratic cost function involving deviations of output, the exchange rate and the policy instruments from their steady state paths and also the rates of change of these variables. One of the findings was that fiscal policy had an important role to play in the stabilization of the model than did monetary policy. A discussion of the important contributions of Barnett and He [1999, 2002] concerning bifurcation analysis in continuous time macroeconometric models is given in Chapter 4. Another important issue in macroeconomic modelling in discrete and continuous time concerns the famous Lucas [1976] critique and also the related issues of deep parameters, structural parameters and rational expectations in macroeconomic modelling. Deep parameters are taken to be the parameters of Euler equations and are directly acquired from the parameters of tastes and technology. The role of rational expectations in macroeconomic modelling had an important influence on the Bergstrom, Nowman

43


and Wymer [1992] model’s formulation. The Bergstrom, Nowman and Wymer [1992] model made systematic use of the assumption of long-run rational expectations in a number of ways. The assumption of long-run rational expectations in many of the equations played an important role in the identification of certain parameters of the model that would not otherwise be identifiable. In particular, it is assumed that the agents in the economy know the trend parameters λ1 and λ2 for productivity and the labour supply and take these into account in forming their expectations. The assumption that the agents in the economy know these parameters is, of course, more realistic than the assumption that they know the parameters affecting the short-run dynamics of the model, which would be necessary to justify the assumption of full rational expectations. It is shown in Bergstrom, Nowman and Wymer [1992] that the steady state growth rate of most of the real variables in the model is λ1 + λ2 . The long-run rational expectations assumption is incorporated in the structural equations by setting the expectations equal to λ1 + λ2 derived from the steady state. The assumption of rational expectations also played an important role in the wage rate adjustment equation and in the foreign exchange

44


market. See Bergstrom, Nowman and Wymer [1992] for full details.

1.7 Stochastic Trends in Econometric Models In further work on the Bergstrom, Nowman and Wymer [1992] model, Nowman [1996] re-estimated the model on more recent data with a different measure of the money supply. One of the conclusions in Nowman [1996] was that the use of single deterministic time trends for technical progress and labour supply had an impact on the forecasting performance of the model. In Nowman [1998b], the trends were modified as segmented trends over 4-year periods and the model re-estimated improving the forecasting performance of the model. Nevertheless, a careful analysis of the estimates and forecasts of these studies suggests that the forecasting performance could be improved by replacing the deterministic trends (simple non-random functions of the time parameter) introduced to allow for such factors as technical progress and changes in the non-accelerating inflation level of employment (which are unobservable) by stochastic trends. Indeed, there are good reasons for believing that

45


the replacement of deterministic trends by stochastic trends could significantly improve the forecasting performance of most macroeconometric models, formulated in either discrete or continuous time, that include deterministic trends to allow for such long-run changes as those mentioned. There has, recently, been extensive work on the problems of statistical inference in cointegrated systems (see, for example, Engle and Granger [1987], Johansen [1988, 1991], Phillips and Durlauf [1986], Park and Phillips [1988, 1989], Phillips [1987, 1991a, 1991b, 1995], Franses [1996] and Chambers [2001, 2003]). Nearly all of this work is concerned with models that are formulated in discrete time and in which the stochastic trend variables are observable. There has, however, been some work by Harvey and Stock [1988] and Bergstrom [1997] on the estimation of the parameters of continuous time dynamic models with unobservable stochastic trends (see also Corradi [1997] and Simos [1996]). The article of Bergstrom [1997] extended the exact Gaussian estimation method developed in the earlier articles, referred to above, so that it is applicable to models incorporating unobservable stochastic trends. It contains a detailed estimation algorithm and formulae for its implementation, when the model comprises either a closed or open system of mixed first- and second-order

46


stochastic differential equations, which incorporate unobservable stochastic trends, while the data is a mixture of stocks (observed at a sequence of equispaced points of time) and flows (observed as integrals over a sequence of adjoining intervals of equal length). The algorithm is applicable, therefore, to models which, in addition to incorporating unobservable stochastic trends, involve all the complications in existing continuous time macroeconometric models. Such a model has been developed for the UK economy by the authors of this monograph. Its parameters have been estimated by the algorithm of Bergstrom [1997], the estimation being the first application of the new algorithm. The model is the first continuous time macroeconometric model incorporating stochastic trends, and its development represents a major step forward in macroeconometric modelling. The purpose of this monograph is to describe the new model, the estimation of its parameters and its application to dynamic analysis and forecasting.

1.8 An Outline of Contents Chapter 2 deals with the econometric methodology used in the later chapters, with emphasis on its application rather than its theoretical derivation. The aim of this chapter is

47


to explain the newly developed exact Gaussian estimation method for continuous time econometric models incorporating unobservable stochastic trends. It is based on the article of Bergstrom [1997], and readers are referred to this article for proofs of the various results used here. An Appendix to the chapter sets out the detailed formulae needed for the present development and this includes previously unpublished formulae for the autocovariance matrices of the innovations in the exact discrete model satisfied by the observations. These formulae play an important role in the estimation procedure. The next two chapters are concerned with the specification of the model and the analysis of its steady state and stability properties. The complete model is set out in Chapter 3, together with a discussion of the specification of the individual equations. The model is, essentially, a development of the model of Bergstrom, Nowman and Wymer [1992]. The most important change is that the deterministic trends incorporated in this model to allow for technical progress and the gradual change in the non-accelerating inflation level of employment are replaced by stochastic trends. Another important change is that, whereas all the equations in the earlier model are specified as second-order differential equations, a few equations in the new model (taking account of

48


the estimated speed of adjustment parameters in the earlier model) are specified as first-order differential equations, all other equations being of the second order. This complicates the estimation procedure, particularly as the data are a mixture of stocks and flows. However, the exact Gaussian estimation algorithm developed by Bergstrom [1997] and described in Chapter 2 allows for the combined complications of mixed-order differential equations and mixed stock and flow data, as well as unobservable stochastic trends. For the model developed in this monograph a full steady state and stability analysis is carried out in Chapter 4. There we derive closed formulae from which the exact steady state paths of the variables and linear (in logarithms) approximations about these steady state paths can be computed. The final chapter (Chapter 5) is concerned with the estimation of the structural parameters of the model from UK data, the computation of the post-sample forecasts and the derivation from the parameter estimates, of the steady state solution and its stability properties.

49

CHAPTER TWO

Continuous Time Econometrics with Stochastic Trends 2.1 Introduction

T

his chapter is concerned with the general econometric methodology used throughout the book. The main

objective is to exposit the newly developed exact Gaussian estimation method for continuous time econometric models that incorporate unobservable stochastic trends. The approach is based on the article of Bergstrom [1997], and readers are referred to this article for proofs of the various results used. Bergstrom [1997] extends the exact Gaussian estimation method developed in earlier work (Bergstrom [1983, 1985, 1986, 1990]) in two ways. Firstly, the algorithm is applicable to models incorporating unobservable stochastic trends. These can provide a more flexible and

50

Continuous Time Econometrics with Stochastic Trends

realistic representation of such unobservable trend variables as technical progress, which have often been represented in econometric models in the past by deterministic trends. Secondly, the algorithm is applicable to models formulated as systems of mixed-order stochastic differential equations, even when the discrete observations of the variables are a mixture of stocks and flows. Hitherto, the problem of estimating mixed-order continuous time systems has been dealt with only for the special case in which all the variables are measured at equispaced points of time, i.e. there are no flow variables (see Agbeyegbe [1984]). The new algorithm is applicable, therefore, to models that, in addition to incorporating unobservable stochastic trends, involve all the complications assumed in the earlier work on the exact Gaussian estimation of continuous time macroeconometric models. The general method used in the new algorithm is similar to that used in the earlier algorithm of Bergstrom [1986, 1990] for the exact Gaussian estimation of systems of higher order stochastic differential equations without stochastic trends. This involves the derivation of an exact discrete model for the observations generated by the underlying continuous time model and the representation of this

51


exact discrete model in the form of a VARMAX (vector autoregressive moving average with exogenous variables) model. The VARMAX model is then used as a basis for the estimation of the parameters of the underlying continuous time model and the generation of post-sample forecasts of the discrete observations. Like the earlier algorithm of Bergstrom [1986, 1990], the new algorithm is also applicable to models containing exogenous variables. In which case, the exact discrete model is also in the form of a VARMAX model. In the remainder of the chapter the new algorithm will be discussed in detail when the model is a mixture of first- and second-order stochastic differential equations and the discrete observations of the variables are a mixture of stocks and flows. The continuous time model is presented and explained in Section 2.2, the exact discrete model and its VARMAX representation are presented in Section 2.3, the use of this VARMAX model for the estimation of the parameters of the continuous time model and the generation of post-sample forecasts of the discrete observations is described in Section 2.4 and Section 2.5 concludes. The formulae used in the exact discrete model are presented in Appendices A and B.

52


2.2 The Continuous Time Model The assumed model is d x1 (t) = [A1 (θ)x 1 (t) + A2 (θ)x2 (t) + A3 (θ)D x2 (t) + B1 (θ )z(t) + C 1 (θ)µ(t) + b1 (θ, λ)]dt + ζ1 (dt)

(t ≥ 0), (2.1)

d[D x 2 (t)] = [A4 (θ)x1 (t) + A5 (θ)x2 (t) + A6 (θ)D x 2 (t) + B2 (θ)z(t) + C 2 (θ)µ(t) + b2 (θ, λ)]dt + ζ2 (dt)

(t ≥ 0),

dµ(t) = λdt + ζ3 (dt)

x 1 (0), x2 (0), D x2 (0) = y (0),

(2.2) (t ≥ 0), µ(0) = 0,

(2.3) (2.4)

where {x 1 (t), t > 0} is a real n1 -dimensional continuous time random process, {x2 (t), t > 0} is a real n2 -dimensional continuous time random process, {µ(t), t > 0} is a real kdimensional continuous time random process, z(t) is an m × 1 vector of non-random functions of t (exogenous variables), y(0) is an (n1 + 2n2 )-dimensional non-random vector, θ is a p-dimensional vector of unknown structural parameters, λ is a k-dimensional vector of unknown drift parameters, A1 , A2 , A3 , A4 , A5 , A6 , B1 , B2 , C 1 and C 2 are matrices whose elements are known functions of θ , b1 and b2 are

53


vectors whose elements are known functions of θ and λ, D is the mean-square differential operator, ζ1 (dt), ζ2 (dt) and ζ3 (dt) are white noise innovation vectors that are, precisely, defined by Assumption 1. Assumption 1:

ζ1 , ζ2 and ζ3 are vectors of random mea-

sures defined on all subsets of the half line 0 < t < ∞ with finite Lebesgue measure, such that E[ζi (dt)] = 0 (i = 1, 2, 3), E[ζi ( 1 )ζ j ( 2 )] = 0 (i, j = 1, 2, 3) for any disjoint sets 1 and 2 on the half line 0 < t < ∞ and E[ζi (dt)ζ j (dt)] = dti j (i, j = 1, 2, 3), where 11 , 22 and 33 are positive definite matrices and 13 and 23 are zero matrices (for a discussion of random measures and their application to continuous time stochastic models, see Bergstrom, [1984a, p. 1157]). In the preceding model, the elements of µ(t) are unobservable stochastic trends, whereas the elements of x 1 (t) and x2 (t) are endogenous variables, which are observable, either at discrete points of time (stock variables) or as integrals over discrete intervals of time (flow variables). At each point of time, x 1 (t) is adjusting through a first-order differential equation system, and x2 (t) is adjusting through a second-order differential equation system to the current values of x1 (t), x2 (t), D x2 (t), z(t), µ(t) and the white noise

54


innovations. As (2.2) is a second-order equation and the variable x2 (t) is not mean differentiable to the second order the appropriate representation is in terms of the increment in the first-order mean square derivative, i.e. d[D x 2 (t)]. Because, under Assumption 1, neither x1 (t) nor D x 2 (t) is differentiable (even in the mean square sense), (2.1) and (2.2) are, more precisely, interpreted as representing the integral equations t x1 (t) − x1 (0) = [A1 x1 (r ) + A2 x2 (r ) + A3 D x2 (r ) 0 t + B1 z(r )+C 1 µ(r )+b1 ]dr + ζ1 (dr ) (t > 0), 0

(2.5)

t

D x 2 (t) − D x 2 (0) =

[A4 x1 (r ) + A5 x 2 (r ) + A6 D x2 (r )

0

+ B2 z(r ) + C 2 µ(r ) + b2 ]dr t ζ2 (dr ) (t > 0), +

(2.6)

0

where

t 0

ζi (dr ) = ζi [0, t] (i, j = 1, 2) and the first integrals

on the right-hand sides of (2.5) and (2.6) are defined in the wide (mean square) sense (see Bergstrom [1984a, p. 1152]). Similarly, (2.3) is interpreted as meaning that t µ(t) − µ(0) = tλ + ζ3 (dr ) (t > 0). 0

55

(2.7)


In the statistical literature dealing with stochastic trends, it is commonly assumed that both the level and drift of the stochastic trend path are stochastic. For example, Harvey and Stock [1988] allowed the drift to evolve according to a Brownian motion with non-random drift. The model comprising (2.1)–(2.4) could be generalized by adding another equation, similar to (2.3), representing the stochastic evolution of the drift vector λ. But, for the sort of variables represented by trends in most economic models, this generalization is unrealistic. It implies, for example, that if one of the elements of λ represents the rate of technical progress, then the rate of technical progress will, almost certainly, exceed any bound at some time in the future. The vector λ in (2.3) is assumed, therefore, to be a vector of constants. Moreover, some empirical support for this assumption has been provided by Harvey, Henry, Peters and Wren-Lewis [1986] in their estimation of a discrete-time employment-output equation for the UK manufacturing data. They incorporated a stochastic trend to represent technical progress and estimated that the drift in this trend was constant; that is the estimated variance of the innovation in the drift was zero. There is also a fundamental difference between the way in which the stochastic trends are introduced into the above model and the way in which they are introduced into the

56


model of Harvey and Stock [1988], who are among the few other econometricians to have dealt with the problem of estimating structural parameters in continuous time models with unobservable stochastic trends. Harvey and Stock [1988] assumed that each of the observable (at discrete intervals) variables is the sum of two unobservable components, the first component being one of a set of completely unobservable variables generated by an underlying stochastic differential equation system and the second being a linear combination of stochastic trends. In the above model the stochastic trends are more deeply embedded in the structure and help to derive the differential equation system, just as, for example, technical progress helps to derive the economy. The essential distinction between the two approaches is that in the above model the stochastic trends play the role of random forcing functions, which help to generate the solution of the differential equation system, whereas in the model of Harvey and Stock [1988] the stochastic trends have no effect on the solution of that system, but are added to the solution (like observation errors) to obtain the observed variables. Finally, it should be noted that λ is introduced as an argument in the vector functions b1 (θ, λ) and b2 (θ, λ) in (2.1) and (2.2). This allows for the incorporation in the model of the assumption of long-run expectations, that is, the assumption

57


that the various agents in the economy know the drifts and take them into account when adjusting the variables that they control.

2.3 The Exact Discrete Model and Its VARMAX Representation We turn now to our precise assumptions concerning the observability of x1 (t), x2 (t) and z(t), (0 < t < ∞), and the definitions of the observable vectors used in the formation of the exact discrete model. To concentrate on the case of practical importance, we assume that x 1 (t) includes flow variables only whereas x2 (t) includes both stock and flow variables. These assumptions are suggested by the parameter estimates for the model of Bergstrom, Nowman and Wymer [1992]. All of the variables whose partial adjustment processes could, on the basis of those estimates, be realistically represented by first-order equations were flow variables such as output, consumption, exports and imports. Stock variables, such as the stock of fixed capital and the level of employment, all adjusted relatively slowly, and the modes of the time lag distributions implied by their estimated speed of adjustment parameters were all quite large (at least 9 months), indicating that a second-order equation

58


was more realistic for these variables. There were also some flow variables, particularly wage and price variables (which were measured as time averages and had, therefore, to be treated as flow variables), for which a second-order adjustment equation appeared to be more realistic. We assume that the stock variables are observed at the time points 0, 1, 2, . . . , T and the flow variables as integrals over the intervals [0, 1], [1, 2], . . . , [T − 1, T ]. We let the elements of x2 (t) and z(t) be ordered (without loss of generality) so that x2 (t) =

x2s (t) f

x2 (t)

,

z(t) =

z s (t) z f (t)

, f

where x2s (t) is an ns2 × 1 vector of stock variables, x2 (t) is f

f

an n2 × 1 vector of flow variables, ns2 + n2 = n2 , z s (t) is an ms × 1 vector of stock variables, z f (t) is an m f × 1 vector of flow variables and ms + m f = m. We then define the observable vectors x¯ 1t , x¯ 2t , x¯ t and z¯t (t = 1, . . . , T ) by t x¯ 1t = x1 (r )dr (t = 1, . . . , T ),

(2.8)

t−1



x¯ 2t

 x2s (t) − x2s (t − 1)   = t f  x 2 (r )dr t−1

59

(t = 1, . . . , T ),

(2.9)


x¯ t = 1 2

 z¯t = 

x¯ 1t x¯ 2t

(t = 1, . . . , T ),

{z s (t) + z s (t − 1)} t

z f (r )dr

(2.10)

  

(t = 1, . . . , T ).

(2.11)

t−1

It has been shown (Bergstrom [1997, Theorems 4 and 5]) that if x1 (t) and x 2 (t)(0 ≤ t ≤ T ) are generated by the model (2.1)–(2.4), then, under certain conditions, the vectors x¯ 1 , x¯ 2 , . . . , x¯ T , z¯1 , z¯2 , . . . , z¯T defined by (2.10) and (2.11) satisfy, exactly, the system: x¯ 1 = G10 + G11 y(0) + E 11 z¯1 + E 12 z¯2 + E 13 z¯3 + η1 ,

(2.12)

x¯ 2 = F21 x¯ 1 + G20 + G21 y(0) + E 21 z¯1 +E 22 z¯2 + E 23 z¯3 + η2 ,

(2.13)

x¯ 3 = F32 x¯ 2 + F31 x¯ 1 + G30 + G31 y(0) + E 31 z¯1 + E 32 z¯2 + E 33 z¯3 + η3 ,

(2.14)

x¯ t = F1 x¯ t−1 + F2 x¯ t−2 + F0 + E 0 z¯t + E 1 z¯t−1 + E 2 z¯t−2 + ηt

where

(t = 4, . . . , T ),

(2.15)

F0 , F1 , F2 , F21 , F32 , F31 , G10 , G11 , G20 , G21 , G30 , G31 ,

E 0 , E 1 , E 2 , E 11 , E 12 , E 13 , E 21 , E 22 , E 23 , E 31 , E 32 , E 33 are matrix functions of the structural parameters of the continuous time model, which are precisely defined in Appendix A,

60


and η1 , η2 , . . . , ηT are unobservable random vectors such that E(ηt ) = 0 (t = 1, . . . , T ), E ηt ηt = 0 (t = 4, . . . , T ), = 1 (t = 5, . . . , T ), E ηt ηt−1 E ηt ηt−2 = 2 (t = 6, . . . , T ), = 3 (t = 7, . . . , T ), E ηt ηt−3 E η2 η1 = 21 , E η2 η2 = 22 , E η1 η1 = 11 , E η3 η2 = 32 , E η3 η3 = 33 , E η3 η1 = 31 , E η4 η1 = 41 , E η4 η2 = 42 , E η4 η3 = 43 , E η5 η3 = 53 , E η6 η3 = 63 , E η5 η2 = 52 , E ηt ηt− j = 0 (t = 5, . . . , T ; j = 4, . . . , T ),

where 0 , 1 , 2 , 3 and the various i j in the above expressions are matrix functions of the structural parameters of the continuous time model that are precisely defined in Appendix B. Sufficient conditions for the discrete observations generated by the continuous time model to satisfy, exactly, the system (2.12)–(2.15) are that matrix A defined by 

A1 A = 0 A4

A2 0 A5

61

 A3 I  A6


is nonsingular, a certain submatrix of e A (see Bergstrom [1997, Assumption 3]) is nonsingular and the elements of z(t) are polynomials in t of degree not exceeding 2. Only the last of these conditions is very restrictive. For a model such as that estimated in Chapter 5, the conditions on A and e A will be satisfied except for combinations of parameters that may arise in areas of the parameter space into which an optimisation algorithm might stray “by chance” that would violate these conditions. The condition that the elements of z(t) are polynomials in t of degree not exceeding 2 is, of course, very restrictive. However, an important property of the system (2.12)–(2.15), which will be referred to as “the exact discrete model”, is that the coefficient matrices E 0 , E 1 , E 2 and E i j (i, j = 1, 2, 3) depend only on the structural parameters of the continuous time model and not on the parameters determining the time path of z(t). Its accuracy depends, therefore, only on the accuracy with which the continuous time path of each exogenous variable can be approximated by a sequence of quadratic functions on the overlapping subintervals [0, 4], [1, 5], . . . , [T − 4, T ] and it will be approximately satisfied, even if the behaviour of the exogenous variables varies greatly over the sample period, provided that their continuous time paths are sufficiently smooth. Moreover, it will be exactly satisfied if the model is

62


closed, that is if the matrices B1 and B2 in the continuous time model are zero matrices. It has been shown (Bergstrom [1997, Theorem 6]) that the system (2.12)–(2.15) can also be represented in the form of a VARMAX model: x¯ 1 − G10 − G11 y(0) − E 11 z¯1 − E 12 z¯2 − E 13 z¯3 = M11 ε1 ,

(2.16)

x¯ 2 − F21 x¯ 1 − G20 − G21 y(0) − E 21 z¯1 − E 22 z¯2 − E 23 z¯3 = M21 ε1 + M22 ε2 ,

(2.17)

x¯ 3 − F32 x¯ 2 − F31 x¯ 1 − G30 − G31 y(0) − E 31 z¯1 − E 32 z¯2 − E 33 z¯3 = M31 ε1 + M32 ε2 + M33 ε3 ,

(2.18)

x¯ t − F1 x¯ t−1 − F2 x¯ t−2 − F0 − E 0 z¯t − E 1 z¯t−1 − E 2 z¯t−2 = Mtt εt + Mt,t−1 εt−1 + Mt,t−2 εt−2 + Mt,t−3 εt−3

(t = 4, . . . , T ), (2.19)

where the random vectors εt (t = 1, . . . , T ) satisfy the conditions E(εt ) = 0 (t = 1, . . . , T ), E εt εt = I (t = 1, . . . , T ), E εs εt = 0 (s = t; s, t = 1, . . . , T ),

(2.20) (2.21) (2.22)

and the coefficient matrices M11 , M21 , M22 , M31 , M32 , M33 , Mtt , Mt,t−1 , Mt,t−2 and Mt,t−3 (t = 4, . . . , T ) are obtained, recursively, from (2.23)–(2.44). M11 M11 = 11 ,

63

(2.23)


−1 M21 = 21 M11 ,

(2.24)

M22 M22 = 22 − M21 M21 ,

(2.25)

−1 M31 = 31 M11 ,

(2.26)

−1 M32 = 32 − M31 M21 , M22

(2.27)

M33 M33 = 33 − M31 M31 − M32 M32 ,

(2.28)

−1 M41 = 41 M11 ,

(2.29)

−1 M22 M42 = 42 − M41 M21 ,

(2.30)

−1 − M42 M32 , M33 M43 = 43 − M41 M31

(2.31)

M44 M44 = 0 − M41 M41 − M42 M42 − M43 M43 ,

(2.32)

−1 M52 = 52 M22 ,

(2.33)

−1 M53 = 53 − M52 M32 , M33

(2.34)

−1 M54 = 1 − M52 M42 − M53 M43 , M44

(2.35)

64


M55 M55 = 0 − M52 M52 − M53 M53 − M54 M54 ,

(2.36)

−1 , M63 = 63 M33

(2.37)

−1 , M64 = 2 − M63 M43 M44

(2.38)

−1 − M64 M54 , M65 = 1 − M63 M53 M55

(2.39)

= 0 − M63 M63 − M64 M64 − M65 M65 , M66 M66

(2.40)

and, for t = 7, . . . , T,

−1 Mt,t−3 = 3 Mt−3,t−3 ,

(2.41)

−1 , Mt,t−2 = 2 − Mt,t−3 Mt−2,t−3 Mt−2,t−2

(2.42)

−1 − Mt,t−2 Mt−1,t−2 , Mt,t−1 = 1 − Mt,t−3 Mt−1,t−3 Mt−1,t−1 (2.43) Mtt Mtt = 0 − Mt,t−3 Mt,t−3 − Mt,t−2 Mt,t−2 − Mt,t−1 Mt,t−1 ,

(2.44)

the matrices Mtt (t = 1, . . . , T ) being lower triangular matrices. The VARMAX model (2.16)–(2.19) is more general than the standard VARMAX model considered in the statistical literature in that the moving average coefficient matrices

65


Mtt , Mt,t−1 , Mt,t−2 and Mt,t−3 are time dependent. It can be shown, however, that they converge to constant matrices as t → ∞ (see Bergstrom [1990, Ch. 7, Theorem 1] and the comments before and after that theorem). As we shall see in this chapter, the VARMAX model (2.16)–(2.19) provides a very convenient basis for both the Gaussian estimation of the structural parameters of the continuous time model (2.1)–(2.4) and the computation of the optimal post-sample forecasts of the discrete observations generated by that model. It is also of considerable interest in itself and, in particular a link with the extensive recent literature on discrete time models with unit roots or stochastic trends (see, especially, Phillips and Durlauf [1986], Phillips [1987, 1991a, 1995], Engle and Granger [1987], Johansen [1988, 1991], Park and Phillips [1988, 1989], Franses [1996]. In this connection, it should be noted that except when one or more variables are affected, either directly or indirectly (via other variables), by the stochastic trends, all of the other elements of the vector x¯ t are integrated of order 1. Then, provided that the k × (n1 + n2 ) matrix [C 1 C 2 ] has rank k, there are n1 + n2 − k linearly independent cointegrating vectors (see Bergstrom [1997, Appendix B]).

66


2.4 Estimation and Forecasting Before deriving the Gaussian likelihood function, it will be useful to review the complete set of parameters to be calculated. The vector θ of basic structural parameters and the vector λ of drift parameters have, already, been explicitly introduced in (2.1)–(2.3), and these parameters will affect all of the matrices and vectors of coefficients on both the left- and right-hand sides of (2.16)–(2.19). It is convenient, at this stage, to introduce the (n1 + 2n2 ) × (n1 + 2n2 ) matrix defined by



11 = 0 21

0 0 0

 12 0 . 22

(2.45)

This matrix and the matrix 33 both occur in formulae for the i and i j matrices in Appendix B. We shall parametrize and 33 by writing them as (α) and 33 (β), where α and β are vectors of structural parameters. If there are no restrictions on these matrices, apart from those implied by Assumption 1, then α will have (n1 + n2 )(n1 + n2 + 1)/2 elements and β will have k(k + 1)/2 elements. In addition to the vector [θ, λ, α, β] of genuine parameters, most of the initial state vector y(0) is unobservable

67


and must be treated as part of the complete vector of parameters to be estimated. The observable part of y(0) is the ns2 × 1 vector x 2s (0), that is the initial levels of the stock variables. f

The unobservable part of y(0) is the (n1 + n2 + n2 ) × 1 vector yn defined by 

x 1 (0)



 f   yn =   x2 (0) . D x 2 (0)

(2.46)

Because the vector yn occurs in only the first three equations of the VARMAX model (2.16)–(2.19), we cannot expect to obtain a consistent estimate of this vector. But, its inclusion as part of the complete vector of parameters to be estimated could have a significant effect on the estimate of [θ, λ, α, β] obtained from a small sample. We shall treat the vector [θ, λ, α, β, yn ] , therefore, as the complete vector of parameters to be estimated. The Gaussian likelihood function can be derived following the general approach of Bergstrom [1983, 1985, 1986, 1997] from (2.16)–(2.19). This system can be regarded as a linear transformation from the vector ε = [ε1 , ε2 , . . . , εT ] to the vector x¯ = [x¯ 1 , x¯ 2 , . . . , x¯ T ] , and the Jacobian of the T transformation is t=1 |Mtt |. Hence, if L(θ, λ, α, β, yn )

68


denotes minus twice the logarithm of the likelihood function (less a constant), then T L= εt εt + 2log|Mtt | ,

(2.47)

t=1

where ε1 , ε2 , . . . , εT are regarded as functions of the observations and the parameters and can be computed, recursively, from (2.16)–(2.19), that is, from the recursive equations −1 ε1 = M11 (x¯ 1 − G10 − G11 y(0) − E 11 z¯1 − E 12 z¯2 − E 13 z¯3 ),

(2.48) −1 (x¯ 2 − F21 x¯ 1 − G20 − G21 y(0) ε2 = M22

− E 21 z¯1 − E 22 z¯2 − E 23 z¯3 − M21 ε1 ),

(2.49)

−1 ε3 = M33 (x¯ 3 − F32 x¯ 2 − F31 x¯ 1 − G30 − G31 y(0)

− E 31 z¯1 − E 32 z¯2 − E 33 z¯3 − M31 ε1 − M32 ε2 ),

(2.50)

εt = Mtt−1 ( x¯ t − F1 x¯ t−1 − F2 x¯ t−2 − F0 − E 0 z¯t − E 1 z¯t−1 − E 2 z¯t−2 − Mt,t−1 εt−1 − Mt,t−2 εt−2 − Mt,t−3 εt−3 ) (t = 4, . . . , T ).

(2.51)

The exact Gaussian estimates of the parameters are obtained by maximizing L with respect to (θ, λ, α, β, yn ). These will be exact maximum likelihood estimates if the innovations in the continuous time model (2.1)–(2.4) are Brownian motion (in which case the innovation vectors

69


ε1 , ε2 , . . . , εT in the VARMAX model (2.16)–(2.19) will be normally distributed) and the continuous time paths of the exogenous variables are quadratic functions of t. Moreover, they can be expected to provide very good estimates under much more general circumstances (see the comments in Bergstrom [1997, p. 486]). The above estimation procedure can also be applied to continuous time models containing the first and second derivatives of the exogenous variables. For this purpose, the vector z(t) is replaced by the vector z ∗ (t) = [z (t), D z (t), D 2 z (t)] and a vector z¯∗t is defined in terms of z ∗ (t) by an equation analogous to (2.11). Although z¯∗t is not directly observable, it can be obtained from z¯t , z¯t−1 and z¯t−2 using quadratic interpolation formulae given in Nowman [1991] (cf. Chambers [1991]), these formulae being exact when the elements of z(t) are quadratic functions of t. As was noted in the previous section, the elements of the vector x¯ t are, in general, integrated of order 1 with n1 + n2 − k cointegrating vectors. Consequently, the classical theory of inference is not applicable to the exact Gaussian estimates discussed above, even when these are exact maximum likelihood estimates.

70


In this connection, it should be noted, first, that the behaviour of the estimates as T → ∞ will differ as between various types of parameter. In a model such as that discussed in the remaining chapters of this monograph, the structural parameter vector θ will include both long-run parameters and speed of adjustment parameters. Whereas the estimates of the speed of adjustment parameters will converge (in probability) at the rate T 1/2 to the true values as T → ∞, the estimates of some of the long-run parameters (those that are uniquely related to the cointegrating vectors) will converge at the rate T . It should be noted, also, that unless C 1 and C 2 are severely restricted, the drift parameters (the elements of λ) are not identifiable. In fact, in a typical macroeconometric model, all or nearly all of the elements of C 1 and C 2 will be known constants. For example, the stochastic trend representing technical progress will enter the employment–output relation and the price–wage relation (each formulated in the logarithms of the variables) with the coefficient −1. The distribution of the estimates discussed above will also be different from the distribution of parameter estimates obtained under classical assumptions. Whereas, under classical assumptions, the deviations of the estimates from the

71


true values of the parameters (each deviation being multiplied by T 1/2 ) are, in general, asymptotically normally distributed, the deviations of the above estimates from the true values of the parameters (each deviation being multiplied by an appropriate power of T ) are asymptotically distributed like very complicated expressions involving integrals of quadratic forms of Brownian motions and integrals of Brownian motions with respect to Brownian motions. This is as we should expect in view of the extensive work of P. C. B. Phillips and his collaborators on the estimation of models with integrated processes (see Phillips and Durlauf [1986], Phillips [1987, 1991a, 1991b, 1995], Park and Phillips [1988, 1989]). The computational cost of using such expressions in assessing the accuracy of the parameter estimates would, however, be very heavy except in very small models. In assessing the accuracy of the parameter estimates for our UK model presented in Chapter 5, we will follow a simpler Bayesian approach. In this connection, it should be noted, first, that, if the innovations in the continuous time model (2.1)–(2.4) are Brownian motion and the exogenous variables are quadratic functions of t, then assuming a uniform prior distribution, L(θ, λ, α, β, yn ) is minus twice the logarithm of the Bayesian posterior probability

72


density function of [θ, λ, α, β, yn ] (less a constant). Then letˆ λ, ˆ α, ˆ yˆ n ) denote the exact Gaussian estimator of ting (θ, ˆ β, ˆ α, ˆ yˆ n ) (θˆ , λ, ˆ α, ˆ yˆ n )]−1 is the (θ, λ, α, β, yn ), 2[∂ 2 L/∂(θˆ , λ, ˆ β, ˆ β, covariance matrix of a normal approximation to this Bayesian posterior probability density function in the neighbourhood of its mode. In Chapter 5, we, formally, present the square roots of the diagonal elements of ˆ λ, ˆ α, ˆ yˆ n ) (θˆ , λ, ˆ α, ˆ yˆ n )]−1 as the “Standard ˆ β, ˆ β, 2[∂ 2 L/∂(θ, Errors” of the parameter estimates, although it must be remembered that they should be interpreted in this approximate Bayesian sense. A final point to be noted in connection with the estimation procedure described above is that the formulation of the algorithm mainly in differences (rather than levels) of the variables does not result in any loss of information or efficiency. All the information in the sample is contained in the observable part x2s (0) of the initial state vector and the observations of the random vectors x¯ 1 , x¯ 2 , x¯ 3 , x¯ 4 , . . . , x¯ T , that is, the set of vectors in terms of which the estimation algorithm is formulated. Moreover, there is a one-to-one correspondence between the vectors [ x¯ 1 , x¯ 2 , x¯ 3 , x¯ 4 , . . . , x¯ T ] and [x¯ 1 , x¯ 2 , x¯ 3 , x¯ 4 , . . . , x¯ T ]. It makes no difference to the estimates, therefore, whether the likelihood function is derived from the probability density function (p.d.f.) of the

73


former vector or the p.d.f. of the latter vector. The estimates obtained by maximizing the two alternative forms of the likelihood function with respect to the unknown parameters are exactly the same. When the innovations in the continuous time model (2.1)–(2.4) are Brownian motion and the elements of z(t) are quadratic functions of t, the estimates obtained by maximizing the function L defined by (2.47) with respect to the parameter vector (θ, λ, α, β, yn ) are exact maximum likelihood estimates taking account of all information in the sample and all restrictions implied by the continuous time model. The VARMAX model (2.16)–(2.19) provides a very convenient basis, not only for the estimation of the parameters of the continuous time model, but also for the generation of optimal post-sample forecasts of the discrete observations. The procedure to be followed is similar to that described in Bergstrom [1989] for obtaining optimal forecasts from a higher order continuous time model without stochastic trends. Suppose that we wish to obtain forecasts of x¯ t for R postsample periods, that is, for t = T + 1, . . . , T + R. Extending the VARMAX model (2.16)–(2.19) up to the period T + R and separating the autoregressive and moving average parts, we have the system (2.52)–(2.59).

74


x¯ 1 = G10 + G11 y(0) + E 11 z¯1 + E 12 z¯2 + E 13 z¯3 + η1 ,

(2.52)

x¯ 2 = F21 x¯ 1 + G20 + G21 y(0) + E 21 z¯1 + E 22 z¯2 + E 23 z¯3 + η2 , (2.53) x¯ 3 = F32 x¯ 2 + F31 x¯ 1 + G30 + G31 y(0) + E 31 z¯1 + E 32 z¯2 + E 33 z¯3 + η3 ,

(2.54)

x¯ t = F1 x¯ t−1 + F2 x¯ t−2 + F0 + E 0 z¯t + E 1 z¯t−1 + E 2 z¯t−2 + ηt

(t = 4, . . . , T + R),

(2.55)

η1 = M11 ε1 ,

(2.56)

η2 = M21 ε1 + M22 ε2 ,

(2.57)

η3 = M31 ε1 + M32 ε2 + M33 ε3 ,

(2.58)

ηt = Mtt εt + Mt,t−1 εt−1 + Mt,t−2 εt−2 + Mt,t−3 εt−3 (t = 4, . . . , T + R).

(2.59)

ˆ λ, ˆ α, From this system, together with the estimator [θ, ˆ ˆ yˆ n ] of the parameter vector, we can obtain a set of optiβ, mal forecasts of the vectors x¯ T +1 , x¯ T +2 , . . . , x¯ T +R . They are optimal in the sense that they will be exact maximum likelihood estimates of the conditional expectations of these vectors, conditional on all information in the sample, when the

75


innovations in the continuous time model (2.1)–(2.4) are Brownian motion and the elements of z(t) are quadratic functions of t. To show this we, first, note that, under the latter conditions, the conditional expectations of the vectors εT +1 , εT +2 , . . . , εT +R , conditional on all information in the sample are zero vectors, that is, E (εT +r /x¯ 1 , x¯ 2 , . . . , x¯ T ) = 0

(r = 1, . . . , R).

(2.60)

Now let ηˆ T +1 , ηˆ T +2 , . . . , ηˆ T +R denote the exact Gaussian estimates of the conditional expectations of ηT +1 , ηT +2 , . . . , ηT +R conditional on all information in the sample. Then, using (2.59), we obtain ηˆ T +1 = Mˆ T +1,T εˆ T + Mˆ T +1,T −1 εˆ T −1 + Mˆ T +1,T −2 εˆ T −2 , (2.61) ηˆ T +2 = Mˆ T +2,T εˆ T + Mˆ T +2,T −1 εˆ T −1 ,

(2.62)

ηˆ T +3 = Mˆ T +3,T εˆ T ,

(2.63)

ηˆ T +r = 0

(r = 4, . . . , R),

(2.64)

where the matrices Mˆ T +1,T , Mˆ T +1,T −1 , Mˆ T +1,T −2 , Mˆ T +2,T , Mˆ T +2,T −1 and Mˆ T +3,T are obtained from the extension of the recursive system (2.41)–(2.44) up to t = T + 3 and the estimated values of the parameters, that is,

76


−1 ˆ 3 Mˆ t−3,t−3 , Mˆ t,t−3 =

(2.65)

−1 ˆ 2 − Mˆ t,t−3 Mˆ t−2,t−3 Mˆ t,t−2 = Mˆ t−2,t−2 ,

(2.66)

−1 ˆ 1 − Mˆ t,t−3 Mˆ t−1,t−3 Mˆ t−1,t−1 − Mˆ t,t−2 Mˆ t−1,t−2 , Mˆ t,t−1 = (2.67) ˆ 0 − Mˆ t,t−3 Mˆ t,t−3 Mˆ tt Mˆ tt = − Mˆ t,t−2 Mˆ t,t−2 − Mˆ t,t−1 Mˆ t,t−1

(2.68) (t = T + 1, . . . , T + 3).

The matrices Mˆ T −2,T −2 , Mˆ T −1,T −2 , Mˆ T −1,T −1 , Mˆ T,T −2 , Mˆ T,T −1 , and Mˆ T T in (2.65)–(2.68) will have been computed in the estimation procedure. Next, let xˆ¯ T +1 , xˆ¯ T +2 , . . . , xˆ¯ T +R denote the exact Gaussian estimators of the conditional expectations of the vectors x¯ T +1 , x¯ T +2 , . . . , x¯ T +R conditional on all information in the sample with the vectors z¯ T +1 , z¯T +2 , . . . , z¯T +R being treated as known. Then, using (2.55), we obtain

xˆ¯ T + 1 = Fˆ 1 x¯ T + Fˆ 2 x¯ T −1 + Fˆ 0 + Eˆ 0 ¯z T +1 + Eˆ 1 ¯z T + Eˆ 2 ¯z T −1 + ηˆ T +1 , (2.69) ˆ x¯ T +1 + Fˆ 2 x¯ T + Fˆ 0 + Eˆ 0 ¯z T +2

xˆ¯ T + 2 = Fˆ 1 + Eˆ 1 ¯z T +1 Eˆ 2 ¯z T + ηˆ T +2 , (2.70)

77


xˆ¯ T +3 = Fˆ 1 xˆ¯ T +2 + Fˆ 2 xˆ¯ T +1 + Fˆ 0 + Eˆ 0 ¯z T +3 + Eˆ 1 ¯z T +2 Eˆ 2 ¯z T +1 + ηˆ T +3 , (2.71)

xˆ¯ T +r = Fˆ 1 xˆ¯ T +r −1 + Fˆ 2 xˆ¯ T +r −2 + Fˆ 0 + Eˆ 0 ¯z T +r + Eˆ 1 ¯z T +r −1 + Eˆ 2 ¯z T +r −2 (r = 4, . . . , R), (2.72)

where Fˆ 0 , Fˆ 1 , Fˆ 2 , Eˆ 0 , Eˆ 1 and Eˆ 2 are obtained by substitutˆ of the parameter vector [θ, λ] into the ing the estimate [θˆ , λ] formulae for F0 , F1 , F2 , E 0 , E 1 and E 2 given in Appendix A. Finally, we obtain the forecasts of the vectors x¯ T +1 , x¯ T +2 , . . . , x¯ T +R from the accumulation formula xˆ¯ T + r = x¯ T +

r

xˆ¯ T +s

(r = 1, . . . , R).

(2.73)

s =1

The forecasts given by (2.73) are optimal post-sample forecasts of the discrete observations of the endogenous variables in the sense mentioned above. Summarizing the forecasting procedure, the main computational steps are as follows. 1. Compute the matrices Mˆ T +1,T , Mˆ T +1,T −1 , Mˆ T +1,T −2 , Mˆ T +2,T , Mˆ T +2,T −1 and Mˆ T +3,T from (2.65)–(2.68). 2. Compute the vectors ηˆ T +1 , ηˆ T +2 and ηˆ T +3 from (2.61)– (2.64).

78


3. Compute the vectors xˆ¯ T +1 , xˆ¯ T +2 , . . . , xˆ¯ T +R from (2.69)–(2.72). 4. Compute the vectors xˆ¯ T +1 , xˆ¯ T +2 , . . . , xˆ¯ T +R from (2.73). In the above forecasting procedure, the post-sample observations of the exogenous variables have been treated as known. In practice they would, of course, be unknown at the time when the post-sample forecasts of the endogenous variables are made, and they would have to be estimated independently. In Chapter 5 we will, however, set aside the last 2 years (8 quarters) observations in the sample for the purpose of testing the predictive performance of the model (the parameters being estimated from the earlier observations) so that the true values of the exogenous variables used in the computation of the forecasts will be known.

2.5 Conclusion In this chapter, we have formulated an open mixed-order continuous time dynamic model with mixed stock and flow variables and unobservable stochastic trends. We have then described a form of VARMAX model which, under certain assumptions, is satisfied, exactly, by the discrete observations generated by the continuous time model, and we have

79


shown how to compute the coefficient matrices of this model for any given values of the parameters of the continuous time model. Finally, we have shown how to use the VARMAX model to compute the exact Gaussian estimates of the parameters of the continuous time model and optimal forecasts of the post-sample discrete observations. When the innovations in the continuous time model are Brownian motion and the exogenous variables are quadratic functions of the time parameter, the computational procedures described produce exact maximum likelihood estimates of the parameters of the continuous time model, and the forecasts are exact maximum likelihood estimates of the conditional expectations of the post-sample discrete observations generated by the continuous time model. Moreover, both the parameter estimates and the forecasts can be expected to be very accurate under much more general conditions.

Appendix A: Formulae for the Coefficient Matrices of Exact Discrete Model In this Appendix we present formulae for the coefficient matrices of the discrete time model (2.12)–(2.15) derived in Bergstrom [1997]. These formulae make extensive use of

80


certain selection matrices, each of which transforms a vector of appropriate dimension into a vector of lower dimension, whose elements are a selection of the elements of the original vector. The selection matrices S1 , S2 , S3 , S4 , S5 and S are defined as follows. S1 = [I 0 0 0 S2 = 0 0 0 I S3 = 0 0

0 0 0], 0 I 0 , I 0 0 0 0 0 , 0 0 I

(2.74)

S4 = [I 0], S5 = [0 I ], S1 . S= S2

S1 is an n1 × (n1 + 2n2 ) matrix with an n1 × n1 identity matrix in columns 1 to n1 . S2 is an n2 × (n1 + 2n2 ) matrix with an ns2 × ns2 identity matrix in rows 1 to ns2 and columns n1 + n2 + 1 to n1 + n2 + ns2 identity matrix in rows ns2 + 1 to n2 and columns n1 + ns2 + 1 to n1 + n2 . S3 is an n2 × (n1 + 2n2 ) matrix with an ns2 × ns2 identity matrix in rows 1 f

f

to ns2 and columns n1 + 1 to n1 + ns2 and an n2 + n2 identity matrix in rows ns2 + 1 to n2 and columns n1 + n2 + ns2 + 1 to n1 + 2n2 . S4 is an ms × m matrix with an ms × ms identity

81


matrix in columns 1 to ms . S5 is an m f × m matrix with an m f × m f identity matrix in columns ms + 1 to m. We shall, also, use matrix A defined in Section 3, matrix W defined by −1 W = Se A S3 S3 − Se A S3 S3 e A S3 S2 e A S3 S2 ,

and the (n1 + 2n2 ) × 1 vector b, the (n1 + 2n2 ) × m matrix B and the (n1 + 2n2 ) × k matrix C defined by       B1 C1 b1 b = 0 , B = 0 , C = 0 . b2 B2 C2 The formulae for the coefficient matrices of the discrete time model (2.12)–(2.15) are as follows: F0 = (S + W)A−1 (e A − I )C λ, −1 , F1 = Se A S1 , Se A S2 + Se A S3 S3 e A S3 S2 e A S3 F2 = We A S , F21 = Se A S , F31 = Se A S3 S3 e A S , F32 = F21 , G10 = S{[A−2 (e A − I)− A−1 ]b+[A−3 (e A − I )− A−2 −0.5A−1 ]C λ}, G11 = SA−1 (e A − I ),

82


G20 = S A−1 (e A − I ) + e A S3 S3 [A−2 (e A − I ) − A−1 ]b + S A−2 (e A − I ) + 0.5A−1 (e A − I ) − A−1 + e A S3 S3 [A−3 (e A − I ) − A−2 − 0.5A−1 ] C λ, G21 = Se A S3 S3 A−1 (e A − I ), G30 = S I + e A S3 S3 A−1 (e A − I )

+ e A S3 S3 e A S3 S3 [A−2 (e A − I ) − A−1 ] b + S (A−2 + 1.5A−1 )(e A − I ) − A−1

+ e A S3 S3 [(A−2 + 0.5A−1 )(e A − I ) − A−1 ]

+ e A S3 S3 e A S3 S3 [A−3 (e A − I ) − A−2 − 0.5A−1 ] C λ,

G31 = Se A S3 S3 e A S3 S3 A−1 (e A − I ), E 0 = F1 SL10 + F2 SL20 − SL00 , E 1 = F1 SL11 + F2 SL21 − SL01 , E 2 = F1 SL12 + F2 SL22 − SL02 , E 11 = G11 P1 − SL22 , E 12 = G11 P2 − SL21 , E 13 = G11 P3 − SL20 , E 21 = G21 P1 + F21 SL22 − SL12 , E 22 = G21 P2 + F21 SL21 − SL11 , E 23 = G21 P3 + F21 SL20 − SL10 , E 31 = G31 P1 + F32 SL12 + F31 SL22 − SL02 , E 32 = G31 P2 + F32 SL11 + F31 SL21 − SL01 , E 33 = G31 P3 + F32 SL10 + F31 SL20 − SL00 ,

83


where

 11 3 S 4 = A−1 B  12  + A−2 B + A−3 B, 2 S5   1 S 4 = A−1 B  6  − 2A−2 B − 2A−3 B, 0   1 1 − S 4 = A−1 B  12  + A−2 B + A−3 B, 2 0 

L 00

L 01

L 02

 1 1 S 4 = A−1 B 12  + A−2 B + A−3 B, 2 0   7 S 4 = A−1 B  6  − 2A−3 B, S5   1 1 − S 4 = A−1 B  12  − A−2 B + A−3 B, 2 0 

L 10

L 11

L 12

−



L 20

L 21

 1 1 S 4 = A−1 B 12  − A−2 B + A−3 B, 2 0   1 S 4 = A−1 B  6  + 2A−2 B − 2A−3 B, 0 −

84


 11 3 S 4 L 22 = A−1 B  12  − A−2 B + A−3 B, 2 S5   7 S4   4 P1 = A−1 B   − 2A−2 B + A−3 B, 11 S5 6   −S4 P2 = A−1 B  7  + 3A−2 B − 2A−3 B, − S5 6  1  S4  4  P3 = A−1 B   − A−2 B + A−3 B. 1 S5 3 

Appendix B: Formulae for the Autocovariance Matrices In this Appendix we present formulae for the autocovariance matrices of the unobservable random vectors η1 , η2 , . . . , ηT in the discrete time model (2.12)–(2.15), that is for the matrices 0 , 1 , 2 , 3 and the various matrices i j defined below (2.12)–(2.15). It has been shown (Bergstrom [1997, Theorems 1 and 2]) that the vectors η1 , η2 , . . . , ηT in the discrete time model are related to the

85


white noise innovations in the continuous time model (2.1)– (2.4) by the following stochastic integral formulae:

t

ηt =

K 1 (t − r )ζ (dr ) +

t−1

+

t−1

K 2 (t − 1 − r )ζ (dr )

t−2 t−2

t−3

+

t

N1 (t − r )ζ3 (dr ) +

t−1

N2 (t − 1 − r )ζ3 (dr )

t−2

t−2

1

η1 = 1

1

K 11 (1 − r )ζ (dr ) +

N11 (1 − r )ζ3 (dr ),

0

2

K 21 (1 − r )ζ (dr ) +

0

K 22 (2 − r )ζ (dr )

1

1

+

1

K 31 (1 − r )ζ (dr ) +

0

2

N21 (1 − r )ζ3 (dr ) +

0

(t = 4, . . . , T )

0

η3 =

N4 (t − 3 − r )ζ3 (dr )

t−4

t−3

N3 (t − 2 − r )ζ3 (dr ) +

t−3

η2 =

K 4 (t − 3 − r )ζ (dr )

t−4

t−1

+

t−3

K 3 (t − 2 − r )ζ (dr ) +

N22 (2 − r )ζ3 (dr ),

1 2

K 32 (2 − r )ζ (dr )

1

3

+

2

+

1

K 33 (3 − r )ζ (dr ) +

N31 (1 − r )ζ3 (dr )

0 2

N32 (2 − r )ζ3 (dr ) +

1

2

86

3

N33 (3 − r )ζ3 (dr ).


Formulae expressing the weighting functions K 1 , K 2 , K 3 , K 4 , K 11 , . . . , K 33 , N1 , N2 , N3 , N4 , N11 , . . . , N33 in terms of the parameters of the continuous time model are presented in Bergstrom [1997 Theorems 1 and 2], but will not be reproduced here. Instead, we present formulae expressing 0 , 1 , 2 , 3 and the various matrices i j as sums of integrals involving combinations of these weighting functions, followed by formulae expressing these integrals in terms of the parameters of the continuous time model. The latter formulae, which are previously unpublished, greatly, facilitate the computation of the VARMAX representation of the discrete time model and the value of the likelihood function for a given set of values of the parameters of the continu ous time model. The formulae, which involve the matrix defined by (2.45), are as follows:

1

0 = 0

K 1 (r ) K 1 (r )dr +

1

+ 0

+

1

0 1

+ 0

1 0

K 3 (r ) K 3 (r )dr + N1 (r )33 N1 (r )dr N3 (r )33 N3 (r )dr

K 2 (r ) K 2 (r )dr

1

0

+

K 4 (r ) K 4 (r )dr 1

0

+

87

0

1

N2 (r )33 N2 (r )dr N4 (r )33 N4 (r )dr ,


where 1 0

K 1 (r ) K 1 (r )dr

= SA−1 {SA−1 } − {SA−2 (e A − I )}{SA−1 } − {SA−1 } 1 −2 A −1 e rA e rA dr {SA−1 } , ×{SA (e − I )} + {SA } 0

1 0

K 2 (r ) K 2 (r )dr

= {SA−1 (I + e A) − WA−1 }{SA−1 (I + e A) − WA−1 } + {(W − 2S)A−2 (e A − I )}{SA−1 (I + e A) − WA−1 } + {SA−1 (I + e A) − WA−1 }{(W − 2S)A−2 (e A − I )} 1 −1 + {(W − 2S)A } e rA e rA dr {(W − 2S)A−1 } , 0

1 0

K 3 (r ) K 3 (r )dr

= {WA−1 (I + e A) − SA−1 e A}{WA−1 (I + e A) − SA−1 e A} + {(S − 2W)A−2 (e A − I )}{WA−1 (I + e A) − SA−1 e A} + {WA−1 (I + e A) − SA−1 e A}{(S − 2W)A−2 (e A − I )} 1 + {(S − 2W)A−1 } e rA e rA dr {(S − 2W)A−1 } , 0

0 1

K 4 (r ) K 4 (r )dr

= {WA−1 e A}{WA−1 e A} − {WA−2 (e A − I )}

88


×{WA−1 e A} − {WA−1 e A} ×{WA−2 (e A − I )} + {WA−1 }

1

e rA e rA dr {WA−1 } ,

0

1 0

N1 (r )33 N1 (r )dr = {SA−2 }C 33 C {SA−2 } 1 + {SA−1 }C 33 C {SA−2 } 2 1 + {SA−2 }C 33 C {SA−1 } 2 1 + {SA−1 }C 33 C {SA−1 } 3 − {SA−3 (e A − I )}C 33 C {SA−2 } − {SA−2 }C 33 C {SA−3 (e A − I )} 1 −2 rA − SA r e dr C 33 C {SA−1 } 0

0

1 − {SA−1 }C 33 C {SA−2 r e rA dr } 0 1 + {SA−2 } e rA C 33 C e rA dr {SA−2 } , 1

0

N2 (r )33 N2 (r )dr

= {SA−2 (I + e A) − SA−1 − WA−2 } × C 33 C {SA−2 (I + e A) − SA−1 − WA−2 } 1 + {SA−1 (I + e A) − WA−1 } 2 × C 33 C {SA−2 (I + e A) − SA−1 − WA−2 }

89


1 + {SA−2 (I + e A) − SA−1 − WA−2 } 2 × C 33 C {SA−1 (I + e A) − WA−1 } 1 + {SA−1 (I + e A) − WA−1 } 3 × C 33 C {SA−1 (I + e A) − WA−1 } + {(W − 2S)A−3 (e A − I )} × C 33 C {SA−2 (I + e A) − SA−1 − WA−2 } + {SA−2 (I + e A) − SA−1 − WA−2 } + C 33 C {(W − 2S)A−3 (e A − I )} 1 + (W − 2S)A−2 r e rA dr 0

−1

× C 33 C {SA (I + e A) − WA−1 } + {SA−1 (I + e A) − WA−1 } −2 × C 33 C (W − 2S)A −2

rA

r e dr

0

1

+ {(W − 2S)A }

1

e rA C 33 C e rA dr {(W − 2S)A−2 } ,

0 1 0

N3 (r )33 N3 (r )dr

= {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 }C 33 C {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } 1 + {WA−1 (I + e A) − SA−1 e A} 2

90


× C 33 C {S(A−1 − A−2 )e A + WA−2 (I + e A) 1 − WA−1 } + {S(A−1 − A−2 )e A 2 −2 + WA (I + e A) − WA−1 } × C 33 C {WA−1 (I + e A) − SA−1 e A} 1 + {WA−1 (I + e A) − SA−1 e A} 3 × C 33 C {WA−1 (I + e A) − SA−1 e A} + {(S − 2W)A−3 (e A − I )}C 33 C {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } + {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } × C 33 C {(S − 2W)A−3 (e A − I )} 1 −2 rA + (S − 2W)A r e dr 0

× C 33 C {WA−1 (I + e A) − SA−1 e A} + {WA−1 (I + e A) − SA−1 e A} 1 r e rA dr × C 33 C (S − 2W)A−2 + {(S − 2W)A−2 }

0 1

e rA C 33 C e rA dr {(S − 2W)A−2 } ,

0

1 0

N4 (r )33 N4 (r )dr

= {W(A−1 − A−2 )e A}C 33 C {W(A−1 − A−2 )e A} 1 − {WA−1 e A}C 33 C {W(A−1 − A−2 )e A} 2

91


1 − {W(A−1 − A−2 )e A}C 33 C {WA−1 e A} 2 1 + {WA−1 e A}C 33 C {WA−1 e A} 3 + {WA−3 (e A − I )}C 33 C {W(A−1 − A−2 )e A} + {W(A−1 − A−2 )e A}C 33 C {WA−3 (e A − I )} 1 − WA−2 r e rA dr C 33 C {WA−1 e A} 0 −1 A

− {WA e }C 33 C + {WA−2 }

−2

1 rA

WA

r e dr 0

1

e rA C 33 C e rA dr {WA−2 } .

0

1

1 =

K 2 (r ) K 1 (r )dr +

0

+

1

0

1

+ 0

1

K 4 (r ) K 3 (r )dr + N3 (r )33 N2 (r )dr

K 3 (r ) K 2 (r )dr

0

1

0

N2 (r )33 N1 (r )dr 1

+ 0

N4 (r )33 N3 (r )dr,

where 1 K 2 (r ) K 1 (r )dr = −{SA−1 (I + e A) − WA−1 }{SA−1 } 0

− {(W − 2S)A−2 (e A − I )}{SA−1 } + {SA−1 (I + e A)−WA−1 }{SA−2 (e A − I )} 1 + {(W − 2S)A−1 } e rA e rA dr {SA−1 } , 0

92


1

0

K 3 (r ) K 2 (r )dr

= {WA−1 (I + e A) − SA−1 e A}{SA−1 (I + e A) − WA−1 } + {(S − 2W)A−2 (e A − I )}{SA−1 (I + e A) − WA−1 } + {WA−1 (I + e A) − SA−1 e A}{(W − 2S)A−2 (e A − I )} 1 + {(W − 2S)A−1 } e rA e rA dr {(W − 2S)A−1 } , 0

0

1

K 4 (r ) K 3 (r )dr

= − {WA−1 e A}{WA−1 (I + e A) − SA−1 e A} + {WA−2 (e A − I )}{WA−1 (I + e A) − SA−1 e A} − {WA−1 e A}{(S − 2W)A−2 (e A − I )} + {WA−1 } 1 × e rA e rA dr {(S − 2W)A−1 } , 0

0

1

N2 (r )33 N1 (r )dr

= −{SA−2 (I + e A) − SA−1 − WA−2 }C 33 C {SA−2 } 1 − {SA−1 (I + e A) − WA−1 }C 33 C {SA−2 } 2 1 − {SA−2 (I + e A) − SA−1 − WA−2 }C 33 C {SA−1 } 2 1 − {SA−1 (I + e A) − WA−1 }C 33 C {SA−1 } 3 − {(W − 2S)A−3 (e A − I )}C 33 C {SA−2 } + {SA−2 (I + e A) −SA−1 − WA−2 }C 33 C {SA−3 (e A − I )}

93


− (W − 2S)A−2

1

r e rA dr C 33 C {SA−1 }

0 −1

−1

+ {SA (I + e ) − WA }C 33 C A

+ {(W − 2S)A−2 }

SA

−2

1 rA

r e dr 0

1

r e rA C 33 C r e rA dr {SA−2 } ,

0

1 0

N3 (r )33 N2 (r )dr

= {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } × C 33 C {SA−2 (I + e A) − SA−1 − WA−2 } 1 + {WA−1 (I + e A) − SA−1 e A} 2 × C 33 C {SA−2 (I + e A) − SA−1 − WA−2 } 1 + {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } 2 × C 33 C {SA−1 (I + e A) − WA−1 } 1 + {WA−1 (I + e A) − SA−1 e A} 3 × C 33 C {SA−1 (I + e A) − WA−1 } + {(S − 2W)A−3 (e A − I )} × C 33 C {SA−2 (I + e A) − SA−1 − WA−2 } + {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } × C 33 C {(W − 2S)A−3 (e A − I )} 1 −2 rA + (S − 2W)A r e dr 0

94


× C 33 C {SA−1 (I + e A) − WA−1 } + {WA−1 (I + e A) − SA−1 e A} 1 −2 rA × C 33 C (W − 2S)A r e dr + {(S − 2W)A−2 }

0

1

r e rA 0

× C 33 C e rA dr {(W − 2S)A−2 } , 0

1

N4 (r )33 N3 (r )dr

= {W(A−1 − A−2 )e A}C 33 C {S(A−1 − A−2 ) 1 × e A + WA−2 (I + e A) − WA−1 } − {WA−1 e A} 2 × C 33 C {S(A−1 − A−2 )e A +WA−2 (I + e A) − WA−1 } 1 + {W(A−1 − A−2 )e A}C 33 C {WA−1 (I + e A)−SA−1 e A} 2 1 − {WA−1 e A}C 33 C {WA−1 (I + e A) − SA−1 e A} 3 + {WA−3 (e A − I )}C 33 C {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } + {W(A−1 − A−2 )e A} × C 33 C {(S − 2W)A−3 (e A − I )} 1 + WA−2 r e rA dr C 33 C {WA−1 (I + e A) − SA−1 e A} 0 −1 A

− {WA e }C 33 C + {WA−2 }

−2

(S − 2W)A

1 rA

r e dr 0

1

e rA C 33 C e rA dr {(S − 2W)A−2 } .

0

95


1

1

K 4 (r ) K 2 (r )dr 0 0 1 1 + N3 (r )33 N1 (r )dr + N4 (r )33 N2 (r )dr,

2 =

K 3 (r ) K 1 (r )dr

+

0

0

where

1 0

K 3 (r ) K 1 (r )dr

= − {WA−1 (I + e A) − SA−1 e A}{SA−1 } − {(S − 2W)A−2 (e A − I )}{SA−1 } + {WA−1 (I + e A) − SA−1 e A}{SA−2 (e A − I )} 1 −1 + {(S − 2W)A } e rA e rA dr {SA−1 } , 0

0

1

K 4 (r ) K 2 (r )dr

= −{WA−1 e A}{SA−1 (I + e A) − WA−1 } + {WA−2 (e A − I )}{SA−1 (I + e A) − WA−1 } − {WA−1 e A}{(W − 2S)A−2 (e A − I )} 1 + {WA−1 } e rA e rA dr {(W − 2S)A−1 } , 0

1 0

N3 (r )33 N1 (r )dr

= −{S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } 1 × C 33 C {SA−2 } − {WA−1 (I + e A) − SA−1 e A} 2

96


1 {S(A−1 − A−2 )e A 2 + WA−2 (I + e A) − WA−1 }C 33 C {SA−1 } 1 − {WA−1 (I + e A) − SA−1 e A}C 33 C {SA−1 } 3 − {(S − 2W)A−3 (e A − I )}C 33 C {SA−2 } × C 33 C {SA−2 } −

+ {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } 1 r e rA dr × C 33 C {SA−3 (e A − I )} − (S − 2W)A−2

−1

−1

0 −1 A

× C 33 C {SA } + {WA (I + e ) − SA e } 1 r e rA dr + {(S − 2W)A−2 } × C 33 C SA−2 A

0

1

r e rA C 33 C e rA dr {SA−2 } ,

0

1 0

N4 (r )33 N2 (r )dr

= {W(A−1 − A−2 )e A}C 33 C {SA−2 (I + e A) − SA−1 −WA−2 } 1 − {WA−1 e A}C 33 C {SA−2 (I + e A) − SA−1 − WA−2 } 2 1 + {W(A−1 − A−2 )e A}C 33 C {SA−1 (I + e A) − WA−1 } 2 1 − {WA−1 e A}C 33 C {SA−1 (I + e A) − WA−1 } 3 + {WA−3 (e A − I )}C 33 C {SA−2 (I + e A) − SA−1 − WA−2 } + {W(A−1 − A−2 )e A}C 33 C {(W − 2S)A−3 (e A − I )} 1 −2 rA + WA r e dr C 33 C {SA−1 (I + e A) − WA−1 } 0

97


−1 A

− {WA e }C 33 C

−2

(W − 2S)A

1 rA

r e dr 0

+ {WA−2 }

1

e rA C 33 C e rA dr {(W − 2S)A−2 } .

0

1

3 = 0

where

1

0

K 4 (r ) K 1 (r )dr +

1 0

N4 (r )33 N1 (r )dr,

K 4 (r ) K 1 (r )dr

= {WA−1 e A}{SA−1 } − {WA−2 (e A − I )}{SA−1 } − {WA−1 e A}{SA−2 (e A − I )} + {WA−1 } 1 e rA e rA dr {SA−1 } , × 0

0 1

N4 (r )33 N1 (r )dr

= − {W(A−1 − A−2 )e A}C 33 C {SA−2 } 1 + {WA−1 e A}C 33 C {SA−2 } 2 1 − {W(A−1 − A−2 )e A}C 33 C {SA−1 } 2 1 + {WA−1 e A}C 33 C {SA−1 } 3 − {WA−3 (e A − I )}C 33 C {SA−2 } + {W(A−1 − A−2 )e A}C 33 C {SA−3 (e A − I )}

98


− WA−2

1

r e rA dr C 33 C {SA−1 }

0 −1 A

− {WA e }C 33 C −2

SA

−2

1 rA

r e dr 0

1

+ {WA }

e rA C 33 C e rA dr {SA−2 } .

0

11 = 0

1

K 1 (r ) K 1 (r )dr +

0

1

N1 (r )33 N1 (r )dr,

where 1 1 0 K 1 (r ) K 1 (r )dr and 0 N1 (r )33 N1 (r )dr are evaluated as in the formulae for 0 . 1 1 (r )dr + 0 N21 (r )33 N11 (r )dr 21 = 0 K 21 (r ) K 11 where

1 0

K 21 (r ) K 11 (r )dr = − SA−1 e A − Se A S3 S3 A−1 {SA−1 }

− {(Se A S3 S3 − S)A−2 (e A − I )}{SA−1 } + SA−1 e A − Se A S3 S3 A−1 {SA−2 (e A − I )} 1 rA rA + (Se A S3 S3 − S)A−1 e e dr {SA−1 } , 0

1 0

N21 (r )33 N11 (r )dr

= − SA−2 e A − SA−1 − Se A S3 S3 A−2 C 33 C {SA−2 }

99


1 −1 A SA e − Se A S3 S3 A−1 C 33 C {SA−2 } 2 1 −2 A SA e − SA−1 − Se A S3 S3 A−2 C 33 C {SA−1 } − 2 1 −1 A − SA e − Se A S3 S3 A−1 C 33 C {SA−1 } 3 − (Se A S3 S3 − S)A−3 (e A − I ) C 33 C {SA−2 } + SA−2 e A − SA−1 − Se A S3 S3 A−2 C 33 C {SA−3 (e A − I )} 1 − (Se A S3 S3 − S)A−2 r e rA dr C 33 C {SA−1 } −

0

+ SA−1 e A − Se A S3 S3 A−1 C 33 C SA−2

+

Se A S3 S3 − S A−2

22 =

1

r e rA dr

0

1


0

1

K 22 (r ) K 22 (r )dr 1 1 + N21 (r )33 N21 (r )dr + N22 (r )33 N22 (r )dr, 0

K 21 (r ) K 21 (r )dr

1

+

0

0

0

where 0

1

K 21 (r ) K 21 (r )dr

= SA−1 e A − Se A S3 S3 A−1 SA−1 e A − Se A S3 S3 A−1 + Se A S3 S3 − S A−2 (e A − I ) SA−1 e A − Se A S3 S3 A−1

100


+ SA−1 e A − Se A S3 S3 A−1 Se A S3 S3 − S A−2 (e A − I ) 1 rA rA A + Se A S3 S3 − S A−1 e e dr Se S3 S3 − S A−1 , 0

1 0

K 22 (r ) K 22 (r )dr =

0

1

K 1 (r ) K 1 (r )dr ,

the latter integral being evaluated as in the formula for 0 , 1 N21 (r )33 N21 (r )dr 0 = SA−2 e A − SA−1 − Se A S3 S3 A−2 × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 1 −1 A + SA e − Se A S3 S3 A−1 2 × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 1 −2 A + SA e − SA−1 − Se A S3 S3 A−2 2 × C 33 C SA−1 e A − Se A S3 S3 A−1 1 −1 A SA e − Se A S3 S3 A−1 + 3 × C 33 C SA−1 e A − Se A S3 S3 A−1 + Se A S3 S3 − S A−3 (e A − I ) × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 + SA−2 e A − SA−1 − Se A S3 S3 A−2 × C 33 C Se A S3 S3 − S A−3 (e A − I )

101


+

Se A S3 S3 − S A−2

1

r e rA dr 0

× C 33 C SA−1 e A − Se A S3 S3 A−1 + SA−1 e A − Se A S3 S3 A−1 1 r e rA dr × C 33 C Se A S3 S3 − S A−2 0

1 rA + Se A S3 S3 − S A−2 e 0 × C 33 C e rA dr Se A S3 S3 − S A−2 ,

1 0

N22 (r )33 N22 (r )dr =

1

0

N1 (r )33 N1 (r )dr ,

the latter integral being evaluated as in the formula for 0 . 31 = 0

1

K 31 (r ) K 11 (r )dr +

1 0

N31 (r )33 N11 (r )dr ,

where

1 0

K 31 (r ) K 11 (r )dr A −1 A = − Se S3 S3 A e − e A S3 S3 A−1 {SA−1 } − Se A S3 S3 e A S3 S3 − I A−2 (e A − I ) {SA−1 } + Se A S3 S3 A−1 e A − e A S3 S3 A−1 {SA−2 (e A − I )} 1 rA rA + Se A S3 S3 e A S3 S3 − I A−1 e e dr {SA−1 } , 0

102


0

1

N31 (r )33 N11 (r )dr

= − SA−1 (e A − I ) + Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2 1 A −1 A Se S3 S3 A e − e A S3 S3 A−1 2 1 −1 A × C 33 C {SA−2 } − SA (e − I ) 2 + Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2 C 33 C {SA−1 } × C 33 C {SA−2 } −

1 A −1 A Se S3 S3 A e − e A S3 S3 A−1 C 33 C {SA−1 } 3 − Se A S3 S3 e A S3 S3 − I A−3 (e A − I ) C 33 C {SA−2 } + SA−1 (e A − I ) + Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2 × C 33 C {SA−3 (e A − I )} − Se A S3 S3 (e A S3 S3 − I )A−2 −

r e rA dr C 33 C {SA−1} + Se A S3 S3 A−1 e A −e A S3 S3 A−1

1

× 0

1 −2 rA × C 33 C SA r e dr + Se A S3 S3 e A S3 S3 − I A−2

0

1

×


0

1

32 = 0

K 31 (r ) K 21 (r )dr +

1

+ 0

1

0

N31 (r )33 N21 (r )dr +

103

K 32 (r ) K 22 (r )dr

0

1

N32 (r )33 N22 (r )dr,


where

1 0

K 31 (r ) K 21 (r )dr = Se A S3 S3 A−1 e A −e A S3 S3 A−1 SA−1 e A − Se A S3 S3 A−1 + Se AS3 S3 e AS3 S3 −I A−2(e A−I ) SA−1e A−Se A S3 S3 A−1 + Se A S3 S3 A−1e A−e A S3 S3 A−1 Se A S3 S3−S A−2 (e A−I ) 1 + Se A S3 S3 e AS3 S3 − I A−1 e rA e rA dr Se AS3 S3−S A−1 ,

0

0

1

K 32 (r )33 K 22 (r )dr =

0

1

K 21 (r ) K 11 (r )dr ,

the latter integral being evaluated as in the formula for 21 , 1 N31 (r )33 N21 (r )dr 0 = SA−1 (e A − I ) + Se A S3 S3 (A−2 e A − A−1 − e A S3 S3 A−2 ) × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 1 A −1 A + Se S3 S3 A e − e A S3 S3 A−1 2 × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 1 −1 A + SA (e − I )+ Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2 2 × C 33 C SA−1 e A − Se A S3 S3 A−1 1 A −1 A + Se S3 S3 A e − e A S3 S3 A−1 3 × C 33 C SA−1 e A − Se A S3 S3 A−1 A A −3 A + Se S3 S3 e S3 S3 − I A (e − I )

104


× C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 + SA−1 (e A − I ) + Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2 × C 33 C Se A S3 S3 − S A−3 (e A − I ) 1 A −2 A rA + Se S3 S3 e S3 S3 − I A r e dr 0 × C 33 C SA−1 e A − Se A S3 S3 A−1 + Se A S3 S3 A−1 e A − e A S3 S3 A−1 A −2 1 rA Se S3 S3 − S A r e dr × C 33 C 0

1 rA + Se A S3 S3 e A S3 S3 − I A−2 e 0 × C 33 C e rA dr Se A S3 S3 − S A−2 , 1 1 N32 (r )33 N22 (r )dr = N21 (r )33 N11 (r )dr ,

0

0

the latter integral being evaluated as in the formula for 21 . 1 1 33 = K 31 (r ) K 31 (r )dr + K 32 (r ) K 32 (r )dr , 0 0 1 1 + K 33 (r ) K 33 (r )dr + N31 (r )33 N31 (r )dr 0 0 1 1 + N32 (r )33 N32 (r )dr + N33 (r )33 N33 (r )dr, 0

where

0

1 0

K 31 (r ) K 31 (r )dr A −1 A = Se S3 S3 A e − e A S3 S3 A−1

105


× Se A S3 S3 A−1 e A − e A S3 S3 A−1 + Se A S3 S3 e A S3 S3 − I A−2 (e A − I ) × Se A S3 S3 A−1 e A − e A S3 S3 A−1 + Se A S3 S3 A−1 e A − e A S3 S3 A−1 × Se A S3 S3 e A S3 S3 − I A−2 (e A − I ) 1 rA + Se A S3 S3 e A S3 S3 − I A−1 e 0 ×e rA dr Se A S3 S3 e A S3 S3 − I A−1 ,

1 0

K 32 (r ) K 32 (r )dr =

1 0

K 21 (r ) K 21 (r )dr ,

the latter integral being evaluated as in the formula for 22 , 1 1 K 33 (r ) K 33 (r )dr = K 1 (r ) K 1 (r )dr , 0

0

the latter integral being evaluated as in the formula for 0 , 1 N31 (r )33 N31 (r )dr 0 = SA−1 (e A − I ) + Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2 × C 33 C SA−1 (e A − I ) + Se A S3 S3 × A−2 e A − A−1 − e A S3 S3 A−2 1 A −1 A + Se S3 S3 A e − e A S3 S3 A−1 2 × C 33 C SA−1 (e A − I ) + Se A S3 S3 × A−2 e A − A−1 − e A S3 S3 A−2

106


1 −1 A SA (e − I )+ Se A S3 S3 A−2 e A − A−1 −e A S3 S3 A−2 2 × C 33 C Se A S3 S3 A−1 e A − e A S3 S3 A−1 1 A −1 A + Se S3 S3 A e − e A S3 S3 A−1 3 × C 33 C Se A S3 S3 A−1 e A − e A S3 S3 A−1 + Se A S3 S3 e A S3 S3 − I A−3 (e A − I ) × C 33 C SA−1 (e A − I )+ Se AS3 S3 A−2 e A−A−1−e A S3 S3 A−2 + SA−1 (e A − I ) + Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2 × C 33 C Se A S3 S3 e A S3 S3 − I A−3 (e A − I ) 1 + Se A S3 S3 e A S3 S3 − I A−2 r e rA dr +

0

× C 33 C Se A S3 S3 A−1 e A − e A S3 S3 A−1 + Se A S3 S3 A−1 e A − e A S3 S3 A−1 1 r e rA dr × C 33 C Se A S3 S3 e A S3 S3 − I A−2 0 A A −2 + Se S3 S3 e S3 S3 − I A 1 × e rA C 33 C e rA dr Se A S3 S3 e A S3 S3 − I A−2 , 0

1 0

N32 (r )33 N32 (r )dr =

1

0

N21 (r )33 N21 (r )dr ,

the latter integral being evaluated as in the formula for 22 , 1 1 N33 (r )33 N33 (r )dr = N1 (r )33 N1 (r )dr , 0

0

107


the latter integral being evaluated as in the formula for 0 . 41 = 3 . 42 =

1

K 4 (r ) K 21 (r )dr +

1

K 3 (r ) K 22 (r )dr 0 0 1 1 + N4 (r )33 N21 (r )dr + N3 (r )33 N22 (r )dr, 0

where

0

1 0

K 4 (r ) K 21 (r )dr = −{WA−1 e A} SA−1 e A − Se A S3 S3 A−1 + {WA−2 (e A − I )} SA−1 e A − Se A S3 S3 A−1 − {WA−1 e A} Se A S3 S3 − S A−2 (e A − I ) 1 −1 + {WA } e rA e rA dr Se A S3 S3 − S A−1 , 0

0

1

K 3 (r ) K 22 (r )dr =

1 0

K 3 (r ) K 1 (r )dr ,

the latter integral being evaluated as in the formula for 2 , 1 N4 (r )33 N21 (r )dr 0 = {W(A−1 − A−2 )e A}C 33 C SA−2 e A −SA−1 − Se A S3 S3 A−2 1 − {WA−1 e A}C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 2 1 + {W(A−1 − A−2 )e A}C 33 C SA−1 e A − Se A S3 S3 A−1 2

108


1 − {WA−1 e A}C 33 C SA−1 e A − Se A S3 S3 A−1 3 + {WA−3 (e A − I )}C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 + {W(A−1 − A−2 )e A}C 33 C Se A S3 S3 − S A−3 (e A − I ) 1 + WA−2 r e rA dr C 33 C SA−1 e A − Se A S3 S3 A−1 0

−1 A

− {WA e }C 33 C + {WA−2 }

1

Se

A

S3 S3

e rA C 33 C e rA dr

0

1

N3 (r )33 N22 (r )dr

1

=

1 rA

r e dr 0

−S A

0

−2

0

Se A S3 S3 − S A−2 , N3 (r )33 N1 (r )dr ,

the latter integral being evaluated as in the formula for 2 . 1 1 43 = K 4 (r ) K 31 (r )dr + K 3 (r ) K 32 (r )dr 0 0 1 1 + K 2 (r ) K 33 (r )dr + N4 (r )33 N31 (r )dr 0 0 1 1 + N3 (r )33 N32 (r )dr + N2 (r )33 N33 (r )dr, 0

0

where

1

K 4 (r ) K 31 (r )dr 0 = −{WA−1 e A} Se A S3 S3 A−1 e A − e A S3 S3 A−1 + {WA−2 (e A − I )} Se A S3 S3 A−1 e A − e A S3 S3 A−1

109


− {WA−1 e A} Se A S3 S3 e A S3 S3 A−1 − I A−2 (e A − I ) 1 −1 + {WA } e rA e rA dr Se A S3 S3 e A S3 S3 A−1 − I A−1 , 0

0

1

K 3 (r ) K 32 (r )dr

= {WA−1 (I + e A) − SA−1 e A} SA−1 e A − Se A S3 S3 A−1 + {(S − 2W)A−2 (e A − I )} SA−1 e A − Se A S3 S3 A−1 + {WA−1 (I + e A)−SA−1 e A} Se A S3 S3 − S A−2 (e A − I ) 1 + {(S − 2W)A−1 } e rA e rA dr Se A S3 S3 − S A−1 , 0

0

1

K 2 (r ) K 33 (r )dr =

1 0

K 2 (r ) K 1 (r )dr ,

the latter integral being evaluated as in the formula for 1 , 1 N4 (r )33 N31 (r )dr 0 = {W(A−1 − A−2 )e A}C 33 C SA−1 (e A − I ) + Se A S3 S3 1 × A−2 e A − A−1 − e A S3 S3 A−2 − {WA−1 e A} 2 ×C 33 C SA−1 (e A−I)+ Se A S3 S3 A−2 e A−A−1 −e A S3 S3 A−2 1 + {W(A−1 − A−2)e A}C 33 C Se AS3 S3 A−1e A −e A S3 S3 A−1 2 1 − {WA−1 e A}C 33 C Se A S3 S3 A−1 e A − e A S3 S3 A−1 3 + WA−3 (e A − I )}C 33 C {SA−1 (e A − I ) + Se A S3 S3 A−2 e A − A−1 − e A S3 S3 A−2

110


+ {W(A−1 − A−2)e A}C 33 C Se A S3 S3 e AS3 S3 − I A−3 (e A − I ) 1 −2 rA + WA r e dr C 33 C Se A S3 S3 A−1 e A−e A S3 S3 A−1 0

1 − {WA−1 e A}C 33 C Se A S3 S3 e A S3 S3 − I A−2 r e rA dr + {WA−2 }

0

0

0

1

e rA C 33 C e rA dr Se A S3 S3 e A S3 S3 − I A−2

1

N3 (r )33 N32 (r )dr

= {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 1 + {WA−1 (I + e A) − SA−1 e A} 2 × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 1 + {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } 2 × C 33 C SA−1 e A − Se A S3 S3 A−1 1 + {WA−1 (I + e A) − SA−1 e A} 3 × C 33 C SA−1 e A − Se A S3 S3 A−1 + {(S−2W)A−3 (I +e A)} × C 33 C SA−2 e A − SA−1 − Se A S3 S3 A−2 + {S(A−1 − A−2 )e A + WA−2 (I + e A) − WA−1 } × C 33 C Se A S3 S3 A−1 − S A−3 (e A − I ) 1 −2 rA + (S −2W)A r e dr C 33 C SA−1 e A − Se A S3 S3 A−1 0

111


−1

−1 A

+ {WA (I + e ) − SA e }C 33 C

A

1

×

rA

r e dr

Se A S3 S3 − S A−2

+ {(S − 2W)A−2 }

0

1

×

r e rA C 33 C e rA dr

0

1

Se A S3 S3 − S A−2 ,

N2 (r )33 N33 (r )dr =

0

1

N2 (r )33 N1 (r )dr ,

0

the latter integral being evaluated as in the formula for 1 , 52 = 3 .

1

1

K 3 (r ) K 33 (r )dr 1 1 + N4 (r )33 N32 (r )dr + N3 (r )33 N33 (r )dr,

53 =

0

K 4 (r ) K 32 (r )dr

+

0

0

where

1

0

0

K 4 (r ) K 32 (r )dr =

1 0

K 4 (r ) K 21 (r )dr ,

the latter integral being evaluated as in the formula for 42 , 1 1 K 3 (r ) K 33 (r )dr = K 3 (r ) K 1 (r )dr , 0

0


0

112



0

the latter integral being evaluated as in the formula for 2 , 63 = 3 .

The integrals

1 0

e rA e rA dr and

1 0

e rA C 33 C e rA dr in the

above formulae can be evaluated by the method of Van Loan 1 [1978] and the integral 0 r e rA dr from the formula 1 r e rA dr = A−1 e A − A−2 (e A − I ), 0

which is, easily, obtained using the method of integration of parts.

113

CHAPTER THREE

Model Specification

3.1 Introduction In this chapter we discuss the new mixed-order continuous time macroeconometric model of the UK economy with stochastic trends. The model is based on the Bergstrom, Nowman and Wymer [1992] second-order continuous time macroeconometric model. The model is the first continuous time macroeconometric model incorporating stochastic trends, and its development represents a major step forward in general continuous time macroeconometric modelling over the last 30 years. The detailed specification of the structural equations and economic theory of previous continuous time macroeconomic models of various economies of the world is presented, for example, in Bergstrom and Wymer [1976], Knight and Wymer [1978], Bergstrom, Nowman

114

Model Specification

and Wymer [1992], Jonson, Moses and Wymer [1977], Knight and Mathieson [1979], Gandolfo and Padoan, [1982, 1984, 1987a, 1990], Kirkpatrick [1987], Bailey, Hall and Phillips [1987], Donaghy [1993], Sjo¨ o¨ [1993], Nieuwenhuis [1995] and Stavrev [2001], amongst others. In this chapter we provide a rigorous analysis and discussion of each of the macroeconomic structural relationships in the model and a formal derivation of them as the solution of a dynamic optimization problem, which takes account of adjustment costs. We also provide a discussion of the underlying assumptions concerning firm behaviour and market structure. This chapter is organised as follows. Section 3.2 present the complete set of equations of the model where we shall, also, discuss their general properties. Sections 3.3 and 3.4 present the private consumption and residential fixed capital equations as they have similar specifications. Sections 3.5 to 3.8 discuss the equations relating to the production sector of the economy, in particular, employment, private non-residential fixed capital, output and the price level. Section 3.9 presents the wage rate adjustment equation and Section 3.10 the interest rate equation. The last group of equations for international flows are presented in Sections 3.11 to 3.16 to explain imports, non-oil exports, transfers abroad, real profits, interest and dividends from

115


abroad, cumulative net real investment abroad and the exchange rate. The stock identity is presented in Section 3.17 and conclusions in Section 3.18. This chapter, also, contains two technical Appendices presenting the formal derivation of the adjustment equations and the distributed lag relations.

3.2 Equations and General Properties of the Model The model, in its deterministic form, is set out below. It is a system of 18 mixed first- and second-order nonlinear differential equations with 63 structural parameters, including a vector β of 33 long-run parameters, a vector γ of 27 speed of adjustment parameters, and a vector λ of 3 trend or drift parameters. The variables and equations of the model, in its deterministic form, are as follows.

Endogenous Variables C = real private consumption E n = real non-oil exports F = real current transfers abroad I = volume of imports K h = residential fixed capital K = private non-residential fixed capital

116

Model Specification

K a = cumulative net real investment abroad (excluding change in official reserves) L = employment P = real profits, interest and dividends from abroad p = price level Q = real net output q = exchange rate (price of sterling in foreign currency) r = interest rate S = stocks w = wage rate

Exogenous Variables B = stock of bonds dx = dummy variable for exchange controls (dx = 1 for 1974–79, dx = 0 for 1980 onwards) E o = real oil exports Gc = real government consumption K p = public non-residential fixed capital pf = price level in leading foreign industrial countries pi = price of imports (in foreign currency) M = money supply rf = foreign interest rate T1 = total taxation policy variable ((Q + P )/T1 is real private disposable income)

117


T2 = indirect taxation policy variable (Q /T2 is real output at factor cost) Y f = real income of leading foreign industrial countries

Unobservable Trend Variables µ1 = productivity trend variable (dµ1 /dt is the proportional rate of decrease in the amount of labour required to produce a given output with a given amount of capital) µ2 = labour supply trend variable (dµ2 /dt is the proportional rate of growth in the non-accelerating inflation level of employment) µ3 = trend variable allowing for the growth in the use of credit and charge cards (plastic money) (dµ3 /dt is the proportional rate of growth in the use of cards)

Structural Equations

β1 e −{β2 (r −Dlog p)+β3 Dlog p} (Q + P ) DlogC = λ1 + λ2 + γ1 log T1 C (3.1) D 2 logL = γ2 (λ2 − DlogL) β4 e −µ1 {Q −β6 − β5 K −β6 }−1β6 + γ3 log L

118

(3.2)

Model Specification

D 2 logK h = γ4 (λ1 + λ2 − DlogK h ) β7 e −{β8 (r −Dlog p)+β9 Dlog p} (Q + P ) + γ5 log T1 K h (3.3) D 2 logK = γ6 (λ1 + λ2 − DlogK ) β5 (Q /K )(1+β6 ) + γ7 log r − β10 Dlog p + β11

(3.4)

DlogQ = λ1 + λ2 + γ8 log   {1−β12 (qp/pi )β13 }{1+β14 (λ1 +λ2 )}  × (C +Gc +DK+DKh +DK p + En + E o )    ×    Q + γ9 log

β14 (C + Gc + D K + DKh + DK p + E n + E o ) S (3.5)

D 2 log p = γ10 (Dlog(w/p) − λ1 ) β15 β4 T2 we −µ1 {1 − β5 (Q /K )β6 }−(1+β6 )/β6 + γ11 log p (3.6) D 2 logw = γ12 (λ1 − Dlog(w/p)) + γ13 Dlog( pi /qp) β4 e −µ1 {Q −β6 − β5 K −β6 }−1/β6 + γ14 log β16 e µ2

119

(3.7)


D 2 r = −γ15 Dr + γ16 [β17 + β18rf − β19 Dlogq + β20 { p(Q + P )/Me µ3 } + β21 (B/M) − r ]

(3.8)

DlogI = λ1 + λ2

  β12 (qp/pi )β13 {1 + β14 (λ1 + λ2 )}  × (C + Gc + DK + DKh + DK p + E n + E o )    + γ17 log     ( pi /qp)I

β14 (C + Gc + DK + DKh + DK p + E n + E o ) + γ18 log S

(3.9)

DlogE n = λ1 + λ2 + γ19 log

β

β22 Y f 23 ( pf /qp)β24 En

(3.10)

DF = γ20 {β25 (Q + P ) − F }

(3.11)

DP = γ21 [{β26 + β27 (rf − Dlog pf )}K a − P ]

(3.12)

D 2K a = −γ22 DKa + γ23 [{β28 + β29 (rf − r )−β30 Dlogq −β31 dx } × (Q + P ) − K a ]

β32 pf + γ25 log qp E n + E o + P − F − DKa + γ26 (r − β33 ) + γ27 log ( pi /qp)I (3.14)

pf D logq = γ24 Dlog qp 2

(3.13)

120

Model Specification

DS = Q + ( pi /qp)I − C − DK − DKh −DK p − E n − E o − Gc

(3.15)

Dµ1 = λ1

(3.16)

Dµ2 = λ2

(3.17)

Dµ3 = λ3

(3.18)

The above model can be simplified by solving (3.16)–(3.18) subject to the normalization conditions µ1 (0) = 0, µ2 (0) = 0, µ3 (0) = 0, to obtain µ1 = λ1 t, µ2 = λ2 t, µ3 = λ3 t, and substituting these expressions into (3.2), (3.6), (3.7) and (3.8). But, this simplification will not be possible in the stochastic form of the model, where (3.16), (3.17) and (3.18) will be replaced by stochastic differential equations and white noise innovations will be added to each of the other equations. The main innovative feature of the above model is the introduction of stochastic trends to represent unobservable variables that, in earlier models, have been represented by deterministic trends. But, the model also retains several important features of the earlier models of Bergstrom and Wymer [1976] and Bergstrom, Nowman and Wymer [1992].

121


One of these features is that nearly every equation has the form of a partial adjustment (error correction) equation, in which the dependent variable is adjusting, continuously, in response to the deviation of its current level from its partial equilibrium level, which is a function of other variables in the model. These partial adjustment equations are a mixture of first- and second-order differential equations. The firstorder equations are of the form d x(t) = λ + γ { f (z(t)) − x(t)}, dt

(3.19)

where z(t) is a vector of other variables in the model that, directly, influence x(t), f (z(t)) (a function of z(t)) is the partial equilibrium level of x(t) at time t, γ is a speed of adjustment parameter and λ is a parameter representing the expected rate of increase in f (z(t)), which depends on the expected rates of increase in the elements of z(t). The second-order equations are of the form d 2 x(t) d x(t) = γ1 λ − + γ2 { f (z(t)) − x(t)} dt 2 dt

(3.20)

in which γ1 and γ2 are speed of adjustment parameters and λ and f (z(t)) have the same interpretation as in (3.19). The main reason why economic variables adjust gradually, rather than instantaneously, to their partial equilibrium level is that there are adjustment costs, which depend on the

122

Model Specification

rate of change and, possibly, the acceleration of the adjusting variable. Moreover, equations of the form (3.19) and (3.20) can be, formally derived as the solution of a dynamic optimization problem, which takes account of these adjustment costs. This is done in Appendix A of this chapter. Equation (3.19) (which is identical with (3.A15) of Appendix A) is obtained by minimizing the integral over future time of a quadratic cost function, whose value, at any future point of time, depends on both the expected deviation of the adjusting variable from its partial equilibrium value and its expected rate of change at that point of time. Equation (3.20) (which identical with (3.A37) of Appendix A) is obtained in a similar way by using a cost function that takes account of second-order adjustment costs, the value of the function, at any future point of time, depending on the expected acceleration of the adjusting variable at that point of time, as well as its expected rate of change and expected deviation from its partial equilibrium value. Another general feature of the model, which was also a feature of the model of Bergstrom, Nowman and Wymer [1992], is that it incorporates the assumption of long-run rational expectations. In particular, it is assumed that the agents in the economy know the trend parameters λ1 and λ2 for productivity and the labour supply and take these

123


into account in forming their expectations. The assumption that the agents in the economy know the parameters is, of course, more realistic than the assumption that they know the parameters affecting the short-run dynamics of the model, which would be necessary to justify the assumption of full rational expectations. It is shown in Chapter 4 that the steady state growth rate of most of the real variables in the model is λ1 + λ2 . The long-run rational expectations assumption is incorporated in equations of the form (3.19) or (3.20) relating to these variables, therefore, by setting the expectations parameter λ equal to λ1 + λ2 . The reference, in the preceding paragraph, to the steady state growth rate leads us, naturally, to the discussion of a third general feature of the model. Like the model of Bergstrom and Wymer [1976] and the many models for which that model served as a prototype (including the model of Bergstrom, Nowman and Wymer [1992]), it is designed in such a way as to permit a rigorous mathematical analysis of its steady state and stability properties, thus providing a check on the capacity of the model to generate plausible long-run behaviour. Because of the smallness of the samples available for the estimation and testing of macroeconometric models, it is very important to ensure, preferably by the mathematical analysis of their dynamic properties, that

124

Model Specification

they are capable of generating plausible long-run behaviour under realistic assumptions about the behaviour of the exogenous variables. As was pointed out by Bergstrom and Wymer [1976, p. 268], failure to generate plausible longrun behaviour could indicate a structural defect, such as the omission of an important feedback. Such a defect could seriously affect the predictive power of the model and its usefulness for either medium-term forecasting or policy analysis. The final feature of the model that we shall mention is that, like the models of Bergstrom and Wymer [1976] and Bergstrom, Nowman and Wymer [1992], it makes intensive use of economic theory to obtain a parsimonious parameterization and cross equation restrictions. This will be evident from the discussion of the individual equations of the model, to which we now turn.

3.3 Private Consumption Real private consumption C is one of six variables in the model whose adjustment equations take the form of firstorder differential equations, the adjustment equations of the remaining equations being of the second order. In choosing the variables whose adjustment equations were to take the

125


form of first-order differential equations, we relied not only on our a priori knowledge of the ease with which the speed of adjustment of each variable can be changed, but also on the estimates of the speed of adjustment parameters in the model of Bergstrom, Nowman and Wymer [1992]. Since the time lag distribution generated by a first-order differential equation is of the exponential form, as is shown in Appendix B of this chapter, it can be a good approximation to the true unknown adjustment equation only if the mode of the time lag distribution generated by that equation is either small or nonexistent. The six variables whose adjustment equations take the form of first-order differential equations in this model are those for which the estimated modal time lags in the model of Bergstrom, Nowman and Wymer [1992] are smallest (C, Q, E, I, F, P). The estimated modal time lag of C in that model is only 0.78 quarters. The consumption adjustment equation (3.1) assumes that the proportional rate of increase in consumption depends on the ratio of the partial equilibrium level of consumption β1 e −{β2 (r −Dlog p)+β3 Dlog p} (Q + P )/T1 to the current level of consumption. The taxation policy variable T1 is defined in such a way that (Q + P )/T1 is real private disposable income (as normally measured); that is T1 is the ratio of the net real national income at market prices to real private

126

Model Specification

disposable income (as normally measured). The partial equilibrium level of consumption is assumed, therefore, to depend on real private disposable income (Q + P )/T1 , the real interest rate r − Dlog p and the inflation rate Dlog p. The reason for assuming that inflation has an independent effect on consumption (in addition to its effect through the real interest rate) is that inflation reduces the real disposable income of holders of government bonds (like a hidden tax), and, hence, the variable (Q + P )/T1 overstates the true level of real disposable income. The partial equilibrium level of consumption is really assumed, therefore, to be a function of only the real interest rate and real disposable income correctly measured. We expect both β2 and β3 to be positive, implying that the propensity to consume is lower the higher are both the real interest rate and the inflation rate. The parameter β1 can be interpreted as the limit of the propensity to consume as the real interest rate and inflation rate each tend to zero. Consumption is assumed to adjust gradually, rather than instantaneously, to its partial equilibrium level because of adjustment costs. These costs can be either physical or psychological. Equation (3.1) implies that when consumption is at its partial equilibrium level, it will increase at the rate λ1 + λ2 , which is its steady state growth rate, as will be shown in

127


Chapter 4. It is implicitly assumed, therefore, that consumers expect the variables affecting consumption to change in such a way that the partial equilibrium level of consumption increases at its steady state. In this way, the equation incorporates the assumption of long-run rational expectations. It should be noted, also, that the equation has the [β1 e

form

of

(3.19),

−{β2 (r −Dlog p)+β3 Dlog p}

with

x(t) = logC , f (z(t)) = log

(Q + P )/T1 ] and λ = λ1 + λ2 . It can

be formally derived, therefore, as the solution of a dynamic optimization problem, which takes account of first-order adjustment costs, as in Appendix A of this chapter.

3.4 Residential Fixed Capital It is convenient, at this stage, to discuss (3.3), which is the residential (housing) fixed capital adjustment equation. Although it is a capital adjustment equation, it is, essentially, a demand equation for residential services, since the consumption of residential services is, implicitly, assumed to be proportional to the stock of residential capital. The equation is similar in form, therefore, to (3.1), except that it is a second-order differential equation. It assumes that the acceleration of the logarithm of the stock of residential

128

Model Specification

capital depends on the excess of its steady state growth rate λ1 + λ2 over its current growth rate and on the ratio of the partial equilibrium stock of residential capital to the current stock. The formulation as a second-order differential equation allows for the fact that adjustment costs will have a more important influence on the adjustment of residential capital and the consumption of the services of this capital than on the general level of consumption. The equation implies that, when the stock of residential capital is at its partial equilibrium level and growing at its steady state rate, its acceleration is zero. It is implicitly assumed, therefore, that the partial equilibrium stock of residential capital is expected to grow at its steady state rate, thus incorporating the assumption of long-run rational expectations. It should be noted, also, that the equation has the [β7 e

form

of

(3.20)

−{β8 (r −Dlog p)+β9 Dlog p}

with

x(t) = logK h , f (z(t)) = log

(Q + P )/T1 ] and λ = λ1 + λ2 . It can

formally derived, therefore, as the solution of a dynamic optimization problem, which takes account of both firstand second-order adjustment costs, as in Appendix A of this chapter. Like (3.1) and (3.3), most of the other equations in the model can be expressed in the form of either (3.19) or

129


(3.20) with λ being chosen so that the equation is consistent with the assumption of long-run rational expectations. Since this will be fairly obvious, it will be unnecessary to comment on this aspect of the equation in each case.

3.5 Employment In the next few sections, we shall discuss the equations relating to the production sector of the economy. These are the adjustment equations for employment, private nonresidential fixed capital, output and the price level. It will be useful, before discussing these equations in detail, to say something about the underlying assumptions concerning firm behaviour and market structure. It is assumed that the economy is made up of a large number of monopolistic competitors with identical production functions and uniformly differentiated products. Since the prices charged by the different firms in the economy are equal, it can be treated as a single product economy, and it tends to a perfectly competitive economy as the degree of differentiation between the products of the different firms tends to zero, that is as the price elasticity of the partial demand function for each firm’s product tends to minus infinity.

130

Model Specification

It would be possible to derive a set of adjustment equations for employment, output, capital and the price level by a multidimensional generalization of the analysis in Appendix A of this chapter, assuming that the agent has control of a vector of variables, rather than a single variable. But, to limit the number of parameters to be estimated, we followed a simpler approach, which takes account of our a priori knowledge of the relative speeds at which different factors of production can be adjusted (or the relative costs of adjusting them at a given speed). We know that it is easier to adjust output by varying the intensity with which the employed labour force is used (varying the number of hours per week worked by each employee) than to vary the number of persons employed, and it is easier to vary the number of persons employed than to vary the stock of fixed capital. We assume, therefore, that, at each point of time, output is adjusting in response to sales, the number of persons employed is adjusting in response to output and the stock of capital is adjusting in response to the marginal product of capital and the real interest rate. The remainder of this section is devoted to a more detailed discussion of the employment adjustment equation (3.2). This is formulated as a second-order differential equation,

131


partly for a priori reasons mentioned in the preceding paragraph, and partly on the basis of the estimated speed of adjustment parameters in the employment equation in the model of Bergstrom, Nowman and Wymer [1992]. These estimates imply that the time lag distribution with which employment adjusts to output and capital has a mode of 3.48 quarters (the modal time lag). The exponential time lag distribution generated by a first-order differential equation would, therefore, be a poor approximation to that distribution. Equation (3.2) assumes that the acceleration of the logarithm of employment depends on the excess of its steady state growth rate λ2 over its current growth rate and on the ratio of the partial equilibrium level of employment to the current level. The function β4 e −µ1 {Q −β6 − β5 K −β6 }−1/β6 defining the partial equilibrium level of employment is a constant elasticity of substitution production function, which incorporates the productivity stochastic trend variable µ1 as well as output and capital. This function plays a central role in the model. Its parameters occur in several other equations, resulting in a very parsimonious parametrization and cross equation restrictions. The elasticity of substitution between labour and capital is 1/(1 + β6 ) and can be derived as follows.

132

Model Specification

When the production sector of the economy is in a state of partial equilibrium, employment is related to output and capital by the equation L = β4 e −µ1 {Q −β6 − β5 K −β6 }−1/β6 ,

(3.21)

from which we obtain 1+β6 ∂L β4 e −µ1 −β6 − {β5 β6 K −(1+β6 ) } {Q − β5 K −β6 } β6 = − ∂K β6 (1+β6 ) −(1+β6 ) = −β5 (β4 e −µ1)−β6 β4 e −µ1{Q −β6 −β5 K −β6}−1/β6 K (1+β6 ) L = −β5 (β4 e −µ1 )−β6 , K and hence L = K

(β4 e −µ1 )β6 β5

∂L − ∂K

!

1 1+β6

"

.

(3.22)

The term −∂L/∂K (which is, of course, positive) is the marginal technical rate of substitution between labour and capital and is, approximately, equal to the number of units of labour released as a result of the addition of one unit of capital, when output is held constant. Equation (3.22) shows that the ratio of labour to capital employed is related to the marginal rate of substitution between labour and capital through a function with constant elasticity 1/(1 + β6 ). This elasticity is called the elasticity

133


of substitution between labour and capital. It is shown in Chapter 4 that the steady state growth rate of employment is λ2 , which is the rate of growth of the non-accelerating inflation level of employment. Equation (3.2) implies, therefore, that, when employment is at its partial equilibrium level and growing at its steady state rate, its acceleration is zero.

3.6 Private Non-Residential Fixed Capital Equation (3.4) assumes that the acceleration of the logarithm of the stock of private non-residential fixed capital depends on the excess of its steady state growth rate λ1 + λ2 over its current growth rate and on the ratio of the marginal product of capital β5 (Q /K )(1+β6 ) to the real rate of interest plus a risk premium. The marginal product of capital is obtained from the relation ∂L/∂K ∂Q =− , ∂K ∂L/∂Q where ∂L/∂K and ∂L/∂Q are the partial derivatives of the production function β4 e −µ1 {Q −β6 − β5 K −β6 }−1/β6 . We have

134

Model Specification

1+β6 β4 e −µ1 −β6 ∂L −β6 − β6 = − β5 β6 K −(1+β6 ) , {Q − β5 K } ∂K β6 1+β6 β4 e −µ1 −β6 ∂L − = − −β6 Q −(1+β6 ) , {Q − β5 K −β6 } β6 ∂Q β6 and hence {β5 β6 K −(1+β6 ) } ∂Q =− = β5 ∂K {−β6 Q −(1+β6 ) }

Q K

(1+β6 ) .

We assume that investors are risk averse and that the minimum value of the marginal product of capital required to induce further investment in fixed capital must exceed the real interest rate r − Dlog p by a risk premium. Moreover, we assume that the risk premium depends on the rate of inflation, since the degree of uncertainty concerning the average rate of inflation over the life of fixed capital, and concerning, therefore, the relative rates of return from investment in fixed capital and bonds, is likely to depend on the current rate of inflation. The sum of the real interest rate and the risk premium is represented in (3.4) by the term r − β10 Dlog p + β11 . In the special case where β10 = 1, the risk premium is a constant β11 . Equation (3.4) implies that, when the marginal product of capital

135


equals the real interest rate plus the risk premium and capital is growing at its steady state rate, its acceleration is zero.

3.7 Output Equation (3.5) assumes that the proportional rate of increase in output depends on the ratio of the partial equilibrium level of output to the current level of output and the ratio of the partial equilibrium level of stocks (inventories) to the current level of stocks. The partial equilibrium level of stocks is assumed to equal β14 (C + Gc + DK + DKh + DK p + E n + E 0 ), where the term in brackets equals the total sales for private and public consumption, capital formation and exports, while the parameter β14 is the optimal ratio of stocks to sales. It is assumed that, in the state of partial equilibrium, the total supply from output and imports {Q + ( pi /pq)I } is just sufficient to meet current sales and maintain the constant ratio β14 of stocks to sales, when sales are growing at their steady state rate λ1 + λ2 . The supply required to maintain the constant ratio β14 of stocks to sales, when sales are growing at the rate λ1 + λ2 , is β14 (λ1 + λ2 )(C + Gc + DK + DKh + DK p + E n + E 0 ), and the total supply required to meet current sales and maintain

136

Model Specification

the ratio β14 of stocks to sales is, therefore, {1 + β14 (λ1 + λ2 )} (C + Gc + DK + DKh + DK p + E n + E 0 ). We assume that the proportion of the total supply that is provided by imports depends on the ratio of the price of imports to the price of domestic goods and, in particular that it equals β12 (q p/pi )13 . In the state of partial equilibrium imports and output must, therefore, satisfy (3.23) and (3.24). ( pi /pq)I = β12 (qp/pi )β13 {1 + β14 (λ1 + λ2 )} × (C + Gc + DK + DKh + DK p + E n + E 0 ), (3.23) Q = {1 − β12 (qp/pi )β13 }{1 + β14 (λ1 + λ2 )} × (C + Gc + DK + DKh + DK p + E n + E 0 )

(3.24)

From (3.23) we obtain I = β12 (qp/pi )(1+β13 ) {1 + β14 (λ1 + λ2 )} × (C + Gc + DK + DKh + DK p + E n + E 0 ),

(3.25)

which implies that the price elasticity of demand for imports is −(1 + β13 ). The partial equilibrium level of output is given by the term on the right-hand side of (3.24). Equation (3.5) implies that, when both output and stocks are at their

137


partial equilibrium levels, output will grow at its steady state rate λ1 + λ2 .

3.8 Price Level Equation (3.6) assumes that the acceleration of the logarithm of the price level depends on the excess of the current rate of increase in the real wage over the rate of technical progress λ1 and on the ratio of the partial equilibrium price level to the current price level. The partial equilibrium price level β15 β4 T2 e −µ1 {1 − β5 (Q /K )β6 }−(1+β6 )/β6 is equal to β15 T2 w(∂L/∂Q ), where ∂L/∂Q is the partial derivative, with respect to Q, of the production function β4 e −µ1 {Q −β6 − β5 K −β6 }−1/β6 . It follows from the definition of the indirect tax variable T2 that it can be written as T2 = 1 + t2 , where t2 is the average rate of indirect taxation. The term T2 w(∂L/∂Q ) is equal, therefore, to marginal cost including indirect taxation. The parameter β15 is a measure of the degree of imperfection of competition, and it is assumed to satisfy the condition β15 ≥ 1, with β15 = 1 in the limiting case of perfect competition. In that limiting case, therefore, the partial equilibrium price level equals marginal cost including indirect taxation.

138

Model Specification

In the more general case, β15 is defined by β15 = e/(1 + e), where e is the own-price elasticity of the partial demand function for each firm’s product. The partial equilibrium condition p = β15 T2 w(∂L/∂Q ) is then equivalent to the profit maximization condition that the marginal revenue of the representative firm is equal to its marginal cost. To show this let pi , Q i and L i denote the price, output and amount of labour employed by firm i. The marginal revenue ∂( pi Q i )/∂ Q i of firm i is then given by ∂( pi Q i ) ∂pi = Qi + pi ∂Q i ∂Q i Q i ∂pi = pi +1 pi ∂Q i (1 + e) = pi e pi = . β15 The condition that the marginal revenue of firm i equals its marginal cost is given, therefore, by ∂L i pi = T2 w β15 ∂Q i from which we obtain

∂L i pi = β15 T2 w ∂Q i

139

.


Hence, since the price, output and amount of labour employed are assumed to be the same for all firms, we obtain

∂L p = β15 T2 w ∂Q

.

The first term on the right-hand side of (3.6) differs from the corresponding term in most of the second-order equations in the model. For most of the second-order equations, this term is the deviation of the rate of change of the variable, whose acceleration is being explained by the equation, from the steady state rate of change of that variable. The steady state rate of change in the price level does not exist, however, unless the money supply grows at a constant rate, and past variations in the rate of growth of the money supply have been so great that it is unrealistic to assume that agents in the economy have any firm expectations concerning the future rate of change in the money supply. The first term on the right-hand side of (3.6) can be written as γ10 {(Dlogw − λ1 ) − Dlog p}, in which the term Dlogw − λ1 can be regarded as the expected rate of change of the partial equilibrium price level in the medium future. Then (3.6) implies that, when the price variable is at its partial equilibrium level and growing at the expected

140

Model Specification

rate of increase of its partial equilibrium level, its acceleration is zero. Finally, it should be mentioned that the speed of adjustment parameters in (3.6) depends, not only on adjustment costs (which, for price variables, are relatively minor), but also on the time lags in the competitive process, in which each firm is adjusting its price in small steps, taking account of the current prices of its competitors’ products (see Chamberlin [1946, Ch. 5, Section 3]).

3.9 Wage Rate Equation (3.7) assumes that the acceleration of the logarithm of the wage rate depends on the excess of the rate of technical progress λ1 over the current rate of increase in the real wage, on the current rate of increase in the ratio of the import price level to the domestic price level (both measured in sterling), and on the ratio of the partial equilibrium level of employment to the labour supply β16 e µ2 . The labour supply β16 e µ2 can be interpreted as the non-accelerating inflation level of employment, since (3.7) implies that, when the real wage is growing at its steady state rate λ1 and the real exchange rate is constant, the acceleration of the nominal wage rate will be zero if, and only if, the partial equilibrium level of employment (which depends on the current

141


levels of output and capital) equals β16 e µ2 . The labour supply, so defined, is not directly measurable, since it depends, not only on demographic factors, but also on conditions in the labour market. It is assumed, therefore, to be an exponential function of the unobservable trend variable µ2 with constant drift parameter λ2 and, in the stochastic version of the model, constant volatility parameter. The first term on the right-hand side of (3.7) has the same form as the first term on the right-hand side of (3.6) (discussed above) with the opposite sign. The two equations together imply that, if the rate of increase in the real wage rate exceeds the rate of technical progress and all other terms on the right-hand side of each equation are zero, then there will be an acceleration of the price level and a deceleration of the wage rate, while, if the rate of increase in the real wage rate is less than the rate of technical progress and all other terms on the right-hand side of each equation are zero, then there will be an acceleration of the wage rate and a deceleration of the price level. The term Dlog( pi /q p) allows for pressure (by trade unions and other organizations) for higher wages to compensate for the loss of welfare caused by the fall in the real exchange rate (since, although w/p is the measure of the real wage that is relevant for firms employing labour, it is not a measure of welfare).

142

Model Specification

Equation (3.7), together with the other equations in the model, implies that there is no long-run trade off between unemployment and inflation. This can be seen by comparing two steady states, in one of which the money supply M is growing at a faster rate than in the other. The only variables in (3.7) that will change as between the two steady states are Dlogw, Dlog p and Dlogq. As we move to the steady state with the higher rate of growth of M, Dlogw and Dlog p will increase by the same amount as DlogM, while Dlogq will decrease by the same amount. The first two terms on the right-hand side of (3.7) will be unchanged, therefore, and, since the left-hand side will be zero, the third term on the right-hand side must also be unchanged. In both steady states, all 3 terms on the right-hand side of (3.7) are zero, and employment equals β16 e µ2 (see Chapter 4). A final point to be noted concerning (3.7) is that we have used the concept of the non-accelerating level of employment, rather than the more commonly used concept of the non-accelerating inflation level of unemployment. The advantage of this is that employment is a more precisely defined variable than unemployment, since the latter depends on the somewhat arbitrary definition of the labour force.

143


3.10 Interest Rate The interest rate adjustment equation (3.8) represents the dynamic behaviour of the market for long-dated bonds. It is, essentially, a portfolio balance equation, which allows for substitution between money, domestic bonds and foreign bonds. The acceleration of the interest rate (the yield on long-dated bonds) is assumed to depend on the current rate of change of the interest rate and the excess of the partial equilibrium interest rate over the current rate. The partial equilibrium interest rate is the rate at which the demand for domestic bonds equals the current stock of these bonds, given the return on the two alternative assets money and foreign bonds. The return from holding money comes mainly from the facilitation of transactions. The marginal rate of return from holding money is assumed, therefore, to depend on the ratio of income to the stock of money and is represented by the term β20 { p(Q + P )/Me µ3 }, in which the stochastic trend variable µ3 allows for the growing use of plastic money (credit and charge cards). The rate of return on foreign bonds is represented by the term β17 + β18rf − β19 Dlogq, in which Dlogq is a proxy for the expected rate of change in the exchange rate.

144

Model Specification

If there were no uncertainty, we would expect to have β18 = β19 = 1. Indeed, the partial equilibrium condition for (3.8) includes, as a very special case, the uncovered interest rate parity condition r = rf − Dlogq. But this cannot be expected to hold in a world of uncertainty. The inclusion of the term β21 (B/M) in (3.8) ensures, together with the other terms, that, if the stock of bonds, the money stock and income all increase in the same proportion and the foreign interest rate and the rate of increase in the exchange rate all remain unchanged, then the partial equilibrium level of r will also remain unchanged.

3.11 Imports Equation (3.9) assumes that the proportional rate of increase in the volume of imports depends on the ratio of the partial equilibrium real value of imports to the current real value of imports and the ratio of the partial equilibrium level of stocks to the current level of stocks. It implies that, when stocks and the real value of imports are at their steady state levels, the volume of imports will increase at its steady state rate λ1 + λ2 . The partial equilibrium real value of imports has been discussed in Section 3.7, where it was shown that the price elasticity of demand for imports is −(1 + β13 ).

145


Equations (3.5) and (3.9) together ensure that, in the steady state, imports and domestic output are just sufficient to meet total sales and keep stocks growing at the rate λ1 + λ2 . It should be noticed that we have distinguished between the volume of imports I and the real value of imports pi I /q. To avoid introducing two many price variables (and to be consistent with our assumption of, essentially, a single product economy), all other real variables in the model (for example, C ) have been defined as values at current prices deflated by p the implicit price deflator of the gross domestic product.

3.12 Non-Oil Exports Equation (3.10) assumes that the proportional rate of increase of non-oil exports depends on the ratio of the partial equilibrium level of non-oil exports to the current level. It implies that, when non-oil exports are at their partial equilibrium level, they will increase at their steady state rate λ1 + λ2 . The partial equilibrium level of non-oil exports is assumed to be a function of foreign real income Yf and the ratio of the foreign price level to the domestic price level (both measured in sterling) pf /q p. This is a standard constant

146

Model Specification

elasticity form of demand function, in which the parameters β23 and −β24 can be interpreted as the foreign income and price elasticities of demand for UK products.

3.13 Transfers Abroad Equation (3.11)–(3.13) are the adjustment equations for the real transfers abroad F, real profits, interest and dividends from abroad P and cumulative net real investment abroad K a . These equations are formulated in natural values rather than logarithms, because each of the variables F, P and K a can be either positive or negative. Equation (3.11) assumes that the rate of increase of real transfers abroad F is proportional to the excess of the partial equilibrium level of this variable over its current level and that the partial equilibrium level is proportional to real income Q + P .

3.14 Real Profits Interest and Dividends from Abroad Equation (3.12) has the same general form as (3.11). It assumes that the rate of increase of real profits, interest and dividends from abroad P is proportional to the excess of the

147


partial equilibrium level of this variable over its current level. The partial equilibrium level is assumed to equal cumulative real net investment abroad K a multiplied by a linear function of the real foreign interest rate rf − Dlog pf . This is an approximation, which takes account of the several different types of income in the variable P. Consider, first, the real dividend component of P. This equals the real market value of the shares on which these dividends are paid multiplied by the dividend yield. The market value of the shares will, generally, be somewhat greater than the share component of K a , because of the growth in the market value of the shares since their purchase by the UK owners. On the other hand, the dividend yield, although closely related to the real interest rate rf − Dlog pf , will, normally, be less than the latter variable, because of the potential growth of dividends. These approximations are reflected in the parameters β26 and β27 in (3.12). Consider, next, the real interest component of P . This equals the real market value of the bonds on which the interest is paid multiplied by the foreign interest rate rf . Because of inflation, the real market value of the bonds will be less than the bond component of K a , which is the real historic cost of the bonds. Since the nominal interest paid on the bonds remains constant over their life, their

148

Model Specification

real market value falls during a period of rising prices. This effect of inflation will be allowed for by multiplying the bond component of K a by a linear function of the real foreign interest rate rf − Dlog pf , as in (3.12), rather than by rf .

3.15 Cumulative Net Real Investment Abroad Equation (3.13) assumes that the acceleration of K a depends on its current rate of change and the excess of its partial equilibrium level over its current level. The partial equilibrium level of K a is assumed to equal real income multiplied by a linear function of the difference between the foreign and domestic interest rates, the rate of change in the exchange rate and a dummy variable to allow for exchange controls (which were abolished in 1979). This is, essentially, a portfolio balance relation representing investors’ allocation of capital between domestic and foreign assets, taking account of the difference between the rates of return on these assets and the expected change in the exchange rate. The parameter β30 reflects the speed with which the rate of change in the exchange rate is expected to revert to the steady state rate of change.

149


3.16 Exchange Rate The exchange rate adjustment equation (3.14) represents the dynamic behaviour of the foreign exchange market. The acceleration of the logarithm of the exchange rate is assumed to depend on the rate of decrease in the real exchange rate Dlog( pf /qp), on the ratio of the expected steady state real exchange rate β32 to the current real exchange rate, on the excess of the current interest rate r over the expected steady state interest rate β33 and on the ratio of the balance of payments surplus (including capital items other than the change in the official reserves of gold and foreign exchange) to the real value of imports. The last term allows for the effect of the excess supply or demand for sterling resulting from a surplus or deficit in the balance of payments, while the previous two terms allow for the effect of the speculative excess supply or demand for sterling resulting from the deviations of the interest rate and real exchange rate from their expected steady state levels. As in the case of the price and wage adjustment equations (3.6) and (3.7), the first term on the right-hand side of (3.14) differs from the corresponding term in most of the secondorder equations in the model. Instead of the deviation of

150

Model Specification

the rate of change in the nominal exchange from its steady rate of change (which depends on the rate of change in the money supply), we have the deviation of the rate of change of the real exchange rate from its steady state rate of change (which is zero). The reason for this variation from the form of most of the second-order equations is, as in the case of (3.6) and (3.7), that it is unrealistic to assume that the agents in the economy have any firm expectations concerning the future rate of change in the money supply. When solving the complete model for the steady state in Chapter 4, we shall use the assumption of long-run rational expectations by assuming that β32 and β33 are equal to the actual steady state values of the real exchange rate and interest rate. But this assumption will not be imposed when estimating the parameters of the model.

3.17 Stocks The stock adjustment equation (3.15) is, formally, an identity, although it is not exactly satisfied by the data, because of errors in measurement. It will be treated as a stochastic differential equation with white noise innovation, therefore, in the estimation of the parameters of the model.

151


3.18 Conclusion In this chapter we have specified a new continuous time econometric model of the United Kingdom. The main innovative feature of the new model is the incorporation of stochastic trends to represent unobservable variables that, in earlier models, have been represented by deterministic trends. This and other general properties of the model have been discussed in Section 3.2 and the specification of the individual equations in Sections 3.3–3.17. Another important general property of the model is that, like the earlier UK models of Bergstrom and Wymer [1976] and Bergstrom, Nowman and Wymer [1992], it is designed in such a way as to permit a rigorous mathematical analysis of its steady state and stability properties. This analysis will be carried out in Chapter 4, to which we now turn.

Appendix A: Derivation of General Adjustment Equations In this Appendix, we show how adjustment equations of the form taken by most of the structural equations in the model can be derived as the solution of a dynamic optimization problem, which takes account of adjustment costs. We deal

152

Model Specification

first with the simplest case, in which the adjustment cost depends only on the rate of change of the adjusting variables. It will be shown, that, in this case, the optimal adjustment equation is a first-order differential equation. We assume that an economic agent has control of a single variable x(t) (where t is the time parameter), which can be adjusted, continuously, in response to a vector z(t) of other variables. The vector z(t) can include both exogenous variables and other endogenous variables in the model. We define a partial equilibrium function f (z(t)), to which x(t) would adjust instantaneously and follow continuously if there were no adjustment costs; that is we would have x(t) = f (z(t)) at all points of time if the cost of adjusting x(t) were zero. The function f (z(t)) could be the value of x(t) that maximizes some objective functiong(x(t), z(t)). Assume, now, that there is an adjustment cost, which depends only on the rate of change of x(t), but not on its acceleration rate or any of its higher derivatives with respect to t; that is we assume that the adjustment cost depends only on d x(t)/dt. Then, if at time t, x(t) differs from f (z(t)), the agent plans a path of x(t) that moves, gradually, towards the expected path of f (z(t)), in such a way as to minimize the integral of a cost function, which takes account of both adjustment costs and costs associated

153


with the deviation of x(t) from f (z(t)). These costs can be either monetary or psychological (loss of utility). We assume that the cost function can be approximated by a quadratic function, so that, if the future path of f (z(t)) were known, the integral to be minimized would be ∞ 2 2 0 [{x(t + r ) − f (z(t + r ))} + a{d x(t + r )/dr } ]dr, where a is a parameter reflecting the importance of adjustment costs. Since the future path of z(t) is unknown, it is necessary to make some assumption about expectations. We shall assume that z(t) is expected to change in such a way that f (z(t)) increases at a constant linear rate λ. Then, the expected value, at time t, of f (z(t + r )) is given by E[ f (z(t + r ))] = f (z(t)) + λr.

(3.A1)

This assumption is consistent with the formulation of the model in Section 3.2 of this chapter. There, nearly all of the variables are measured in logarithms, and the formulation of the adjustment equations, implicitly, assumes that their partial equilibrium values are expected to increase at constant linear rates equal to their steady state growth rates. In particular, if, at time t, a variable is at its partial equilibrium level, it will not be held constant, but will be increased at a

154

Model Specification

rate equal to its steady state growth rate. This is consistent with the assumption of long-run rational expectations. Taking account of the above assumption with respect to expectations and the assumption of a quadratic cost function, the optimal planned path, at time t, of x(t + r ) (r > 0) is that which minimizes the integral ∞ C {x(t + r ), d x(t + r )/dr }dr, where 0

C {x(t + r ), d x(t + r )/dr } = {x(t + r ) − f (z(t)) − λr }2 + a{d x(t + r )/dr }2 .

(3.A2)

A necessary condition for x(t + r )(r ≥ 0) to be the path of x ∞ that minimizes 0 C {x(t + r ), d x(t + r )/dr }dr, is that it satisfies Euler’s differential equation d Cx − C x = 0 (r ≥ 0), dr

(3.A3)

where C x and C x are the partial derivatives of the function C with respect to x(t + r ) and d x(t + r )/dr , respectively (see Courant [1936, Vol. II, Ch. 7]). From (3.A2) we obtain C x = 2{x(t + r ) − f (z(t)) − λr },

(3.A4)

C x = 2a{d x(t + r )/dr }.

(3.A5)

155


Substituting from (3.A4) and (3.A5) into (3.A3) and dividing by −2a, we obtain the differential equation d 2 x(t + r ) 1 − {x(t + r ) − f (z(t)) − λr } = 0 dr 2 a

(r ≥ 0)

(3.A6)

which must be satisfied by the optimal planned path of x(t + r ) (r ≥ 0). Now let y(t + r ) be defined by y(t + r ) = x(t + r ) − f (z(t)) − λr,

(3.A7)

so that y(t + r ) is the deviation of x from its partial equilibrium level at time t + r. Then, from (3.A6) and (3.A7), we obtain d 2 y(t + r ) 1 − y(t + r ) = 0 dr 2 a

(r ≥ 0),

(3.A8)

which can be written in the factorized form (D − γ )(D + γ )y(t + r ) = 0

(r ≥ 0),

(3.A9)

√ where γ = 1/ a and D = d/dr. The general solution of the differential equation (3.A9) is y(t + r ) = c 1 e −γ r + c 2 e γ r

(r ≥ 0),

(3.A10)

where c 1 and c 2 are arbitrary constants. However, since the optimal planned path of y(t + r ) must converge to zero as

156

Model Specification

r → ∞, we must put c 2 = 0 and c 1 = y(t), so that the optimal planned path of y(t + r ) is given by y(t + r ) = y(t)e −γ r

(r ≥ 0).

(3.A11)

The solution given by (3.A11) satisfies the differential equation dy(t + r ) = −γ y(t + r ) (r ≥ 0), (3.A12) dr and, from (3.A7) and (3.A12), we obtain d x(t + r ) = λ + γ { f (z(t)) + λr − x(t + r )} (r ≥ 0), dr

(3.A13)

which is the differential equation satisfied by the optimal planned path of x(t + r ). The actual path of x(t) will not, of course, follow the optimal planned path, since z(t) is changing continuously. However, d x(t)/dt will change continuously in response to changes in z(t), in such a way that, at each point of time t, x(t) will be starting to move along the optimal planned path corresponding to the value of z(t) at that point of time. That is

d x(t) d x(t + r ) , = dt dr r =0

(3.A14)

where the derivative on the left-hand side of (3.A14) is that of the actual path of x, while the derivative on the

157


right-hand side is that of the optimal planned path. It follows from (3.A13) and (3.A14) that the actual path of x satisfies the differential equation d x(t) = λ + γ { f (z(t)) − x(t)}. dt

(3.A15)

Nearly all of the first-order differential equations in the model formulated in Section 3.2 of this chapter have the form of equation (3.A15). It says that the dependent variable x increases at a rate equal to the expected rate of increase in its partial equilibrium level plus a speed of adjustment parameter times the current excess of its partial equilibrium level over its actual level. It follows from the definition of γ below (3.A9) that the speed of adjustment is inversely related to the adjustment cost, as we should expect. We turn now to the case in which the adjustment cost depends on both the rate of change and acceleration of x(t); that is on both d x(t)/dt and d 2 x(t)/dt 2 . Making the same assumption about expectations as was made in the simpler case and assuming, again, that the cost function can be approximated by a quadratic function, the optimal planned path, at time t, of x(t + r ) is that which minimizes the integral

158

Model Specification

∞

C {x(t + r ), d x(t + r )/dr, d 2 x(t + r )/dr 2 }dr

0

where C {x(t + r ), d x(t + r )/dr, d 2 x(t + r )/dr 2 } = {x(t + r ) − f (z(t) − λr )}2 + a{d x(t + r )/dr }2 + b{d 2 x(t + r )/dr 2 }2

(3.A16)

and a and b are parameters measuring the importance of first- and second-order adjustment costs. A necessary condition for x(t + r ) (r ≥ 0) to be the path of ∞ x that minimizes 0 C {x(t + r ), d x(t + r )/dr, d 2 x(t + r )/dr 2 } dr , is that it satisfies Euler’s differential equation d d2 C x + 2 C x = 0 (r ≥ 0), dr dr

Cx −

(3.A17)

where C x , C x and C x are the partial derivatives of the function C with respect to x(t + r ), d x(t + r )/dr and d 2 x(t + r )/ dr 2 , respectively (see Courant [1936, Vol. II, Ch. 7]). From (3.A16) we obtain C x = 2{x(t + r ) − f (z(t)) − λr },

(3.A18)

C x = 2a{d x(t + r )/dr },

(3.A19)

C x = 2a{d 2 x(t + r )/dr }.

(3.A20)

159


Substituting from (3.A18), (3.A19) and (3.A20) into (3.A17) and dividing by 2b, we obtain the differential equation d 4 x(t + r ) a d 2 x(t + r ) − dr 4 b dr 2 1 + {x(t + r ) − f (z(t)) − λr } = 0 (r ≥ 0), (3.A21) b which must be satisfied by the optimal planned path of x(t + r ) (r ≥ 0). Then, from (3.A7) and (3.A21) we obtain d 4 y(t + r ) a d 2 y(t + r ) 1 + y(t + r ) = 0 (r ≥ 0), − dr 4 b dr 2 b (3.A22) which is the differential equation that must be satisfied by the optimal planned path of y(t + r ) the deviation of x(t + r ) from its expected partial equilibrium level. The solution of (3.A22) depends on the roots of the equation u4 −

a 2 1 u + = 0. b b

(3.A23)

We shall assume that all four roots of (3.A23) are real, in which case the convergence of the optimal planned path of x(t + r ) to its expected partial equilibrium path f (z(t)) + λr, as r → ∞, will be monotonic rather than oscillatory. The assumption that the roots of (3.A23) are real, obviously, implies that they have the form α, −α, β, −β, where α and β

160

Model Specification

are positive numbers, and the differential equation (3.A22) can be written in the factorized form (D + α)(D − α)(D + β)(D − β)y(t + r ) = 0

(r ≥ 0), (3.A24)

where D = d/dr. The general solution of the differential equation (3.A24) is y(t + r ) = C 1 e −αr + C 2 e −βr + C 3 e αr + C 4 e βr,

(3.A25)

where C 1 , C 2 , C 3 and C 4 are arbitrary constants. However, since the optimal planned path of y(t + r ) must converge to zero as r → ∞, we must have C 3 = C 4 = 0. Moreover, C 1 and C 2 must satisfy the initial conditions: [C 1 e −αr + C 2 e −βr ]r =0 = y(t),

d {C 1 e −αr + C 2 e −βr } dr

r =0

=

dy(t) , dt

(3.A26) (3.A27)

or C 1 + C 2 = y(t), − αC 1 − βC 2 =

dy(t) . dt

(3.A28) (3.A29)

Solving (3.A28) and (3.A29) for C 1 and C 2 , we obtain C1 =

{βy(t) + dy(t)/dt} , (β − α)

161

(3.A30)


C2 =

{αy(t) + dy(t)/dt} . (α − β)

(3.A31)

The optimal planned path of y(t + r ) is given, therefore, by y(t + r ) = C 1 e −αr + C 2 e −βr

(r ≥ 0),

(3.A32)

where C 1 and C 2 are given by (3.A30) and (3.A31). The path of y(t + r ) given by (3.A32) satisfies the differential equation (D + α)(D + β)y(t + r ) = 0

or

(r ≥ 0),

(3.A33)

d 2 y(t + r ) dy(t + r ) = −γ1 − γ2 y(t + r ), (3.A34) dr 2 dr

where γ1 = α + β and γ2 = αβ. From (3.A7) and (3.A4) we obtain d 2 x(t + r ) d x(t + r ) λ − = γ 1 dr 2 dr + γ2 { f (z(t)) + λr − x(t + r )}

(r ≥ 0),

(3.A35)

which is the differential equation satisfied by the optimal planned path of x(t + r ). As in the case of the first-order adjustment process discussed earlier in this Appendix, the actual path of x(t) will not follow the optimal path, because z(t) is changing

162

Model Specification

continuously. However, d 2 x(t)/dt 2 will change continuously in response to changes in z(t), in such a way that, at each point in time, x(t) will be starting to move along the optimal planned path corresponding to the value of z(t) at that time. That is

2 d 2 x(t) d x(t + r ) = dt 2 dr 2 r =0,

(3.A36)

where the second derivative on the left-hand side of (3.A36) is that of the actual path of x and that on the right-hand side is that of the optimal planned path. It follows from (3.A35) and (3.A36) that the actual path of x satisfies the differential equation

d 2 x(t + r ) d x(t + r ) + γ2 { f (z(t)) − x(t)}. = γ1 λ − dr 2 dr (3.A37)

Nearly all of the second-order differential equations in the model formulated in Section 3.2 of this chapter have the form of (3.A37). It says that the dependent variable x accelerates at a rate equal to the speed of adjustment parameter γ1 times the excess of the expected rate of increase of the partial equilibrium level of x over its actual rate of increase plus the speed of adjustment parameter γ2 times the excess its partial equilibrium level over its actual level.

163


Appendix B: Distributed Lag Relations The first- and second-order adjustment equations (3.A15) and (3.A37) derived in Appendix A each imply that the value of the dependent variable at time t depends on all past values of the explanatory variables; that is they depend on z(t − r ) (r ≥ 0). In this Appendix we investigate the precise form of these distributed lag relations. See also Bergstrom [1967], Bergstrom and Wymer [1976], Bergstrom, Nowan and Wymer [1992] and Gandolfo [1981] for these distributed lag relations. Let the trend path of the partial equilibrium f (z(t)) in (3.A15) be µ + λt, so that, over a long period, f (z(t)) and x(t) fluctuate about µ + λt . We now define y(t) and g(z(t)) by y(t) = x(t) − (µ + λt),

(3.B1)

g(z(t)) = f (z(t)) − (µ + λt).

(3.B2)

Then, (3.A15) implies that dy(t) = γ {g(z(t)) − y(t)}. dt

(3.B3)

We shall now show that the solution of (3.B3) is given by ∞ y(t) = γ e −γ r g(z(t − r ))dr (3.B4) 0

164

Model Specification

Substituting s = t − r, we obtain from (3.B4) t y(t) = γ e −γ (t−s ) g(z(s ))ds −∞ t −γ t e γ s g(z(s ))ds. = γe

(3.B5)

−∞

Then, differentiating (3.B5), we obtain t t dy(t) 2 −γ t γs −γ t d γs e g(z(s ))ds +γ e e g(z(s ))ds = −γ e dt dt −∞ −∞ t e −γ (t−s ) g(z(s ))ds + γ e −γ t e γ t g(z(t)) = −γ 2 −∞

= γ {g(z(t)) − y(t)},

which shows that the expression for y(t) given by (3.B4) is the solution (3.B3). A verbal interpretation of (3.B4) is that y(t) depends with an exponentially distributed time lag on the past values of g(z(t)). The density function of the time lag distribution is ∞ γ e −γ r , and the mean of the distribution is 0 r γ e −γ r dr = γ1 . We can derive the distributed lag relation implied by the second-order adjustment equation in a similar way. From (3.A37), (3.B1) and (3.B2), we obtain d 2 y(t) dy(t) = − γ1 + γ2 {g(z(t)) − y(t)}. dt 2 dt

(3.B6)

The solution of (3.B6) depends on the roots of the equation w 2 + γ1 w + γ2 = 0.

165

(3.B7)


We shall assume that the roots of (3.B7) are real and negative, in which case y(t) converges monotonically to g(z(t)) as t → ∞. Letting −α and −β denote the two roots, we have α + β = γ1 and αβ = γ2 , so that (3.B6) can be written dy(t) d 2 y(t) + (α + β) + αβy(t) = αβg(z(t)). dt 2 dt

(3.B8)

The form of the time lag distribution with which y(t) responds to variations in g(z(t)) depends on whether the roots −α and −β are equal or different. We shall deal, first, with the case in which they are different, and assume that β > α > 0. We shall now show that, in this case, the solution of (3.B8) is given by

∞

y(t) =

φ(r )g(z(t − r ))dr,

(3.B9)

0

where φ(r ) =

αβ {e −αr − e −βr }. β −α

(3.B10)

Substituting s = (t − r ), we obtain from (3.B9) and (3.B10) y(t) =

αβ β −α

t

−∞

e −α(t−s ) g(z(s ))ds −

t −∞

e −β(t−s ) g(z(s ))ds . (3.B11)

166

Model Specification

Then, differentiating (3.B11), we obtain dy(t) dt =

t αβ d t −β(t−s ) d e −α(t−s ) g(z(s ))ds − e g(z(s ))ds . β − α dt −∞ dt −∞ (3.B12)

But, d dt

t d e −α(t−s ) g(z(s ))ds = e −αs g(z(s ))ds e −αt dt −∞ −∞ t t d e αs g(z(s ))ds + e −αt e αs g(z(s ))ds = −αe −αt dt −∞ −∞ t = −α e −α(t−s ) g(z(s ))ds + e −αt e αt g(z(t)) −∞ t e −α(t−s ) g(z(s ))ds + g(z(t)), (3.B13) = −α

t

−∞

and, similarly, t t d e −β(t−s ) g(z(s ))ds = −β e −β(t−s ) g(z(s ))ds + g(z(t)). dt −∞ −∞ (3.B14) From (3.B12), (3.B13) and (3.B14), we obtain t dy(t) αβ e −β(t−s ) g(z(s ))ds = β dt β −α −∞ t −α(t−s ) −α e g(z(s ))ds . −∞

167

(3.B15)


Differentiating (3.B15) and using (3.B13) and (3.B14), we obtain t d 2 y(t) αβ = e −α(t−s ) g(z(s ))ds α2 dt 2 β −α −∞ t e −β(t−s ) g(z(s ))ds + αβg(z(t)). − β2

(3.B16)

−∞

From (3.B11), (3.B15) and (3.B16), we obtain d 2 y(t) dy(t) + αβy(t) = αβg(z(t)), + (α + β) dt 2 dt which shows that y(t) as defined by (3.B9) and (3.B10) is the solution of (3.B8) and, hence, of (3.B6). It follows that y(t) depends on z(t) through the distributed time lag relation (3.B9), the density function φ(r ) of the lag distribution being given by (3.B10). The mean of this distribution (the mean time lag) is given by

∞

r φ(r )dr =

0

= = = =

∞ ∞ 1 −αr −βr r αe dr − α rβe dr , β β −α 0 0 1 β α − , β −α α β 2 β − α2 1 , β −α αβ β +α , αβ 1 1 + . (3.B17) α β

168

Model Specification

The mode of the lag distribution (the modal time lag) is given by arg maxφ(r ) =

log β − log α . β −α

(3.B18)

To prove (3.B18), we note that argmaxφ(r ) is the value of r that satisfies the condition dφ(r ) = 0. dr

(3.B19)

It follows from (3.B10) that (3.B19) will be satisfied if and only if αe −αr = βe −βr ,

(3.B20)

log α − αr = log β − βr.

(3.B21)

and hence,

Solving (3.B21) for r, we obtain the expression on the righthand side of (3.B18). We shall deal, finally, with the case in which the roots of (3.B7) are equal, each being equal to −α. In this case, (3.B6) can be written d 2 y(t) dy(t) + 2α + α 2 y(t) = α 2 g(z(t)). dt 2 dt

169

(3.B22)


We shall now show that the solution of (3.B22) is given by ∞ y(t) = ψ(r )g(z(t − r ))dr, (3.B23) 0

where ψ(r ) = α 2 r e −αr .

(3.B24)

Substituting s = (t − r ), we obtain from (3.B23) and (3.B24) y(t) =

t

−∞

α 2 (t − s )e −α(t−s ) g(z(s ))ds

t te −αt e αs g(z(s ))ds − α 2 s e −αt e αs g(z(s ))ds −∞ −∞ t t e αs g(z(s ))ds − α 2 e −αt s e αs g(z(s ))ds. = α 2 te −αt = α2

t

−∞

−∞

(3.B25)

Then, differentiating (3.B25), we obtain t dy(t) e αs g(z(s ))ds {α 2 e −αt − α 3 te −αt } = dt −∞ t s e αs g(z(s ))ds {α 3 e −αt } + −∞ 2

+ α tg(z(t)) − α 2 tg(z(t)) t e αs g(z(s ))ds {α 2 e −αt − α 3 te −αt } = −∞ t s e αs g(z(s ))ds {α 3 e −αt }. (3.B26) + −∞

170

Model Specification

Then, differentiating (3.B26), we obtain t d 2 y(t) = e αs g(z(s ))ds {−2α 3 e −αt + α 4 te −αt } dt 2 −∞ t s e αs g(z(s ))ds {−α 4 e −αt } + −∞

+ α 2 g(z(t)) − α 3 tg(z(t)) + α 3 tg(z(t)) t e αs g(z(s ))ds {−2α 3 e −αt + α 4 te −αt } = −∞ t s e αs g(z(s ))ds {−α 4 e −αt } + α 2 g(z(t)). + −∞

(3.B27)

From (3.B25), (3.B26) and (3.B27), we obtain dy(t) d 2 y(t) + α 2 y(t) = α 2 g(z(t)), + 2α dt 2 dt which shows that y(t), as defined (3.B23) and (3.B24), is the solution of (3.B22) and, hence, of (3.B6). It follows that y(t) depends on z(t) through the distributed time lag relation (3.B23), the density function ψ(r ) of the lag distribution being given by (3.B24). The distribution with density function given by (3.B24) is a special case of the Gamma distribution. The density function f (x) of the Gamma distribution, in the general case, is given by f (x) =

α λ λ−1 −αx x e , (λ)

171


where α and λ are parameters and (λ) is the Gamma function defined by

(λ) =

∞

x λ−1 e −x d x.

0

In the special case where λ = 2, we obtain the density function defined by (3.B24). The mean of the Gamma distribution is λ/α, and hence the mean of the distributed lag defined by (3.B24) is 2/α. Moreover, it is easily verified that the mode of the latter distribution is 1/α.

172

CHAPTER FOUR

Steady State and Stability Analysis 4.1 Introduction

A

n important feature of the continuous time macroeconometric model specified in this monograph is that,

like the model of Bergstrom and Wymer [1976] and the many models for which that model has served as a prototype, it is designed in such a way as to permit a rigorous mathematical analysis of its steady state and stability properties, thus providing an important check on its capacity to generate plausible long-run behaviour. Such a steady state and stability analysis is carried out in this chapter. In Section 4.2, we show that, when the exogenous variables satisfy certain conditions and the stochastic innovations have their expected value zero, the model has a steady state solution in which all variables grow at constant exponential rates

173


(in some cases zero). Closed formulae from which the exact steady state paths of the variables can be computed are derived in Appendix A. In Section 4.3 and Appendix 4.3, we derive, from the original model, an explicit differential equation system in the logarithms of the ratios of the variables to their steady state paths and show that the nonlinear ´ part of this system satisfies the Poincare–Liapounov–Perron conditions (see Bellman [1953, p. 93] and Coddington and Levinson [1955, p. 314]). The latter conditions ensure that the asymptotic stability of the steady state solution derived in Section 4.2 depends on the eigenvalues of the linear part of this system; that is it depends on the eigenvalues of the matrix of coefficients of the linear part of this system when it is written in the form of a first-order system of higher dimension. Section 4.4 reviews some previous work on fiscal and monetary policy in the Bergstrom, Nowman and Wymer [1992] model and results concerning bifurcation analysis. Section 4.5 concludes. The results obtained in this chapter will be applied to the empirical estimates obtained from UK data in Chapter 5. There we shall derive the steady state solution of the estimated model and investigate the stability and cyclical properties of the estimated model in the neighbourhood of the steady state.

174


4.2 The Steady State The steady state solution of the model is derived under the assumptions represented by (4.1)–(4.17). B = B ∗ e λ4 t ,

(4.1)

dx = 0,

(4.2)

E 0 = E 0∗ e (λ1 +λ2 )t ,

(4.3)

Gc = g ∗ (Q + P ),

(4.4)

K p = K ∗p e (λ1 +λ2 )t ,

(4.5)

M = M ∗ e λ4 t ,

(4.6)

pf = pf∗ e λ5 t ,

(4.7)

pi = pi∗ e λ5 t = pf∗ e λ5 t ,

(4.8)

rf = rf∗ ,

(4.9)

T1 = T1∗ ,

(4.10)

T2 = T2∗ ,

(4.11)

!

Yf =

Y ∗f e

λ1 +λ2 β23

175

" t

,

(4.12)


β32 =

q ∗ p∗ pf∗

(4.13)

β33 = r ∗ ,

(4.14)

µ1 (0) = 0,

(4.15)

µ2 (0) = 0,

(4.16)

µ3 (0) = 0,

(4.17)

where B ∗, E 0∗ , g ∗, K ∗p , M ∗, pf∗ , pi∗ , rf∗ , T1∗ , T2∗ , Y ∗f , λ4 and λ5 are given constants, while q ∗ , p ∗ and r ∗ are functions of the parameters of the model given by the long-run rational expectations steady state solution. The assumptions concerning β32 and β33 can be interpreted as long-run rational expectations in the foreign exchange markets. It can easily be verified that, under the above assumptions, the system of (3.1)–(3.18) has a steady state solution of the form: C = C ∗ e (λ1 +λ2 )t ,

(4.18)

E n = E n∗ e (λ1 +λ2 )t ,

(4.19)

F = F ∗ e (λ1 +λ2 )t ,

(4.20)

I = I ∗ e (λ1 +λ2 )t ,

(4.21)

176


K = K ∗ e (λ1 +λ2 )t ,

(4.22)

K a = K a∗ e (λ1 +λ2 )t ,

(4.23)

K h = K h∗ e (λ1 +λ2 )t ,

(4.24)

L = L ∗ e λ2 t ,

(4.25)

p = p∗ e (λ3 +λ4 −λ1 −λ2 )t ,

(4.26)

P = P ∗ e (λ1 +λ2 )t ,

(4.27)

q = q ∗ e (λ1 +λ2 +λ5 −λ3 −λ4 )t ,

(4.28)

Q = Q ∗ e (λ1 +λ2 )t ,

(4.29)

r = r ∗,

(4.30)

S = S ∗ e (λ1 +λ2 )t ,

(4.31)

w = w ∗ e (λ3 +λ4 −λ2 )t ,

(4.32)

µ1 = λ1 t,

(4.33)

µ2 = λ2 t,

(4.34)

µ3 = λ3 t,

(4.35)

177


where C ∗, E n∗ , F ∗, I ∗, K ∗, K a∗ , K h∗ , L ∗, p∗, P ∗, q ∗, Q ∗ , r ∗ , S ∗ and w ∗ are constants (hereafter referred to as steady state level parameters) satisfying the nonlinear simultaneous equation system (4.36)–(4.50). ∗

C = β1 e

Q∗ + P∗ , T1∗ −1/β6 − β5 (K ∗ )−β6 ,

{(β2 −β3 )(λ3 +λ4 −λ1 −λ2 )−β2 r ∗ }

L ∗ = β4 (Q ∗ )−β6

(4.36) (4.37)

Q∗ + P∗ , (4.38) = β7 e T1∗ ∗ 1+β6 Q ∗ , (4.39) r = β10 (λ3 + λ4 − λ1 − λ2 ) − β11 + β5 K∗   β 1 − β12 q ∗ p ∗ /pi∗ 13 {1 + β14 (λ1 + λ2 )}    × C ∗ +G∗ +(λ1 +λ2 ) K ∗ + K ∗ + K ∗ + E ∗ + E ∗  p n o c   h γ8 log   ∗   Q   K h∗

   + γ9 log  

{(β8 −β9 )(λ3 +λ4 −λ1 −λ2 )−β8 r ∗ }

 β14 C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p  +E n∗ + E o∗   = 0,  S∗

p∗ = β15 β4 T2∗ w ∗ {1 − β5 (Q ∗ /K ∗ )β6 }−(1+β6 )/β6 , ∗ −β6

(Q )

∗ −β6

− β5 (K )

=

178

β16 β4

(4.40) (4.41)

−β6 ,

(4.42)


r ∗ = β18 rf∗ + β20 { p∗ (Q ∗ + P ∗ )/M ∗ } + β21 (B ∗ /M ∗ ) + β17 + β19 (λ3 + λ4 − λ1 − λ2 − λ5 ),

(4.43)

β  {β12 q ∗ p ∗ /pi∗ 13 }{1 + β14 (λ1 + λ2 )}  × C ∗ +G∗ +(λ1 +λ2 ) K ∗ + K ∗ + K ∗ + E ∗ + E ∗  p n o  c  ∗ h γ17 log   ∗ ∗ ∗   pi /q p I 

β14 C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p  +E ∗ + E ∗  n o + γ18 log   S∗ 

    = 0,  (4.44)

E n∗ β25

=

β β22 Y ∗f 23

Q∗ + P∗ F∗

=

pf∗ q ∗ p∗

β24 ,

γ20 + λ1 + λ2 , γ20

! " K ∗ γ21 + λ1 + λ2 a = , β26 + β27 rf∗ − λ5 P∗ γ21

(4.45)

(4.46)

(4.47)

! " Q∗ + P∗ ∗ ∗ β28 + β29 rf − r + β30 (λ3 + λ4 − λ1 − λ2 − λ5 ) K a∗ 2 γ23 + (λ1 + λ2 ) + γ22 (λ1 + λ2 ) = , (4.48) γ23

179


(The last equation is obtained by using the relation D 2K a /K a = D 2 logK a + (DlogK a )2 = (DlogK a )2 when K a = K a∗ e (λ1 +λ2 )t ).

pi∗ I ∗ = E n∗ + E o∗ + P ∗ − F ∗ − (λ1 + λ2 )K a∗ , q ∗ p∗

(4.49)

p∗ I ∗ (λ1 + λ2 )S ∗ = Q ∗ + ∗i ∗ − C ∗ + G∗c + (λ1 + λ2 ) q p × K ∗ + K h∗ + K ∗p + E n∗ + E o∗ . (4.50)

A procedure for solving the equation system (4.36)–(4.50) is derived in Appendix A. The first step is to derive from (4.36)–(4.50) a pair of simultaneous equations (4.A12) and (4.A19) in the steady state interest rate r ∗ and the steady state real exchange rate x ∗ = ing

pi∗

=

p ∗f ).

q ∗ p∗ p ∗f

=

q ∗ p∗ pi∗

(implicitly, assum-

These must be solved, numerically, for r ∗ and

x ∗ . Then, the remaining steady state level parameters can be computed, recursively, from a sequence of closed formulae set out in Appendix A.

4.3 Stability Analysis The stability analysis is based on a transformed model in the variables y1 (t), y2 (t), . . . , y15 (t), defined as follows.

180


y1 (t) = y3 (t) = y5 (t) = y7 (t) = y9 (t) = y11 (t) = y13 (t) = y15 (t) =

C (t) L(t) log (t) = log , y , 2 C ∗ e (λ1 +λ2 )t L ∗ e λ2 t K h (t) K (t) , y4 (t) = log , log K h∗ e (λ1 +λ2 )t K ∗ e (λ1 +λ2 )t Q (t) p(t) , y , log (t) = log 6 Q ∗ e (λ1 +λ2 )t p ∗ e (λ3 +λ4 −λ1 −λ2 )t w(t) , y8 (t) = r (t) − r ∗ , log w ∗ e (λ3 +λ4 −λ2 )t I (t) E n (t) log ∗ (λ +λ )t , y10 (t) = log , I e 1 2 E n∗ e (λ1 +λ2 )t F (t) P (t) , y , log (t) = log 12 F ∗ e (λ1 +λ2 )t P ∗ e (λ1 +λ2 )t K a (t) q(t) , y , log (t) = log 14 K a∗ e (λ1 +λ2 )t q ∗ e (λ1 +λ2 +λ5 −λ3 −λ4 )t S(t) log ∗ (λ +λ )t . S e 1 2

The transformed model is derived from the original model (3.1)–(3.18) in Appendix B and comprises the system of nonlinear differential equations (4.51)–(4.65). D y1 = γ1 {log(Q ∗ e λ5 + P ∗ e y12 ) − log(Q ∗ + P ∗ ) − β2 y8 + (β2 − β3 )D y6 − y1 }, (4.51) 1 log{(Q ∗ )−β6 − β5 (K ∗ )−β6 } D 2y2 = −γ2 D y2 + γ3 β6 1 ∗ y5 −β6 ∗ y4 −β6 − log{(Q e ) − β5 (K e ) } − y2 , (4.52) β6

181


D 2y3 = −γ4 D y3 + γ5 {log(Q ∗ e λ5 + P ∗ e y12 ) − log(Q ∗ + P ∗ ) − β8 y8 + (β8 − β9 )D y6 − y3 },

(4.53)

D 2y4 = −γ6 D y4 + γ7 [(1 + β6 ) (y5 − y4 ) + log{r ∗ − β10 (λ3 + λ4 − λ1 − λ2 ) + β11 } − log{y8 + r ∗ − β10 (Dλ6 + λ3 + λ4 − λ1 − λ2 ) + β11 }],

∗ ∗ y6 +y14

(4.54)

β13

q p e pi∗ ∗ ∗ β13 q p − log 1 − β12 pi∗ + log C∗ e y1 + G∗c + K ∗ e y4 (D y4 + λ1 + λ2 )

D y5 = γ8 log 1 − β12

+ K h∗ e y3 (D y3 + y1 + y2 ) +K ∗p (λ1 + λ2 ) + E n∗ e y10 + E o∗ − log C ∗ +G∗c +(λ1 +λ2 )× K ∗ + K h∗ + K ∗p + E n∗ + E o∗ −y5 + γ9 log C∗ e y1 + G∗c + K ∗ e y4 (D y4 + λ1 + λ2 ) + K h∗ e y3 (D y3 + y1 + y2 )+ K ∗p (λ1 + λ2 )+ E n∗ e y10 + E o∗ − log C ∗ +G∗c +(λ1 +λ2 ) K ∗ + K h∗ + K ∗p + E n∗ + E o∗ −y15 , (4.55) 1 + β 6 D 2y6 = γ10 (D y7 − D y6 ) + γ11 y7 − y6 − β6 ∗ y5 β6 1 + β6 Q e × log 1 − β5 + K ∗ e y4 β6 ∗ β6 Q × log 1 − β5 , (4.56) K∗

182


D 2y7 = γ12 (D y6 − D y7 ) − γ13 (D y6 + D y14 ) 1 β6 log (Q ∗ )−β6 − β5 (K ∗ )− + γ14 β6 1 ∗ y5 −β6 ∗ y4 −β6 − log{(Q e ) − β5 (K e ) } , (4.57) β6

p ∗ e y6 (Q ∗ e y5 + P ∗ e y12 ) D y8 = −γ15 D y8 + γ16 β20 M∗ ∗ ∗ ∗ p (Q + P ) − β20 −β (4.58) D y − y 19 14 8 , M∗ 2

D y9 = γ17 (1 + β13 )(y6 + y14 ) − y9 + log C∗ e y1 + G∗c + K ∗ e y4 (D y4 + λ1 + λ2 )

+ K h∗ e y3 (D y3 + y1 + y2 ) + K ∗p (λ1 + λ2 ) + E n∗ e y10 + E o∗ − log C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p + E n∗ + E o∗ + γ18 log C∗ e y1 + G∗c + K ∗ e y4 (D y4 + λ1 + λ2 ) + K h∗ e y3 (D y3 + y1 + y2 ) + K ∗p (λ1 + λ2 ) + E n∗ e y10 + E o∗ − log C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p (4.59) + E n∗ + E o∗ − y15 , D y10 = −γ19 [β24 (y6 + y14 ) + y10 ],

D y11 = γ20 β25

Q ∗ e y5 + P ∗ e y12 F ∗ e y11

183

−

Q∗ + P∗ F∗

(4.60) ,

(4.61)


! " K ∗ e y13 K a∗ a D y12 = γ21 β26 + β27 rf∗ − λ5 − , P ∗ e y12 P∗

(4.62)

D 2y13 = −{γ22 + 2(λ1 + λ2 )}D y13 − (D y13 )2 ! " + γ23 β28 + β29 rf∗ − r ∗ − y8 − β30 (D y14 + λ1 + λ2 + λ5 − λ3 − λ4 ) ∗ y5 ! " Q e + P ∗ e y12 − β28 + β29 rf∗ − r ∗ × ∗ y K a e 13 ∗ Q + P∗ − β30 (λ1 + λ2 + λ5 − λ3 − λ4 ) × , K a∗ (4.63) D 2y14 = − γ24 (D y6 + D y14 ) − γ25 (y6 + y14 ) + γ26 y8 + γ27 log E n∗ e y10 + E o∗ + P ∗ e y12 − F ∗ e y11 − K ∗ e y13 (D y13 + λ1 + λ2 ) − log E n∗ + E o∗ + P ∗ − F ∗ − (λ1 + λ2 )K a∗ (4.64) + y6 + y14 − y9 , D y15 = Q ∗ e y5 + pi∗ /q ∗ p ∗ I ∗ e y9 −y6 −y14 − C ∗ e y1 − G∗c − K ∗ e y4 (D y4 + y1 + y2 ) − K h∗ e y3 (D y3 + y1 + y2 ) − K ∗p (λ1 +λ2 )− E n∗ e y10 − E o∗ /S ∗ e y15 − Q ∗ + pi∗ /q ∗ p ∗ I ∗ − C ∗ − G∗c − (λ1 + λ2 ) K ∗ + K h∗ + K ∗p (4.65) − E n∗ − E o∗ /S ∗ .

184


It can, easily, be verified that the system (4.51)–(4.65) has a steady state solution yi (t) = 0

(i = 1, . . . , 15).

(4.66)

Moreover, it is clear from the definitions of the variables yi (t) = 0(i = 1, . . . , 15) that the solution (4.66) corresponds to the steady state solution of the original model. For, it implies that the variables C(t), E n (t), . . . , w(t) satisfy (4.18)– (4.32). Next, for each of equations (4.51)–(4.65), we shall express the function on the right-hand side of the equation as the sum of two terms, the first being a linear function of the variables y1 , y2 , . . . , y15 and the second a nonlinear function of lower order of smallness as yi → 0(i = 1, . . . , 15). For this purpose, we define y = [y1 , y2 , . . . , y15 ] and |y| = 15 i=1 |yi |. Then, by a Taylor series expansion to the first order, about the origin, of the functions on the righthand side of (4.51)–(4.65), we obtain the system (4.67)– (4.81), in which the functions f i (y)(i = 1, . . . , 15) satisfy ´ the Poincare–Liapounov–Perron conditions f i (y)/|y| → 0 as |y| → 0 (i = 1, . . . , 15).

D y1 = −γ1 y1 − γ1 β2 y8 − γ1 (β3 − β2 )D y6 + γ1 P ∗ + y12 + f 1 (y), Q∗ + P∗

185

γ1 Q ∗ Q∗ + P∗

y5 (4.67)


γ3 (Q ∗ )−β6 y5 D 2y2 = − γ2 D y2 − γ3 y2 + (Q ∗ )−β6 − β5 (K ∗ )−β6 γ3 β5 (K ∗ )−β6 y4 + f 2 (y), − (4.68) (Q ∗ )−β6 − β5 (K ∗ )−β6 D 2y3 = −γ4 D y3 + γ5 (β8 − β9 )D y6 − γ5 y3 − γ5 β8 y8 γ5 Q ∗ γ5 P ∗ y y12 + f 3 (y), (4.69) + + 5 Q∗ + P∗ Q∗ + P∗ D 2y4 = − γ6 D y4 +

γ7 β10 r ∗ − β10 (λ3 + λ4 − λ1 − λ2 ) + β11

D y6

− γ7 (1 + β6 )y4 + γ7 (1 + β6 )y5 γ7 y8 + f 4 (y), − r ∗ − β10 (λ3 + λ4 − λ1 − λ2 ) + β11 (4.70) γ8 β12 β13 (x ∗ )β13 D y5 = − γ5 y5 −γ9 y15 − (y6 + y14 ) + (γ8 + γ9 ) 1 − β12 (x ∗ )β13  ∗  C y1 + (λ1 + λ2 )K ∗ y4 + K ∗ D y4 + (λ1 + λ2 )K h∗ y3         + K h∗ D y3 + E n∗ y10 ×   C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + E n∗ + E o∗       ∗ + (λ1 + λ2 )K p + f 5 (y), (4.71) D 2y6 = −γ10 (D y6 − D y7 ) − γ11 (y6 − y7 )  ! ∗ "β6     γ11 β5 (1 + β6 ) KQ ∗  + (y5 − y4 ) + f 6 (y), ∗ β6     1 − β5 KQ ∗

186

(4.72)


D 2y7 = γ12 (D y6 − D y7 ) − γ13 (D y6 + D y14 ) ∗ −β6 (Q ) y5 − β5 (K ∗ )−β6 y4 + γ14 + f 7 (y), (Q ∗ )−β6 − β5 (K ∗ )−β6

(4.73)

D 2y8 = −γ15 D y8 − γ16 β19 D y14 − γ16 y8 ∗ ∗ ∗ ∗ p (Q + P ∗ ) p Q y6 + γ16 β20 y5 + γ16 β20 ∗ M M∗ ∗ ∗ p Q y12 + f 8 (y), + γ16 β20 (4.74) M∗ D y9 = −γ17 y9 − γ18 y15 + γ17 (1 + β13 )(y6 + y14 ) + (γ17 + γ18 )   ∗ C y1 + (λ1 + λ2 )K ∗ y4 + K ∗ D y4 + (λ1 + λ2 )K h∗ y3         + K h∗ D y3 + E n∗ y10 ×  C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p + E n∗ + E o∗        + f 9 (y),

(4.75)

D y10 = −γ19 y10 − γ19 β24 (y6 + y14 ),

D y11

∗ Q∗ P y5 + γ20 β25 y12 = γ20 β25 F∗ F∗ ∗ Q + P∗ y11 + f 11 (y), − γ20 β25 F∗

(4.76)

(4.77)

! " K ∗ a (y13 − y12 ) + f 12 (y), D y12 = γ21 β26 + β27 rf∗ − λ5 P∗ (4.78)

187


2

D y13

Q∗ + P∗ y8 = −{γ22 + 2(λ1 + λ2 )}D y13 − γ23 β29 K a∗ ∗ ! " Q + P∗ ∗ ∗ D y β r − γ23 β30 + γ + β − r 14 23 28 29 f K a∗ ∗ Q + β30 (λ3 + λ4 − λ1 − λ2 − λ5 )} y5 K a∗ ! " + γ23 β28 + β29 rf∗ − r ∗ + β30 (λ3 + λ4 − λ1 ! " P∗ ∗ ∗ β r − λ2 − λ5 ) − γ + β − r y 12 23 28 29 f K a∗ + β30 (λ3 + λ4 − λ1 − λ2 − λ5 ) ∗ Q + P∗ × y13 + f 13 (y), (4.79) K a∗

D 2y14 = −γ24 (D y6 + D y14 ) − γ25 (y6 + y14 ) + γ26 y8 + γ27 (y6 + y14 − y9 )  ∗ E n y10 + P ∗ y12 − F ∗ y11 − (λ1 + λ2 )K a∗ y13     −K ∗ D y13 a + γ27  E n∗ + E o∗ + P ∗ − F ∗ − (λ1 + λ2 )K a∗    + f 14 (y),

   

(4.80)

∗ ∗ ∗ pi I Q∗ C y5 + (y9 − y6 − y14 ) − y1 = S∗ q ∗ p∗ S ∗ S∗ (λ1 + λ2 )K h∗ (λ1 + λ2 )K ∗ K∗ − y4 − D y4 − y3 S∗ S∗ S∗

D y15

    

188


−

K h∗ S∗

D y3 −

E n∗ S∗

y10

  p∗ I ∗  Q ∗ + q ∗i p∗ − C ∗ − G∗c − (λ1 + λ2 )         × K ∗ + K ∗ + K ∗ − E ∗ − E ∗  p n o h y15 + f 15 (y). −   S∗         (4.81)

Finally, we represent the mixed-order system (4.67)–(4.81) as a first-order system Du = C u + φ(u),

(4.82)

where C is a 23 × 23 matrix, whose elements are functions of the parameters of the model (3.1)–(3.18) and u = [u1 , u2 , . . . , u23 ], φ = [φ1 , φ2 , . . . , φ23 ], ui (i = 1, . . . , 23) and φi (i = 1, . . . , 23) are defined as follows. u1 (t) = y1 (t), u2 (t) = y5 (t), u3 (t) = y9 (t), u4 (t) = y10 (t), u5 (t) = y11 (t), u6 (t) = y12 (t), u7 (t) = y15 (t), u8 (t) = y2 (t), u9 (t) = y3 (t), u10 (t) = y4 (t), u11 (t) = y6 (t), u12 (t) = y7 (t), u13 (t) = y8 (t), u14 (t) = y13 (t), u15 (t) = y14 (t), u16 (t) = D y2 (t), u17 (t) = D y3 (t), u18 (t) = D y4 (t), u19 (t) = D y6 (t), u20 (t) = D y7 (t), u21 (t) = D y8 (t), u22 (t) = D y13 (t), u23 (t) = D y14 (t), φ1 (u) = f 1 (y), φ2 (u) = f 5 (y), φ3 (u) = f 9 (y), φ4 (u) = f 10 (y),

189


φ5 (u) = f 11 (y), φ6 (u) = f 12 (y), φ7 (u) = f 15 (y), φi (u) = 0(i = 8, . . . , 15), φ16 (u) = f 2 (y), φ17 (u) = f 3 (y), φ18 (u) = f 4 (y), φ19 (u) = f6 (y), φ20 (u) = f 7 (y), φ21 (u) = f 8 (y), φ22 (u) = f 13 (y), φ23 (u) = f 14 (y).

It should be noted that the ordering of the variables ui (i = 1, . . . , 23) differs from that of the variables yi to which they correspond. For example, u2 = y5 (not y2 ). The variables u1 , u2 , . . . , u7 are transformations of the variables in the original model (3.1)–(3.18) that adjust through a first-order differential equation, the variables u8 , u9 , . . . , u15 are transformations of the variables that adjust through a second-order differential equation, and the variables u16 , u17 , . . . , u23 are the rates of change of the variables u8 , u9 , . . . , u15 . Similarly, the ordering of the functions φi (i = 1, . . . , 23) differs from that of the functions f i to which they correspond. It should be noted, also, that each of the rows 8–15 of the matrix C has one element equal to 1 and all other elements equal to 0. This is because rows 8–15 contain the coefficients of the identity relations Dui = ui+8

(i = 8, . . . , 15).

190


It is for the same reason that φi (u) = 0 (i = 8, . . . , 15). Since ui = 0 (i = 1, . . . , 23) implies that yi = 0 (i = 1, . . . , 15) and the system (4.51)–(4.65) has a zero solution, the system (4.82) has a zero solution u(t) = [0, . . . , 0] . The latter solution is said to be asymptotically stable if there exists a δ > 0 such that, for any solution u(t) of the system (4.82) satisfying the condition |u(0)| < δ, |u(0)| → 0 as t → ∞ (see Coddington and Levinson [1955, p. 314]). As was mentioned earlier, the properties of the higher order terms in the Taylor series expansion imply that ´ the functions f i (y)(i = 1, . . . , 15) satisfy the Poincare– Liapounov–Perron

conditions

f i (y)/|y| → 0

as

|y| →

0 (i = 1, . . . , 15). Hence, the functions φi (u) (i = 1, . . . , 23) must, also, satisfy these conditions: that is φi (u)/|u| → 0 as |u| → 0 (i = 1, . . . , 23). (The conditions are, trivially, satisfied from the functions φ8 (u), . . . , φ15 (u), which are equal to zero.) It follows, therefore, by a theorem of Perron (see Coddington and Levinson [1955, p. 314]), and by earlier results of Poincare´ and Liapounov (see Bellman [1953, p. 93]), that the zero solution of the system (4.82) is asymptotically stable if the eigenvalues of C all have negative real parts. If the eigenvalues of C all have negative real parts, we shall, also, describe the steady state solution (4.18)–(4.35)

191


of the original model (3.1)–(3.18) as asymptotically stable. For, convergence of the variables ui (t) (i = 1, . . . , 23) to zero implies that the proportional deviations of the variables C(t), E n (t), . . . , w(t) from their steady state paths converge to zero. The steady state level parameters are of interest in themselves, and, assuming they have, already, been computed, the additional cost of computing the unknown elements of C and the eigenvalues of this matrix is comparatively light. Finally, it should be noted that the eigenvalues of C provide information not only about the stability of the steady state solution but also about the cyclical properties of a solution of the model in the neighbourhood of the steady state.

4.4 Stability and Bifurcations In this section we discuss some work on stability and bifurcation analysis, using the Bergstrom, Nowman and Wymer [1992] model. As the Bergstrom, Nowman and Wymer [1992] model’s steady state computed from the estimated parameters was slightly unstable (although, statistically they could not reject the hypothesis that it is stable) the investigation of methods to reduce or eliminate the instability

192


was undertaken by Bergstrom, Nowman and Wandasiewicz [1994]. They investigated the use of a simple fiscal policy feedback and the use of complicated control theory considered in Bergstrom [1987] that could reduce or eliminate the instability. The simple fiscal policy feedback was introduced by Bergstrom [1984b] in which tax rates are reduced and (or) transfer payments, such as social security benefits, increased during a recession and vice versa during a boom. Simple Keynesian analysis suggests that such a policy is likely to have a stabilizing influence. The policy feedback used was specified as follows: DlogT1 = γ β log

Q Q ∗ e (λ1 +λ2 )t

− log

T1 T1∗

.

It allows at each point in time the value of the fiscal policy variable T1 to adjust towards a partial equilibrium level that is an increasing function of the ratio of output to its steady state level. The parameters can be interpreted as follows: β measures the strength of the feedback, γ the speed of adjustment parameter that reflects delays in obtaining new information about the level of output (and administrative delays in changing tax and benefit rates) and T1∗ is the steady state level of the fiscal policy variable. They also considered

193


the limiting case of γ → ∞ in which case the feedback relation becomes instantaneous. Bergstrom, Nowman and Wandasiewicz [1994] found that this simple fiscal policy feedback in cases of either instantaneously or with an exponential time lag had no definite stabilizing influence on the dynamic behaviour of the model. In Bergstrom, Nowman and Wandasiewicz [1994] they also investigated stability analysis, using the more advanced approach of optimal control theory. They proved the existence of stabilizing feedbacks involving the fiscal policy variable and the money supply as combined policy instruments. They also derived optimal feedbacks by minimizing an infinite horizon quadratic cost function involving deviations of output, the exchange rate and the monetary and fiscal policy instruments from their steady state paths (and also, the rates of change of the policy instruments). One of the findings was that fiscal policy had an important role to play in the stabilization of the model than does monetary policy (see Bergstrom, Nowman and Wandasiewicz [1994] for full details). In a more recent study, Bergstrom [2001] also investigated the stability of the Bergstrom, Nowman and Wymer [1992] model, using the estimates of the model in Nowman [1996] by looking at the wage-acceleration parameter that determines the effect of labour demand

194


on wage acceleration in the model. It was shown that there was a bounded range of positive values of the wage acceleration parameter in which the steady state was asymptotically stable. We finally discuss a series of important papers by Barnett and He [1998, 1999, 2002] (see references in these articles for related literature). In Barnett and He [1999] they investigated bifurcations in the Bergstrom, Nowman and Wymer [1992] model. Using the estimates of the Bergstrom, Nowman and Wymer [1992] model, they used the gradient method to find a set of parameters where all the eigenvalues had strictly negative real parts and the system is stable. They then developed a numerical algorithm to obtain stability boundaries and applied it to particular cases of parameters of the model. Two types of boundaries were considered, which could occur according to the way unstable eigenvalues are created. They found that the Bergstrom, Nowman and Wymer [1992] model displayed an interesting set of bifurcations that included transcritical bifurcations, Hopf bifurcations and the important class of co-dimension two bifurcations was confirmed. In Barnett and He [1998], which is a continuation of the research in Barnett and He [1999], they investigated the dynamic behaviour on bifurcation boundaries in the Bergstrom,

195


Nowman and Wymer [1992] model and found that the model in certain cases is unstable on bifurcation boundaries. They also derived a new formula for determining the analytical forms of the transcritical bifurcation boundaries and investigated the statistical significance of the instability inference. Barnett and He [1998, 2002] also investigated the use of the simple ad hoc fiscal policy feedback rule used in Bergstrom, Nowman and Wandasiewicz [1994] and found that the model’s stable region became smaller and argued that the policy may be counter-productive. See Barnett and He [1998, 1999, 2002] for further discussions on the issues of the Lucas critique, time inconsistency (see Kydland and Prescott [1977]), use of control theory and a number of interesting numerical examples and figures illustrating bifurcation boundaries. Barnett and He [2001] try to resolve some controversies with respect to nonlinearity, chaos and bifurcation in macroeconomic models and also used the Bergstrom, Nowman and Wymer [1992] model as part of their analysis. Finally a more recent series of interesting papers by Barnett and He [2004, 2005a, b, 2006] investigates singularity-induced bifurcations that they introduce into economics and its relevance to macroeconometric models.

196


4.5 Conclusion We have shown, in this chapter, that, under certain assumptions concerning the paths of the exogenous variables, the continuous time macroeconometric model specified in Chapter 3 has a steady state solution in which all variables grow at constant proportional rates (in some cases, zero). We have, also, derived sufficient conditions for the asymptotic stability of the steady state solution in terms of the coefficients of the linear part of a transformed model. We now turn to some empirical results from UK data in the next chapter.

Appendix A: Steady State Level Parameters In this Appendix, we derive a computational procedure for solving the simultaneous equation system (4.36)–(4.50) for the steady state level parameters. As was mentioned in Section 4.2, the first step in this computational procedure is to solve a pair of simultaneous equations in the steady state interest rate r ∗ and the steady state real exchange rate x ∗ . This pair of simultaneous equations, (4.A12) and (4.A19), is derived as follows.

197


From (4.47) and (4.48) we obtain P∗ = h(r ∗ ) Q∗ + P∗ where

h(r ∗ ) =

! " ! " γ21 γ23 β26 + β27 rf∗ − λ5 {β28 + β29 rf∗ − r ∗ + β30 (λ3 + λ4 − λ1 − λ2 − λ5 )} (γ21 + λ1 + λ2 ){γ23 + (λ1 + λ2 )2 + γ22 (λ1 + λ2 )}

.

Therefore, Q∗ + P ∗ 1 = . Q∗ 1 − h(r ∗ )

From (4.36) and (4.A1), we obtain ∗ C∗ 1 β1 e {(β2 −β3 )(λ3 +λ4 −λ1 −λ2 )−β2 r } , = Q∗ 1 − h(r ∗ ) T1∗

(4.A1)

(4.A2)

From (4.4) and (4.A1), G∗c g∗ , = Q∗ 1 − h(r ∗ )

and from (4.38) and (4.A1), ∗ K h∗ 1 β7 e {(β8 −β9 )(λ3 +λ4 −λ1 −λ2 )−β8 r } . = Q∗ 1 − h(r ∗ ) T1∗ From (4.39), we obtain ∗ r − β10 (λ3 + λ4 − λ1 − λ2 ) + β11 −1(1+β6 ) K∗ = . Q∗ β5

198

(4.A3)

(4.A4)

(4.A5)


Equations (4.40), (4.44) and (4.50) can be regarded as a set of simultaneous equations in Q ∗ , pi∗ I ∗ /q ∗ p ∗ and S ∗ , which can be solved to express each of these variables as a function of C ∗ , G∗c , K ∗ , K h∗ , K ∗p , E n∗ , E o∗ and q ∗ p ∗ /pi∗ . It can, easily, be verified, by substitution into (4.40), (4.44) and (4.50), that the resulting expressions for Q ∗ , pi∗ I ∗ /.q ∗ p ∗ and S ∗ are

∗ ∗ β13 p q {1 + β14 (λ1 + λ2 )} C ∗ + G∗c Q ∗ = 1 − β12 pi∗ + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p + E n∗ + E o∗ , (4.A6)

β13

{1 + β14 (λ1 + λ2 )} C ∗ + G∗c + (λ1 + λ2 ) (4.A7) × (K ∗ + K h∗ + K ∗p ) + E n∗ + E o∗ , S ∗ = β14 C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p + E n∗ + E o∗ .

pi∗ I ∗ = β12 q ∗ p∗

q ∗ p∗ pi∗

(4.A8)

From (4.42), we obtain ∗ −β6 K β16 −β6 ∗ −β6 (Q ) , 1 − β5 = Q∗ β4 and hence ∗

Q =

β16 β4

1 − β5

199

K∗ Q∗

−β6 1/β6

.

(4.A9)


From (4.A5) and (4.A9), we obtain   ∗ β6 /(1+β6 )  β16 − β (λ + λ − λ − λ ) r   10 3 4 1 2        + β11    β 16  ∗  , 1 − β5 Q =     β4  β5        (4.A10)

and from (4.A6)

C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p + E n∗ + E o∗ Q∗ 1 = . ! ∗ ∗ "β13 1 − β12 q pp∗ {1 + β14 (λ1 + λ2 )}

(4.A11)

i

Then, from (4.45), (4.A2), (4.A3), (4.A4), (4.A5), (4.A10) and (4.A11), we obtain φ(r ∗ , x ∗ ) = 0,

(4.A12)

where x∗ = and

q ∗ p∗ q ∗ p∗ = , pf∗ pi∗

β1 e (β2 −β3 )(λ3 +λ4 −λ1 −λ2 ) e −β2 r φ(r , x ) = g + T1∗ ∗ (λ1 + λ2 )β7 e (β8 −β9 )(λ3 +λ4 −λ1 −λ2 ) e −β8 r + T1∗ ∗

∗

1 1 − h(r ∗ )

∗

200

∗


−1/1+β6  ∗ r − β10 (λ3 + λ4 − λ1 − λ2 )         +β 11 + (λ1 + λ2 )   β5       ! "β23 (λ1 + λ2 )K ∗p + β22 Y ∗f (x ∗ )−β24 + E o∗ + ∗ β6 /(1+β6 ) β16 r −β10 (λ3 +λ4 −λ1 −λ2 )+β11 β16 1 − β5 β4 β5 −

1 . {1−β12 (x ∗ )β13 }{1 + β14 (λ1 + λ2 )}

From (4.A6) and (4.A7), we obtain pi∗ I ∗ β12 (x ∗ )β13 = , q ∗ p∗ Q ∗ 1 − β12 (x ∗ )β13

(4.A13)

P∗ h(r ∗ ) = , Q∗ 1 − h(r ∗ )

(4.A14)

from (4.A1),

from (4.47) and (4.A14),     h(r ∗ ) γ21 + λ1 + λ2 K a∗ ! " = ,  ∗ Q∗ 1 − h(r ∗ )  γ 21 β26 + β27 r − λ5

(4.A15)

f

from (4.46), F∗ β25 γ20 = , ∗ ∗ Q +P γ20 + λ1 + λ2

201

(4.A16)


from (4.A1) and (4.A16), F∗ 1 β25 γ20 , = Q∗ 1 − h(r ∗ ) γ20 + λ1 + λ2

(4.A17)

and from (4.49), pi∗ I ∗ E n∗ E o∗ F∗ P∗ (λ1 + λ2 )K a∗ + − − + − = 0. Q ∗ Q ∗ q ∗ p∗ Q ∗ Q∗ Q∗ Q∗

(4.A18)

Then, from (4.45), (4.A10), (4.A13), (4.A14), (4.A15), (4.A17) and (4.A18), we obtain ψ(r ∗ , x ∗ ) = 0

(4.A19)

where ! "β23 ∗ ∗ −β24 ∗ β22 Y f (x ) + Eo

ψ(r ∗ , x ∗ ) = β16 β4

1 − β5

r ∗ −β10 (λ3 +λ4 −λ1 −λ2 )+β11 β5

β6 /(1+β6 ) β16

β12 (x)β13 1 β25 γ20 − − 1−β12 (x ∗ )β13 1 − h(r ∗ ) γ20 + λ1 + λ2    (λ1 + λ2 )(γ21 + λ1 + λ2 )  h(r ∗ ) ! " . 1− +  1 − h(r ∗ )  β +β r∗ − λ γ 21

26

27

f

5

We, also, need a separate equation for p ∗ . This is derived as follows. From (4.43), we obtain

202


p∗ =

 ∗ r − β18rf∗ − β21 (B ∗ /M ∗ ) − β17 − β19 (λ3 + λ4 − λ1     − λ2 − λ5 )

    

   

   

β20

×

M∗ , P∗ + Q∗

(4.A20)

and from (4.A1) and (4.A20),  {1 − h(r ∗ )}{r ∗ − β18 rf∗ − β21 (B ∗ /M ∗ ) − β17     − β19 (λ3 + λ4 − λ1 − λ2 − λ5 )} p∗ =  β20   

     M∗ .  Q∗    (4.A21)

The above equations provide the basis for the computation of the steady state levels of the 15 endogenous variables.

Appendix B: Transformed Model In this Appendix, we derive the transformed model (4.51)– (4.65) in the variables y1 , . . . , y15 defined in Section 4.3. For this purpose, we assume, as in the derivation of the steady state solution, that the exogenous variables and trends satisfy (4.1), . . . ,(4.17). Equations (4.51), . . . , (4.65) are then derived as follows.

203


From (3.1), DlogC = λ1 + λ2 + γ1 logβ1 − β2 (r − Dlog p)

− β3 Dlog p + log(Q + P ) − logT1∗ − logC ,

and, hence, using the relations DlogC = D y1 + λ1 + λ2 , C = C ∗ e (λ1 +λ2 )t e y1 , Q = Q ∗ e (λ1 +λ2 )t e y5 , P = P ∗ e (λ1 +λ2 )t e y12 , r = r ∗ + y8 , Dlog p = D y6 + λ3 + λ4 − λ1 − λ2 ,

we obtain D y1 = γ1 logβ1 − β2 (y8 + r ∗ ) + (β2 − β3 )(D y6 + λ3 + λ4

− λ1 − λ2 )+log(Q ∗ e y5 + P ∗ e y12 ) − logT1∗ − log(C ∗ e y1 ) . (4.B1)

From (4.36), logC ∗ = logβ1 − β2r ∗ + (β2 − β3 )(λ3 + λ4 − λ1 − λ2 ) + log(Q ∗ + P ∗ ) − logT1∗ ,

and, from (4.B1) and (4.B2), we obtain (4.51).

204

(4.B2)


From (3.2) and (4.33) −β6 1 2 D y2 = −γ2 D y2 + γ3 logβ4 − λ1 t − log Q ∗ e (λ1 +λ2 )t e y5 β6 ∗ (λ +λ )t y −β6 ∗ λ2 t y2 1 2 4 − log(L e e ) , − β5 K e e 1 = − γ2 D y2 + γ3 logβ4 − log{(Q ∗ e y5 )−β6 β6 ∗ y4 −β6 ∗ y2 , (4.B3) − β5 (K e ) } − logL e from (4.37), logβ4 = logL ∗ +

1 log{(Q ∗ )−β6 − β5 (K ∗ )−β6 }, β6

(4.B4)

and, from (4.B3) and (4.B4), we obtain (4.52). From (3.3), D 2 logK h = γ4 (λ1 + λ2 − DlogK h ) + γ5 {logβ7 −β8 (r − Dlog p) − β9 Dlog p + log(Q + P ) − logT1∗ − logK h }, = − γ4 D y3 + γ5 logβ7 − β8 (y8 + r ∗ ) + (β8 − β9 )(D y6 + λ3 + λ4 − λ1 − λ2 )

+ log(Q ∗ e y5 + P ∗ e y12 ) − logT1∗ − log(K h∗ e y3 ) , (4.B5)

From (4.38), logK h∗ = logβ7 + (β8 − β9 )(λ3 + λ4 − λ1 − λ2 ) − β8 r ∗ + log(Q ∗ + P ∗ ) − logT1∗ ,


205

(4.B6)


From (3.4),

D y4 = −γ6 D y4 + γ7 logβ5 + (1 + β6 ) log 2

Q ∗ e y5 K ∗ e y4

− log{y8 + r ∗ − β10 (D y6 + λ3 + λ4 − λ1 − λ2 ) + β11 } = − γ6 D y4 +γ7 [logβ5 +(1+β6 )(logQ ∗ − logK ∗ + y5 − y4 ) − log{y8 + r ∗ − β10 (D y6 + λ3 + λ4 − λ1 − λ2 ) + β11 }], (4.B7)

from (4.39), logβ5 + (1 + β6 )(logQ ∗ − logK ∗ ) = log{r ∗ − β10 (λ3 + λ4 − λ1 − λ2 ) + β11 },

(4.B8)

and, from (4.B7) and (4.B8), we obtain (4.54). From (3.5),

∗ ∗ y6+ y14 β13 q p e + log{1+β14 (λ1 +λ2 )} D y5 = γ8 log 1−β12 pi∗ + log C ∗ e y1 + G∗c + K ∗ e y4 (D y4 + λ1 + λ2 ) + K h∗ e y3 (D y3 + λ1 + λ2 ) + K ∗p (λ1 + λ2 ) +E n∗ e y10 + E o∗ ∗ − logQ − y5 + γ9 logβ14 + log C ∗ e y1 + G∗c + K ∗ e y4 × (D y4 + λ1 + λ2 ) + K h∗ e y3 (D y3 + λ1 + λ2 ) + K ∗p (λ1 + λ2 ) + E n∗ e y10 + E o∗ − logS ∗ − y15 ,

206

(4.B9)


from (4.40),

γ8 log 1 − β12

q ∗ p∗ pi∗

β13

+ log{1 + β14 (λ1 + λ2 )}

+ log C ∗ + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p +E n∗ + E o∗ ∗ − logQ + γ9 logβ14 + log {C ∗ + G∗c + (λ1 + λ2 ) × K ∗ + K h∗ + K ∗p + E n∗ + E o∗ − logS ∗ = 0, (4.B10)


D y6 = γ10 (D y7 − D y6 ) + γ11 logβ15 + logβ4 + logT2∗ 2

+ logw + y7 −

1 + β6 β6

log 1 − β5

Q ∗ e y5 K ∗ e y4

β6

∗

− log p − y6 , (4.B11)

from (4.41), log p ∗ = logβ15 + logβ4 + logT2∗ + logw ∗ β6 Q 1 + β6 , (4.B12) log 1 − β5 − β6 K∗


207


From (3.7), D 2y7 = γ12 (D y6 − D y7 ) − γ13 (D y6 + D y14 ) 1 log{(Q ∗ e y5 )−β6 + γ14 logβ4 − logβ16 − β6 ∗ y4 −β6 − β5 (K e ) } , (4.B13)

from (4.42), logβ4 − logβ16 =

1 log{(Q ∗ )−β6 − β5 (K ∗ )−β6 }, β6

(4.B14)


D 2y8 = −γ15 D y8 + γ16 β17 + β18 rf∗ − β19 (D y14 + λ1 + λ2 + λ5 − λ3 − λ4 ) ∗ y6 ∗ y5 ∗ p e (Q e + P ∗ e y12 ) B ∗ +β12 − y8 −r , + β20 M∗ M∗ (4.B15)

from (4.43), r ∗ = β17 − β19 (λ1 + λ2 + λ5 − λ3 − λ4 ) + β8rf∗ ∗ ∗ ∗ p (Q + P ) B + β20 + β21 , (4.B16) M∗ M∗


208


From (3.9), ∗ ∗ q p D y9 = γ17 logβ12 + (1 + β13 ) log pi∗ + (1 + β13 )(y6 + y14 ) + log{1 + β14 (λ1 + λ2 )} + log C ∗ e y1 + G∗c + K ∗ e y4 (D y4 + λ1 + λ2 )

+ K ∗ e y3 (D y3 + λ1 + λ2 ) + K ∗p (λ1 + λ2 ) + E n∗ e y10 + E o∗ − logI ∗ − y9 + γ18 logβ14 + log C ∗ e y1 + G∗c + K ∗ e y4

× (D y4 + λ1 + λ2 ) + K h∗ e y3 (D y3 + λ1 + λ2 ) + K ∗p (λ1 + λ2 ) + E n∗ e y10 + E o∗ − logS ∗ − y15 , (4.B17)

from (4.44), ∗ ∗ q p + log{1 + β14 (λ1 + λ2 )} γ17 logβ12 + (1 + β13 ) log pi∗ ∗ + log C + G∗c + (λ1 + λ2 ) K ∗ + K h∗ + K ∗p + E n∗ + E o∗ − logI ∗ +γ18 logβ14 + log C ∗ + G∗c + (λ1 + λ2 ) (4.B18) × K ∗ + K h∗ + K ∗p + E n∗ + E o∗ −logS ∗ = 0, and, from (4.B17) and (4.B18), we obtain (4.59). From (3.10),

*

D y10 = γ19 logβ22 +

β23 logY ∗f

q ∗ p∗ − β24 log pf∗

− β24 (y6 + y14 ) − logE n∗ − y10 ,

209

+

(4.B19)


from (4.45),

*

q ∗ p∗ logE n∗ = logβ22 + β23 logY ∗f − β24 log pf∗

+ ,

(4.B20)

and, from (4.B19) and (4.B20), we obtain (4.60). From (3.11), Q+P DlogF = γ20 β25 −1 , (4.B21) F from (4.B21), ∗ y5 Q e + P ∗ e y12 − 1 , D y11 = −(λ1 + λ2 ) + γ20 β25 F ∗ e y11 (4.B22) from (4.46),

Q∗ + P∗ = γ20 + λ1 + λ2 , F∗ and, from (4.B22) and (4.B23), we obtain (4.61). γ20 β25

(4.B23)

From (3.12),

! " K ∗ e y13 a ∗ −1 , Dlog p = γ21 β26 + β27 rf − λ5 P ∗ e y12 (4.B24)

from (4.B24), D y12 = −(λ1 + λ2 ) + γ21

! " K ∗ e y13 a β26 + β27 rf∗ −λ5 − 1 , P ∗ e y12 (4.B25)

from (4.47), ! " K ∗ a γ21 β26 + β27 rf∗ − λ5 = γ21 + λ1 + λ2 , (4.B26) P∗

210


and, from (4.B25) and (4.B26), we obtain (4.62). From (3.13), D 2 logK a + (DlogK a )2 = −γ22 DlogK a + γ23 Q+P × {β28 + β29 (rf − r )−β30 Dlogq − β31 dx } −1 , Ka (4.B27)

from (4.B27), D 2y13 = −γ22 (D y13 + λ1 + λ2 ) − (D y13 + λ1 + λ2 )2 ! " + γ23 β28 + β29 rf∗ − r ∗ − y8 − β30 (D y14 + λ1 + λ2 + λ5 − λ3 − λ4 )} ∗ y5 Q e + P ∗ e y12 −1 , (4.B28) × K a∗ e y13

from (4.48),

! " γ22 (λ1 + λ2 ) + (λ1 + λ2 )2 + λ23 = γ23 β28 + β29 rf∗ − r ∗ Q∗ + P∗ + β30 (λ3 + λ4 − λ1 − λ2 − λ5 ) , (4.B29) K a∗

and, from (4.B28) and (4.B29), we obtain (4.63). From (3.14), (4.13) and (4.14), D y14 = −γ24 (D y6 + D y14 ) − γ25 (y6 + y14 ) + γ26 y8 + γ27 log E n∗ e y10 + E o∗ + P ∗ e y12 − F ∗ e y11 ∗ ∗ pi I y9 −y14 −y6 e , − K a∗ e y13 (D y13 + λ1 + λ2 ) − log q ∗ p∗ (4.B30) 2

211


and, from (4.B30) and (4.49) we obtain (4.64). From (3.15), Q + ( pi /qp)I − C − DK − DKh − DK p − E n −E o − Gc DlogS = , S (4.B31) from (4.B31) D y15 = −(λ1 + λ2 )  ∗ y5  ∗ ∗ ∗ ∗ y −y −y ∗ y  Q e + ( pi /q p )I e 9 6 14 − C e 1  − K ∗ e y4 (D y4 + λ1 + λ2 )− K h∗ e y3 (D y3 +λ1 +λ2 )   − K ∗p (λ1 + λ2 ) − E n∗ e y10 − E o∗ − G∗c , + S ∗ e y15 (4.B32)

from (4.A6), (4.A7) and (4.A8), Q ∗ + ( pi∗ /q ∗ p ∗ ) I ∗ − C ∗ − G∗c − (λ1 + λ2 ) K ∗ + K h∗ + K ∗p − E n∗ − E o∗ = (λ1 + λ2 )S ∗ ,


212

(4.B33)

CHAPTER FIVE

Empirical Estimation of the Model and Derived Results 5.1 Introduction

I

n this chapter we now turn our attention to the empirical analysis of the new continuous time macroeconometric

model, using UK data and the exact Gaussian method of Bergstrom [1997]. Since the model of Bergstrom [1997] is linear in the variables, although nonlinear in the parameters, while the continuous time macroeconometric model is non-linear in both the variables and the parameters, it was necessary, in the estimation, to use a linear (in the variables) approximation to the latter model. This approximation is discussed in Section 5.2, where we also present and discuss the parameter estimates. In the subsequent sections we derive and discuss the properties of the estimated model,

213


using the results obtained in earlier chapters. The time lag distributions with which the variables each depend on the past values of other variables on which they are directly dependent have been derived from the estimated parameters using the formulae derived in Appendix B of Chapter 3. They are presented (graphically and through a table of their means and modes) and discussed in Section 5.3. The steady state and stability properties of the estimated model have been derived using the results obtained in Chapter 4 and are discussed in Section 5.4. Finally, in Section 5.5, we discuss the post-sample forecasting performance of the estimated model and compare it with that of an unrestricted second-order VARX model (vector autoregressive model with exogenous variables) and Section 5.6 concludes.

5.2 Estimation from United Kingdom Data As was mentioned earlier, it was necessary to use a linear (in the variables) approximation to the model specified in Chapter 3, in order to obtain estimates of the parameters by the exact Gaussian method described in Chapter 2 (Section 2.4). We have used, for this purpose, a linear approximation about

214


the sample means of the variables. Alternatively, we could have used the linear approximation about the steady state derived in Chapter 4 (Section 4.3 and Appendix B). But, if the variables are not very close to their steady state paths during the sample period, better estimates are likely to be obtained by using the linear approximation about the sample means as is common in most of the literature on continuous time macroeconomic models. It should be emphasized, however, that nothing can be, rigorously, deduced about the stability properties of the model from the linear approximation about the sample means. Indeed, it is possible that, even if all the eigenvalues of the linear approximation about the sample means have negative real parts, the solution of the nonlinear model, starting at the sample means, will be explosive. The stability analysis must, therefore, be based on the linear approximation about the steady state. To put the model into the form used in Chapter 2 (equations (2.1)–(2.3)), we define a 7 × 1 vector x1 (t) of endogenous variables that adjust through first-order differential equations, an 8 × 1 vector x2 (t) of endogenous variables that adjust through second-order differential equations, a 15 × 1 vector z(t) of exogenous variables and a 3 × 1 vector µ(t) of stochastic trends. These vectors are defined as follows.

215


x1 (t) = [logC(t), logE n (t), F (t), logI (t), P(t), logQ (t), logS(t)] , x2 (t) = [logK(t), K a (t), logKh (t), logL(t), log p(t), logq(t), r (t), logw(t)] , z(t) = [logB(t), logK p (t), logM(t), DlogK p (t), dx , logE 0 (t), logGc (t), log pf (t), log pi (t), r f (t), logT1 (t), logT2 (t), logY f (t), Dlog pf (t), Dlog pi (t)] , µ(t) = [µ1 (t), µ2 (t), µ3 (t)] .

By replacing the expression on the right-hand side of each of the equations (3.1), (3.2), . . . , (3.15) of Chapter 3 by a linear approximation (the first-order term of a Taylor series expansion) about the sample means of the variables, and adding a white noise innovation to each of these equations and to each of equations (3.16)–(3.18), we obtain the mixedorder linear stochastic differential equation system d x 1 (t) = [A1 x1 (t) + A2 x2 (t) + A3 D x2 (t) + B1 z(t) + C 1 µ(t) + b1 ] dt + ζ1 (dt)

(t ≥ 0),

(5.1)

d[D x2 (t)] = [A4 x1 (t) + A5 x2 (t) + A6 D x2 (t) + B2 z(t) + C 2 µ(t) + b2 ] dt + ζ2 (dt) dµ(t) = λdt + ζ3 (dt)

(t ≥ 0),

(5.2) (5.3)

where ζ1 , ζ2 and ζ3 are white noise innovation vectors, A1 , A2 , A3 , A4 , A5 , A6 , B1 , B2 , C 1 and C 2 are matrices whose

216


elements are functions of the sample means of the variables and the parameter vector θ = [β, γ ], and b1 and b2 are vectors whose elements are functions of the sample means of the variables and the parameter vector [θ, λ]. The precise forms of the functions defining the elements of these matrices and vectors are given in Appendix A to this chapter. The system (5.1)–(5.3) has the same form as the model (2.1)–(2.3) used in Chapter 2. The estimation algorithm described in that chapter is, therefore, directly applicable in the estimation of the parameters of (5.1)–(5.3), which are also the parameters of the model (3.1)–(3.18) specified in Chapter 3. The estimates are derived by minimizing a function L, which is, formally, equal to minus twice the logarithm of the Gaussian likelihood (less a constant) and is derived from the system (5.1)–(5.3) in the same way as the function L defined by (2.47) is derived from (2.1)–(2.3). (See Bergstrom [1997], for a complete list of steps in the computation of the value of L, for a given set of values of the parameters of the continuous time model, from the discrete observations of the variables.) The estimates obtained by minimizing L with respect to the parameters of the continuous time model would be exact maximum likelihood estimates if the system (5.1)–(5.3) were the true model and ζ1 , ζ2 and ζ3 were Brownian motion vectors, and they will

217


be good estimates under much more general conditions. As in Bergstrom, Nowman and Wymer [1992], we placed prior bounds on all 63 of the structural parameters and minimized L subject to the constraint that the parameter vector belongs to a compact rectangular set defined by these bounds. The parameters were estimated, in the above way, from UK quarterly data for the period 1975–94 inclusive, and the post-sample predictive performance of the model, which will be discussed in Section 5.4, was tested against quarterly data for the 2-year period 1995–96. The data are defined in Appendix B of this chapter. Before being used in the estimation procedure, they were seasonally corrected by the method of Durbin [1963]. The parameter estimates are presented in Table 5.1. They all have plausible values, and only one of them is on a bound. The mark-up parameter β15 is on its lower bound, which implies perfect competition. The column headed “Standard Error” contains the square roots of the diagonal elements of minus the inverse of the Hessian of the logarithm of the Gaussian likelihood function (which is twice the inverse of the Hessian of L), evaluated at the estimated values of the parameters. They cannot, however, be interpreted in the same way as under classical assumptions, since, because of the stochastic trends, the parameter estimates are not

218


Table 5.1. Parameter Estimates from United Kingdom Quarterly Data 1975–94

Parameter β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12 β13 β14 β15 β16 β17 β18 β19 β20 β21 β22 β23 β24

Prior Lower Bound

Prior Upper Bound

0.8000 0.0000 0.0000 0.1000 0.0100 0.0100 1.0000 0.0000 0.0000 0.0000 0.0000 0.1000 −0.5000 0.0000 1.0000 20000 −1.0000 0.5000 0.0000 0.0050 0.0050 10000 0.1000 0.5000

1.0000 4.0000 4.0000 10.000 1.0000 1.0000 10.000 4.0000 4.0000 1.0000 0.0500 0.3000 0.5000 1.0000 1.1000 40000 −0.0100 1.0000 1.0000 0.0500 0.0500 35000 1.0000 1.5000

Estimate 0.9351 0.2014 1.6176 0.1494 0.2630 0.2965 3.5000 0.1815 3.0574 0.1535 0.0077 0.2169 −0.4033 0.7556 1.0000 23150 −0.0832 0.9312 0.1593 0.0073 0.0060 29500 0.5000 1.0000

Standard Error 0.0073 0.0138 0.0203 0.0298 0.0001 0.0043 0.1401 0.0042 0.0945 0.0365 0.0005 0.0102 0.0598 0.0258 44.2813 0.1087 1.6399 0.0224 0.0007 0.0142 451.7023 0.3562 0.0614 (continued)

219


Table 5.1 (continued)

Parameter β25 β26 β27 β28 β29 β30 β31 β32 β33 γ1 γ2 γ3 γ4 γ5 γ6 γ7 γ8 γ9 γ10 γ11 γ12 γ13 γ14 γ15 γ16 γ17

Prior Lower Bound 0.0000 0.0000 0.0000 −2.0000 0.0000 0.0000 0.0000 0.5000 0.0000 0.0010 0.1000 0.0010 0.1000 0.0010 0.1000 0.0001 0.0010 0.0010 0.1000 0.0010 0.1000 0.0010 0.0010 0.1000 0.0010 0.0010

Prior Upper Bound 0.0100 0.0200 1.0000 2.0000 100.00 10.000 0.1000 1.5000 10.000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

220

Estimate 0.0083 0.0081 0.2520 −0.1303 57.0548 0.1577 0.0007 0.7059 0.0089 0.5294 0.2200 0.0100 4.9999 0.2760 0.0976 0.0009 0.1029 0.0944 0.2384 0.0015 4.3549 0.6494 0.0220 1.4070 0.0292 0.1039

Standard Error 0.0021 0.0003 0.0315 0.0057 12.9646 0.0305 0.0001 0.1997 0.4913 0.0735 0.1650 0.0145 0.0336 0.0001 0.0040 0.00001 0.0028 0.0015 0.0195 0.000001 0.0749 0.0056 0.0016 0.5086 0.0039 0.0062


Table 5.1 (continued)

Parameter γ18 γ19 γ20 γ21 γ22 γ23 γ24 γ25 γ26 γ27 λ1 λ2 λ3

Prior Lower Bound

Prior Upper Bound

Estimate

0.0010 0.0010 0.0010 0.0010 0.1000 0.0010 0.1000 0.0010 0.0010 0.0000 0.0000 −0.0100 0.0000

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 0.0100 0.0100 0.0500

0.0917 0.1019 2.8565 0.7942 0.2033 0.0075 3.9345 0.1808 0.1244 0.0009 0.0028 0.0061 0.0098

Standard Error 0.0010 0.0002 0.4912 0.2061 0.0811 0.0014 0.1045 0.0010 0.0047 0.00003 0.0019 0.0025 0.0017

asymptotically normally distributed. They can, however, be given a simple Bayesian interpretation, as was pointed out in Chapter 2. Assuming that the innovations in the continuous time model are Brownian motion and a uniform prior distribution of the parameters, minus the inverse of the Hessian of the logarithm of the likelihood function, is the covariance matrix of a normal approximation to the posterior distribution of the parameters in the neighbourhood of its mode.

221


Turning to the stochastic trend parameters λ1 , λ2 and λ3 , the estimate of λ1 implies that the amount of labour required to produce a given output with a given amount of capital decreases, through technical progress, at a mean instantaneous rate of 1.12 per cent per annum, while the estimate of λ2 implies that the non-accelerating inflation level of employment increases at a mean instantaneous rate of 2.44 per cent per annum. The estimate of λ3 implies that the use of cards (plastic money) increases at a rate equivalent (in its effect) to an increase in the money supply at a mean instantaneous rate of 3.92 per cent per annum. Among the long-run structural parameters β1 , β2 , . . . , β33 , the estimates imply that the long-run marginal propensity to consume β1 is 0.9351. The parameter β6 determines the elasticity of substitution between labour and capital, which, as was mentioned in Chapter 3, equals 1/(1 + β6 ). Substituting the estimated value of β6 into this expression, we obtain an estimate 0.7713 for the elasticity of substitution between labour and capital. Another important elasticity is the price elasticity of demand for imports, which, as was mentioned in Chapter 3, equals −(1 + β13 ). Substituting the estimated value of β13 into this expression, we obtain an estimate −0.5967 for the price elasticity of demand for imports. The direct interpretation of the estimates of the

222


speed of adjustment parameters γ1 , γ2 . . . , γ27 is fairly obvious. Less obvious and, perhaps, of more interest are the time lag distributions with which the variables adjust, each in response to changes in the other variables on which it is directly dependent. These time lag distributions are presented and discussed in Section 5.3, to which we now turn.

5.3 Time Lag Distributions Nearly all of the equations of the model are in the form of one of the continuous adjustment equations (3.A15) and (3.A37) derived in Appendix A of Chapter 3. It is shown in Appendix B of Chapter 3 that each of equations (3.A15) and (3.A37) implies that the dependent variable depends, with a distributed time lag, on all past values of a subset of the other variables in the model. More precisely, it depends, with a distributed time lag, on all past values of its partial equilibrium level, which is a function of these other variables. Appendix B of Chapter 3 includes the derivation of the forms of the time lag distributions in these relations and their means and modes (also presented in Bergstrom, Nowman and Wymer [1992] for the second-order equations). When the adjustment equation is a first-order differential equation

223


of the form (3.A15), the time lag distribution is an exponential distribution given by (3.B4). The density is a decreasing function of the time lag, and the mean of the distribution is the reciprocal of the speed of adjustment parameter γ . When the adjustment equation is a second-order differential equation of the form (3.A37), the time lag distribution is of a more complicated form, depending on the roots of the quadratic equation (3.B7). In the normal case, where these roots are real and negative, it will be a unimodal distribution with a positive mode. But, its exact form depends on whether the roots are equal or unequal. When they are unequal, the density function of the distribution is given by (3.B10), its mean by (3.B17) and its mode by (3.B18), while when they are equal, the density function is given by (3.B24), the mean is 2/α and the mode is 1/α. Estimates of the means and modes of the lag distributions have been derived from the estimated values of the parameters, for all variables in the model that adjust through differential equations of the form (3.A15) or (3.A37) and are presented in Table 5.2. As can be seen, the private non-residential fixed capital stock variable adjusts with the longest time lag and the financial flow variables with the shortest time lags. In particular, the private non-residential fixed capital variable logK

224


Table 5.2. Estimated Time Lag Parameters

Variable logC logEn F logI logKh logK Ka logL P logQ r

Mean Time Lag (Quarters) 1.8889 9.8135 0.3501 9.6246 18.1156 108.4444 27.1067 22.0000 1.2591 9.7182 48.1849

Modal Time Lag (Quarters)

0.9172 27.7487 10.9215 9.6795

3.0672

adjusts with a mean time lag of approximately 27 years and a modal time lag of approximately 7 years; interest rate variable r adjusts with a mean time lag of 12 years and a modal time lag of under 1 year; cumulative net real investment abroad variable K a adjusts with a mean time lag of approximately 7 years and a modal time lag of approximately 3 years; employment variable logL adjusts with a mean time lag of 51/2 years and a modal time lag of approximately 21/2 years and the residential fixed capital variable logKh adjusts with a mean time lag of 41/2 years and a modal time lag of approximately 3 months.

225


0.6

Density

0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

Quarters

Figure 5.1 Lag Distribution Private Consumption: log C

Turning to the first-order equations, we find that the nonoil exports variable logE n , output variable logQ and the import variable logI adjust with mean time lags of approximately 21/2 years; the consumption variable logC and profits, interest and dividends from abroad variable P adjust with mean time lags of approximately 1/2 a year, and finally the international transfers variable F adjusts with a mean time lag of approximately 1 month. The graphs of the density functions of the lag distributions are presented in Figures 5.1 to 5.11.

226

0.04 0.035

Density

0.03 0.025 0.02 0.015 0.01 0.005 0 0

10

20

30

40

Quarters

Figure 5.2 Lag Distribution Employment: log L 0.06

Density

0.05 0.04 0.03 0.02 0.01 0 0

5

10

15

Quarters

Figure 5.3 Lag Distribution Residential Fixed Capital: log Kh

227

20

0.009 0.008 0.007

Density

0.006 0.005 0.004 0.003 0.002 0.001 0 0

20

40

60

80

100

120

140

Quarters

Figure 5.4 Lag Distribution Private Non-Residential Fixed Capital: log K 0.12 0.1

Density

0.08 0.06 0.04 0.02 0 0

5

10

15

Quarters

Figure 5.5 Lag Distribution Output: log Q

228

20

0.025

Density

0.02

0.015

0.01

0.005

0 0

10

20

30

40

Quarters

Figure 5.6 Lag Distribution Interest Rate: r 0.12 0.1

Density

0.08 0.06 0.04 0.02 0 0

5

10

15

Quarters

Figure 5.7 Lag Distribution Imports: log I

229

20

0.12 0.1

Density

0.08 0.06 0.04 0.02 0 0

5

10

15

20

Quarters

Figure 5.8 Lag Distribution Non-Oil Exports: log En 3

Density

2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Quarters

Figure 5.9 Lag Distribution Current Transfers Abroad: F

230

1.8

0.9 0.8 0.7

Density

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

5

4

3

Quarters

Figure 5.10 Lag Distribution Profits Interest and Dividends From Abroad: P 0.007 0.006

Density

0.005 0.004 0.003 0.002 0.001 0 0

10

20

30

40

50

60

Quarters

Figure 5.11 Lag Distribution Investment Abroad: Ka

231

70


5.4 Steady State and Stability Properties In this section we discuss the steady state and stability properties of the estimated model, which have been derived using the results obtained in Chapter 4. For the purpose of this derivation, we have adjusted 4 parameters from their estimated values to values that seem more realistic for the future, bearing in mind that the purpose of the steady state analysis is to provide a check on the capacity of the model to generate plausible long-run behaviour. The parameters adjusted are β13 , which determines the price elasticity of demand for imports, β14 the optimal (partial equilibrium) ratio of stocks to sales, λ2 the mean rate of growth of the labour supply and λ3 the mean rate of growth of the use of plastic money. The parameter β13 has been set at the value 0.2, which implies a price elasticity of demand for imports of −1.2. In view of the greater liberalization of trade, which is now taking place, this seems more realistic for the future than the price elasticity of −0.60 implied by the estimated value of β13 . In view of the growing relative importance of the service sector of the economy, we should expect the value of the parameter β14 to be lower in the future than it was over the sample period. Moreover, the fitting of a trend to the

232


actual ratio of stocks to sales S/(C + Gc + DK + DKh + DK p + E n + E o ) provides strong evidence that there was a significant fall in β14 over the sample period. Since the adjusted value of β14 is used in the generation of post-sample forecasts, to be discussed in Section 5.5, as well as in the steady state derivation, it has been chosen in an objective way, which makes use of no data beyond the end of the sample period. We have fitted a quadratic trend to the ratio S/(C + Gc + DK + DKh + DK p + E n + E o ), using data up to the last quarter of the sample period, and set β14 equal to the trend value in the last quarter of the forecast period, i.e. 2 years beyond the end of the sample period. The value of β14 obtained in this way is 0.5804, which is significantly lower than the estimated value 0.7556 shown in Table 5.1. The parameter λ2 has been set at the value 0.002. This implies a rate of growth of the labour supply of 0.8 per cent per annum, which seems more realistic for the future than the growth rate of 2.4 per cent per annum implied by the estimated value of λ2 . The parameter λ3 has, for simplicity, been set a value 0.0 and the assumed rate of growth of the money supply adjusted accordingly. To obtain the steady state solution, we must, also, assume values of the parameters in (4.1)–(4.17) of Chapter 4, which

233


determine the assumed paths of the exogenous variables. The parameter λ4 , representing the rate of growth of the money supply, has been set at the value 0.0048, which, together with the estimated value 0.0028 of λ1 and the assumed value 0.0020 of λ2 , implies a steady state rate of increase in UK prices of zero. The parameter λ5 , representing the rate of increase in foreign prices, has been set at the value 0.0, which, in conjunction with the zero steady state rate of increase in UK prices, implies a constant steady state exchange rate. The level parameters B ∗ , E 0∗ , K∗p , M ∗ , pf∗ , pi∗ and Y ∗f have been set at values that ensure that the levels of the steady state paths of the corresponding variables are equal to the actual levels of the variables in the base period. The parameters T1∗ , T2∗ and g ∗ are set at the average values of T1 , T2 and Gc /(Q + P ) over the sample period. The numerical values of the parameters of the assumed steady state paths of the exogenous variables are shown in Table 5.3. The steady state growth rates, under the above assumptions, are shown in Table 5.4 and the steady state level parameters in Table 5.5. The actual and steady state levels of the endogenous variables in the last quarters of 1996 are, also, shown in Table 5.5.

234


Table 5.3. Values of Exogenous Variable Time Path Parameters Assumed in Steady State Analysis Parameter

Assumed Value

∗

B E 0∗ g∗ K∗p M∗ pf∗ pi∗ r f∗ T1∗ T2∗ Y ∗f λ4 λ5

178786 1580 0.2400 172434 17143 1.0000 1.0000 0.0098 1.3103 1.1526 0.5514 0.0048 0.0000

Table 5.4. Derived Steady State Growth Rates Instantaneous Growth Rate (per cent per annum)

Variable C , En , F , I , K , Kh , K a , P , Q, S L p w r q

1.92 0.80 0.00 1.12 0.00 0.00

235


Table 5.5. Derived Steady State Levels

Variable C En F I Kh K Ka L P p Q q r S w

Steady State Steady State Level in Actual Level in Last Level Parameter Last Quarter of 1996 Quarter of 1996 65,787 26,456 765 20,218 246,283 741,387 −11,365 23,151 −120 0.5457 92,488 1.5174 0.0100 70,553 1.6230

100,365 40,362 1,168 30,844 375,733 1,131,074 −17,339 27,606 −182 0.5457 141,102 1.5174 0.0100 107,637 2.0764

100,619 42,999 1,096 47,103 579,636 713,621 −18,154 26,244 1,570 1.2449 142,808 0.9139 0.0184 116,833 4.5106

The values of the parameters λ2 and λ4 assumed for the purpose of the steady state analysis are reflected in the large differences between the actual and steady state levels of the variables K and p in the last quarter of 1996. The large excess of the steady state level of the capital variable K over the actual level is a result of the downward adjustment of the parameter λ2 (the rate of growth of the labour supply) from the estimated value 0.0061 to the value 0.002. This

236


increases the steady state ratio of capital to output, because the proportion of output that must be invested to maintain a given ratio of capital to output is smaller the lower is the growth rate. The low steady state price level in the last quarter of 1996, compared with the actual price level in that quarter, is a consequence of setting λ4 (the rate of increase in the money supply) at a level that keeps the steady state price level constant. This reduces the real cost of holding wealth in the form of money and, hence, increases the demand for and steady level of real money balances. But, since M ∗ has been set at the level that makes the steady state level of M equal to the actual level of M in the last quarter of 1996, an increase in the steady state level of real money balances implies a decrease in the steady state price level. We conclude this section with a discussion of the stability and other properties of the solution of the estimated model in the neighbourhood of the steady state. Here, we rely on the results obtained in Chapter 4, Section 4.3 and Appendix B. It is shown in Chapter 4 that the solution of the model is asymptotically stable if the eigenvalues of the matrix C in (4.82) all have negative real parts. If this condition is satisfied, then, any solution of the system (3.1)–(3.18) in which the initial values of the variables are sufficiently close

237


to their steady state paths will converge to the steady state solution if undisturbed. Formulae expressing the elements of the matrix C as functions of the parameters of the model (3.1)–(3.18) are easily implied from Chapter 4. These elements have been evaluated at the estimated values of the parameters, after adjusting the parameter value of β13 to the value 0.2 and β14 to the value 0.5804, as in the computation of the steady state level parameters. (The other parameters adjusted for the computation of the steady state levels have no effect on the asymptotic stability of the solution.) The eigenvalues of the estimated matrix C obtained in this way are presented in Table 5.6. All except 5 of these eigenvalues have negative real parts. The eigenvalues with positive real parts include one real eigenvalue 0.0066 and two complex pairs 0.0012 ± 0.0202i and 0.1695 ± 0.0389i. The long-term solution of the system (4.82) will be dominated by the cycle associated with the complex pair 0.1695 ± 0.0389i, whose real part is greater than that of any of the other eigenvalues in Table 6. If unconstrained, this would be an explosive cycle with a period of, approximately, 40 years. A realistic model must, however, take account of a full employment ceiling or some similar constraint on the productive capacity of the economy. Taking account of such a constraint, the model, in its deterministic

238


Table 5.6. Eigenvalues for Linear Approximation about Steady State Real Part

Imaginary Part

−4.9633 −2.8613 −1.4068 −0.7990 −0.5419 −0.1597 −0.1558 −0.1033 −0.1016 −0.0642 −0.0533 −0.0469 −0.0295 −0.0017 0.0066 −4.2409 −0.0699 0.0012 0.1695

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ±0.1792 ±0.0094 ±0.0202 ±0.0389

form, can be expected to generate a perpetual long cycle with a period of 35–40 years and, approximately, constant amplitude. This could be the well-known Kondratieff cycle observed and discussed by Kondratieff [1926, 1935].

239


The model also generates a cycle with a period of approximately 9 years associated with the pair of eigenvalues −4.2409 ± 0.1792. This could be the well-known economic cycle commonly referred to as the trade cycle or business cycle and generally recognized as having a period of 7–11 years. Although the pair of eigenvalues with which this cycle is associated has a negative real part, the cycle will be prevented from dying out by the stochastic innovations in the model, and its average amplitude will depend on the variances of these innovations. We may conclude, therefore, that the model (3.1)–(3.18) generates plausible long-run behaviour, with fluctuations about the steady state neoclassical growth path including both a trade cycle with a period of about 9 years and a longer (Kondratieff) cycle with a period of about 40 years.

5.5 Post-Sample Forecasting Performance To assess the post-sample forecasting of the model, we have computed the optimal post-sample forecasts of the discrete observations of the endogenous variables for the 8 post-sample quarters 1995Q1–1996Q4, using the method described in Chapter 2, Section 2.4. The method is a recursive procedure, in which the computation of the forecast for

240


each post-sample quarter makes use of the forecasts for the proceeding post-sample quarters. It should be emphasized, however, that the procedure makes no use, either directly or indirectly, of the actual value of any endogenous variable after 1994 quarter 4, the last quarter of the sample period. For the computation of the post-sample forecasts, all parameters of the model except β14 were set equal to their estimated values. For reasons discussed in the preceding section, β14 the optimal (partial equilibrium) ratio of the stocks to sales has been adjusted from its estimated value 0.7556 to the value 0.5804, which was obtained, as explained in the preceding section, by fitting a quadratic trend to the ratio of stocks to sales over the sample period. The latter procedure makes no use of any data relating to quarters after 1994 quarter 4, and it could, therefore, have been used in actual forecasting at the end of the sample period. The root-mean-square errors (RMSE) of the forecasts from the continuous time model for the 8 post-sample quarters 1995Q1–1996Q4 are compared, in Table 5.7, with the RMSE of the corresponding forecasts from a second-order VARX model (vector autoregressive model with exogenous variables). The comparison is confined to the 12 endogenous variables that can assume positive values only (i.e. all except F, P and K a ). These variables can be represented in

241


Table 5.7. Root Mean Square Errors of Post-Sample MultiPeriod Forecasts 1995Q1–1996Q4

Variable logC logEn logI logKh logK logL log p logQ logq r logS logw

RMSE of Forecasts from Continuous Time Model

RMSE of Forecasts from VARX Model

0.0477 0.0797 0.0450 0.0350 0.0024 0.0137 0.0121 0.0395 0.0472 0.0012 0.1396 0.0130

0.0148 0.1131 0.1329 0.0024 0.0077 0.0057 0.0218 0.0181 0.0453 0.0007 0.0096 0.0227

logarithmic form and the forecast errors (when multiplied by 100) interpreted as percentage errors. The forecasts, from the continuous time model, of the sequence of values of each of these variables for the quarters 1995Q1–1996Q4 are shown in Table 5.8 together with the actual values of the variables in the same quarters. Comparing the RMSE of the forecasts from the continuous time and VARX models shown in Table 5.7, we see that for the interest rate variable r the RMSE of the

242


Table 5.8. Post-Sample Multi-Period Forecasts 1995Q1– 1996Q4 Variable

Quarter

Actual Value

Forecast

logC

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

11.4364 11.4306 11.4545 11.4424 11.4579 11.4630 11.4862 11.4761

11.4436 11.4609 11.4739 11.4881 11.4990 11.5165 11.5424 11.5599

logEn

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

10.5943 10.5545 10.6041 10.6157 10.6551 10.6283 10.6432 10.6472

10.5315 10.5325 10.5348 10.5379 10.5418 10.5466 10.5523 10.5586

l ogI

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

10.6232 10.6450 10.6737 10.6913 10.7258 10.7203 10.7384 10.7746

10.6627 10.6566 10.6562 10.6598 10.6671 10.6761 10.6888 10.7010 (continued)

243


Table 5.8 (continued) Variable

Quarter

Actual Value

Forecast

logKh

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

13.2508 13.2540 13.2564 13.2593 13.2616 13.2644 13.2667 13.2700

13.2566 13.2659 13.2753 13.2849 13.2941 13.3053 13.3166 13.3280

logK

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

13.4279 13.4345 13.4415 13.4476 13.4558 13.4627 13.4697 13.4771

13.4273 13.4337 13.4407 13.4481 13.4561 13.4644 13.4732 13.4824

logL

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

10.1576 10.1594 10.1558 10.1639 10.1593 10.1611 10.1680 10.1721

10.1629 10.1675 10.1713 10.1744 10.1772 10.1798 10.1824 10.1852

244



Quarter

Actual Value

Forecast

log p

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

0.1714 0.1817 0.1810 0.1870 0.2023 0.2058 0.2109 0.2199

0.1650 0.1710 0.1766 0.1819 0.1868 0.1917 0.1966 0.2017

logQ

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

11.7785 11.7749 11.7855 11.7857 11.7927 11.8043 11.8117 11.8270

11.7669 11.7767 11.7907 11.8079 11.8275 11.8486 11.8730 11.8967

logq

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

−0.1332 −0.1735 −0.1769 −0.1759 −0.1761 −0.1678 −0.1624 −0.0854

−0.1064 −0.1172 −0.1284 −0.1391 −0.1497 −0.1613 −0.1722 −0.1824 (continued)

245



Quarter

Actual Value

r

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

0.0207 0.0195 0.0195 0.0185 0.0203 0.0199 0.0196 0.0184

0.0192 0.0189 0.0187 0.0186 0.0184 0.0183 0.0182 0.0181

logS

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

11.6071 11.6243 11.6358 11.6506 11.6474 11.6565 11.6566 11.6709

11.6085 11.5754 11.5404 11.5114 11.4908 11.4813 11.4712 11.4822

logw

1995 Q1 Q2 Q3 Q4 1996 Q1 Q2 Q3 Q4

1.4552 1.4489 1.4535 1.4613 1.4983 1.4884 1.4856 1.5049

1.4382 1.4464 1.4542 1.4615 1.4686 1.4776 1.4862 1.4968

246

Forecast


forecasts from the two models differ by only 0.0005. We shall treat the forecasts of this variable from the two models as equally good, therefore. With regard to the remaining 11 variables in Table 5.7, the continuous time model yields forecasts with the lower RMSE for the 5 variables logE n , logI, logK , log p and logw, while the VARX model yields forecasts with the lower RMSE for the six variables, logC , logKh , logL , logQ , logq and logS. The forecasting performance of the VARX model is a little better, therefore, than that of the continuous time model. But, such a small difference could, easily, be a random property of the stochastic innovations in the particular 8 quarters that we have chosen to forecast, even if the continuous time model is correctly specified. We may conclude, therefore, that the forecasting performance of the continuous time model is satisfactory. It should be emphasized, moreover, that our principal aim, in producing this model, was to produce a prototype mixed-order continuous time macroeconometric model with stochastic trends and we have done no experimentation with alternative specifications. For the purpose of practical forecasting, we could use a somewhat less parsimonious specification with a more refined specification of some of the partial equilibrium levels (for example, distinguishing between the effects on the partial

247


equilibrium consumption level of income from employment and other income) and, perhaps, replacing some of the firstorder adjustment equations with second-order adjustment equations.

5.6 Conclusion In this final chapter, we have presented and discussed the estimates of the parameters of the model in Chapter 3, which were obtained from UK data by the exact Gaussian estimation procedure described in Chapter 2. Since the latter procedure is applicable only to a model that is linear in the variables, although, generally, nonlinear in the parameters, it was necessary, for the purpose of estimation, to use a linear (in the variables) approximation to the model specified in Chapter 3. The parameter estimates are, on the whole, very plausible, and all of the time lag distributions derived from the speed of adjustment parameters are very plausible. We have examined the steady state and stability properties of the estimated model, using the results obtained in Chapter 4, and shown that it generates plausible long-run behaviour. More specifically, it generates fluctuations about the steady state neoclassical growth path, including both a trade cycle with a period of about 9 years and a longer (Kondratieff)

248


cycle with a period of about 40 years. Finally, we have examined the post-sample forecasting performance of the estimated model and compared it with that of a second-order VARX model (vector autoregressive model with exogenous variables). We conclude that the forecasting performance is satisfactory, particularly considering the very parsimonious specification of the model.

Appendix A: Linear Approximation about Sample Means In this Appendix we give the formulae for the non-zero elements of the matrices A1 , A2 , . . . , A6 , B1 , B2 , C 1 and C 2 and the vectors b1 and b2 in the system (5.1)–(5.3). Letting [Ah ]i j denote the element in row i and column j of the matrix Ah (h = 1, . . . , 6), using a similar notation for the elements of the matrices Bh and C h (h = 1, 2), and letting [bh ]i denote the ith element of the vector bh (h = 1, 2), the formulae are as follows, the sample mean of any variable x being denoted by x. ¯ [A1 ]11 = −γ1 , 1 [A1 ]15 = γ1 , e logQ + P

249


e logQ

[A1 ]16 = γ1 [A1 ]22

e logQ + P = −γ19 ,

,

[A1 ]33 = −γ20 , [A1 ]35 = γ20 β25 , [A1 ]36 = γ20 β25 e logQ , [A1 ]41 =

(γ17 + γ18 )e logC e logC + e logGc + e logK DlogK + e logK h DlogK h

,

+ e logK p DlogK p + e logEn + e logEo [A1 ]42 =

(γ17 + γ18 )e logEn e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logEn + e logEo

,

[A1 ]44 = −γ17 , [A1 ]47 = −γ18 , [A1 ]55 = −γ21 , [A1 ]61 =

(γ8 + γ9 )e logC e logC + e logGc + e logK DlogK + e logK h DlogK h

,


[A1 ]66

(γ8 + γ9 )e logEn e logC + e logGc + e logK DlogK + e logK h DlogK h

+ e logK p DlogK p + e logEn + e logEo = −γ8 ,

[A1 ]67 = −γ9 ,

250

,


[A1 ]71 = −e [logC −logS] , [A1 ]74 = e [log pi +logI −logq−log p−logS] , [A1 ]76 = e [logQ −logS] , [A1 ]77 = −e [logQ −logS] − e [log pi +logI −logq−log p−logS] + e [logC −logS] + e [logEn −logS] + e [logEo −logS] + e [logGc −logS] + DlogK e [logK −logS] + DlogK h e [logK h −logS] + DlogK p e [logK p −logS] , [A2 ]17 = −γ1 β2 , [A2 ]25 = −γ19 β24 , [A2 ]26 = −γ19 β24 , [A2 ]41 =

[A2 ]43 =

(γ17 + γ18 )e logK DlogK e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logEn + e logEo (γ17 + γ18 )e logK h DlogK h e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logEn + e logEo

,

,

[A2 ]45 = β17 (1 + β13 ), [A2 ]46 = β17 (1 + β13 ), [A2 ]52 = γ21 {β26 + β27 (r f − Dlog pf )}, [A2 ]61 =

(γ8 + γ9 )e logK DlogK e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logEn + e logEo

251

,


(γ8 + γ9 )e logK h DlogK h

[A2 ]63 =

[A2 ]65

[A2 ]66

e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logEn + e logEo γ8 β12 β13 e β13 (logq+log p−log pi ) , =− 1 − β12 e β13 (logq+log p−log pi ) γ8 β12 β13 e β13 (logq+log p−log pi ) =− , 1 − β12 e β13 (logq+log p−log pi )

,

[A2 ]71 = −DlogK e (logK −logS) , [A2 ]73 = −DlogK h e (logK h −logS) , [A2 ]75 = −e (log pi +logI −logq−log p−logS) , [A2 ]76 = −e (log pi +logI −logq−log p−logS) , [A3 ]15 = γ1 (β2 − β3 ), [A3 ]41 =

(γ17 + γ18 )e logK e logC + e logGc + e logK DlogK + e logK h DlogK h

,


(γ17 + γ18 )e logK h e logC + e logGc + e logK DlogK + e logK h DlogK h

,


(γ8 + γ9 )e logK e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logEn + e logEo

252

,


[A3 ]63 =

(γ8 + γ9 )e logK h e logC + e logGc + e logK DlogK + e logK h DlogK h

,

+ e logK p DlogK p + e logE n + e logEo [A3 ]71 = −e {logK −logS} , [A3 ]73 = −e {logK h −logS} , [A4 ]16 = γ7 (1 + β6 ), [A4 ]25 = γ23 β28 + β29 (r f − r ) − β30 Dlogq − β31 dx , [A4 ]26 = γ23 β28 + β29 (r f − r ) − β30 Dlogq − β31 dx e logQ , 1 , [A4 ]35 = γ5 e logQ + P e logQ [A4 ]36 = γ5 , e logQ + P¯ [A4 ]46 = [A4 ]56 = [A4 ]62 = [A4 ]63 [A4 ]64 [A4 ]65

γ3 e −β6 logQ e −β6 logQ − β5 e −β6 logK

,

γ11 β5 (1 + β6 )e β6 (logQ −logK ) 1 − β5 e β6 (logQ −logK ) γ27 e

,

logE n

, e logEn + e logEo + P − F − DKa γ27 =− , logE logE n + e o + P − F − DK e a = −γ27 , γ27 = , logE logE n o e +e + P − F − DKa

253


[A4 ]75 = γ16 β20 e (log p−logM−µ3 ) ,

where µ3 =

λ3 T 2

,

[A4 ]76 = γ16 β20 e (log p+logQ −logM−µ3 ) , [A4 ]86 = [A5 ]11 = [A5 ]17 = [A5 ]22 = [A5 ]27 =

γ14 e −β6 logQ

, e −β6 logQ − β5 e −β6 logK −γ7 (1 + β6 ), γ7 , − r − β10 Dlog p + β11 −γ23 , ! " −γ23 β29 e logQ + P ,

[A5 ]33 = −γ5 , [A5 ]37 = −γ5 β8 , [A5 ]41 = − [A5 ]44 [A5 ]51 [A5 ]55

γ3 β5 e −β6 logK

, e −β6 logQ − β5 e −β6 logK = −γ3 , γ11 β5 (1 + β6 )e β6 (logQ −logK ) , =− 1 − β5 e β6 (logQ −logK ) = −γ11 ,

[A5 ]58 = γ11 , [A5 ]65 = γ27 − γ25 , [A5 ]66 = γ27 − γ25 , [A5 ]67 = γ26 ,

[A5 ]75 = γ16 β20 e (log p+logQ −logM−µ3 ) + P e (log p−logM−µ3 ) ,

254


[A5 ]77 = −γ16 , [A5 ]81 = −

γ14 β5 e −β6 logK

e −β6 logQ − β5 e −β6 logK

,

[A6 ]11 = −γ6 , [A6 ]15 = [A6 ]22 [A6 ]26

γ7 β10

, r − β10 Dlog p + β11 = −γ22 , ! " = −γ23 β30 e logQ + P ,

[A6 ]33 = −γ4 , [A6 ]35 = γ5 (β8 − β9 ), [A6 ]44 = −γ2 , [A6 ]55 = −γ10 , [A6 ]58 = γ10 , [A6 ]62 = − [A6 ]65

γ27

e logEn + e logE o + P − F − DKa = −γ24 ,

[A6 ]66 = −γ24 , [A6 ]76 = −γ16 β19 , [A6 ]77 = −γ15 , [A6 ]85 = γ12 − γ13 , [A6 ]86 = −γ13 , [A6 ]88 = −γ12 , [B1 ]1,11 = −γ1 , [B1 ]1,28 = γ19 β24 ,

255

,


[B1 ]2,13 = γ19 β23 , [B1 ]42 =

(γ17 + γ18 )e logK p DlogK p e logC + e logGc + e logK DlogK + e logK h DlogK h

,

+ e logK p DlogK p + e logE n + e logEo [B1 ]44 =

(γ17 + γ18 )e logK p e logC + e logGc + e logK DlogK + e logK h DlogK h

,


(γ17 + γ18 )e logE o e logC + e logGc + e logK DlogK + e logK h DlogK h

,


[B1 ]49

(γ17 + γ18 )e logGc e logC + e logGc + e logK DlogK + e logK h DlogK h

,

+ e logK p DlogK p + e logE n + e logEo = −γ17 (1 + β12 ),

[B1 ]5,10 = γ21 β27 K a , [B1 ]5,14 = −γ21 β27 K a , [B1 ]62 =

(γ8 + γ9 )e logK p DlogK p e logC + e logGc + e logK DlogK + e logK h DlogK h

,


(γ8 + γ9 )e logK p e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logE n + e logEo

256

,


[B1 ]66 =

(γ8 + γ9 )e logEo e logC + e logGc + e logK DlogK + e logK h DlogK h

,


(γ8 + γ9 )e logGc e logC + e logGc + e logK DlogK + e logK h DlogK h + e logK p DlogK p + e logE n + e logEo

[B1 ]69 =

γ8 β12 β13 e β13 (logq+log p−log pi ) 1 − β12 e β13 (logq+log p−log pi )

,

[B1 ]72 = −DlogK p e (logK p −logS) , [B1 ]74 = −e (logK p −logS) , [B1 ]76 = −e (logEo −logS) , [B1 ]77 = −e (logGc −logS) , [B1 ]79 = e (log pi +logI −logq−log p−logS) , [B2 ]25 = −γ23 β31 e logQ + P , [B2 ]2,10 = γ23 β29 e logQ + P , [B2 ]3,11 = −γ5 , [B2 ]5,12 = γ11 , [B2 ]66 = [B2 ]68

γ27 e logEo

e logEn + e logE o + P − F − DKa = γ25 ,

[B2 ]69 = −γ27 , [B2 ]6,14 = γ24 ,

257

,

,


[B2 ]71 = γ16 β21 e (logB−logM) , [B2 ]73 = −γ16 β20 e (log p+logQ −logM−µ3 ) + β20 P e (log p−logM−µ3 ) + β21 e (logB−logM) ,

where µ3 = λ32T , [B2 ]7,10 = γ16 β18 , [B2 ]8,15 = γ13 , [C 2 ]41 = −γ3 , [C 2 ]51 = −γ11 ,

[C 2 ]73 = −γ16 β20 e (log p+logQ −logM−µ3 ) + P e (log p−logM−µ3 ) ,

[C 2 ]81 = −γ14 , [C 2 ]82 = −γ14 ,

! " [b1 ]1 = λ1 + λ2 + γ1 logβ1 + γ1 log e logQ + P logQ e logQ P − γ1 , − γ1 e logQ + P e logQ + P

[b1 ]2 = λ1 + λ2 + γ19 logβ22 , [b1 ]3 = γ20 β25 e logQ {1 − logQ }, [b1 ]4 = λ1 + λ2 + γ17 logβ12 + γ18 logβ14 + γ17 log{1 + β14 (λ1 + λ2 )} + (λ17 + λ18 ) × log e logC +e logGc +e logK DlogK +e logK h DlogK h + e logK p DlogK p +e logEn +e logEo (λ17 + λ18 )

258


  e logC logC +e logGc logGc +e logK DlogK logK     + e logK h DlogK logK + e logK p DlogK logK h

h

p

p

     

 + e logK DlogK +e logK h DlogK h +e logK p DlogK p          + e logEn logE +e logEo logE n o × , e logC +e logGc + e logK DlogK +e logK h DlogK h + e logK p DlogK p +e logEn +e logE o [b1 ]5 = −γ21 β27 (r f − Dlog pf )K a , [b1 ]6 = λ1 + λ2 + γ8 log{1 + β14 (λ1 + λ2 )} + γ9 logβ14 + γ8 log 1 − β12 e β13 (logq+log p−log pi ) γ8 β12 β13 e β13 (logq+log p−log pi ) + 1 − β12 e β13 (logq+log p−log pi ) ×(logq + log p − log pi ) + (λ8 +λ9 ) × log e logC +e logGc + e logK DlogK +e logK h DlogK h + e logK p DlogK p +e logEn +e logEo − (λ8 +λ9 )     e logC logC +e logGc logGc +e logK DlogK logK         + e logK h DlogK h logK h +e logK p DlogK p logK p     + e logK DlogK +e logK h DlogK h +e logK p DlogK p           logE n logE o +e logE n +e logE o × , e logC +e logGc +e logK DlogK +e logK h DlogK h + e logK p DlogK p + e logEn + e logE o

259


[b1 ]7 = e {logQ −logS} {1 − logQ + logS} + e {log pi +logI −logq−log p−logS} ×{1 − log pi − logI + logq + log p + logS} − e {logC −logS} {1 − logC + logS} − e {logEn −logS} {1 − logE n + logS} − e {logEn −logS} {1 − logE o + logS} − e {logGc −logS} {1 − logGc + logS} + DlogK e {logK −logS} {logK − logS} + DlogK h e {logK h −logS} {logK h − logS} + DlogK p e {logK p −logS} {logK p − logS}, [b2 ]1 = γ6 (λ1 + λ2 )+γ7 logβ5 −γ7 log(r − β10 Dlog p + β11 ) r − β10 Dlog p + γ7 , r − β10 Dlog p + β11 [b2 ]2 = γ23 β28 e logQ (1 − logQ ) − γ23 {β29 (r f − r ) − β30 Dlogq − β31 dx } × e logQ logQ + P , ! " [b2 ]3 = γ4 (λ1 + λ2 ) + γ5 logβ7 + γ5 log e logQ + P logQ e logQ P − γ5 , − γ5 e logQ + P e logQ + P

260


[b2 ]4 = γ2 λ2 + γ3 logβ4 − − +

γ3 log e −β6 logQ − β5 e −β6 logK β6

γ3 e −β6 logQ

e −β6 logQ −β5 e −β6 logK γ3 β5 e −β6 logK

logQ

e −β6 logQ −β5 e −β6 logK

logK ,

[b2 ]5 = γ11 (logβ4 + logβ15 ) − γ10 λ1 γ11 (1 + β6 ) − log 1 − β5 e β6 (logQ −logK ) β6 γ11 β5 (1 + β6 ) e β6 (logQ −logK ) (logQ − logK ), − 1 − β5 e β6 (logQ −logK ) [b2 ]6 = γ25 logβ32 − γ26 β33 + γ27 log e logEn + e logE o + P − F − DKa γ27 e logEn logE n + e logE o logE 0 + P − F − DKa , − e logEn + e logEo + P − F − DKa [b2 ]7 = γ16 β17 + γ16 β20 e (log p+logQ −logM−µ3 ) × {1 − log p − logQ + logM + µ3 } ! " log p−logM−µ3 + γ16 β20 P e {logM − log p + µ3 } + γ16 β21 e (logB−logM) {1 + logM − logB},

261


[b2 ]8 = γ12 λ1 + γ14 (logβ4 − logβ16 ) γ14 − log e −β6 logQ − β5 e −β6 logK β6 γ14 e −β6 logQ − logQ e −β6 logQ −β5 e −β6 logK γ14 β5 e −β6 logK logK . + e −β6 logQ −β5 e −β6 logK

Appendix B: Data Sources of data: BLUE United Kingdom National Accounts: The Blue Book, 1997, ONS BOE

Bank of England

ET

Economic Trends

ETAS

Economic Trends Annual Supplement, 1997, ONS

FS

Financial Statistics

IFS

International Financial Statistics

MD

Monthly Digest of Statistics

UKBP United Kingdom Balance of Payments Definitions of series

262


The series used in this study consisted of quarterly observation for the period 1974–96 and were defined as follows: (C ) Real Private Consumption Consumer’s expenditure at current prices deflated by gross domestic product implicit price deflator, p, as defined below (source: ETAS). (E n ) Real Non-Oil Exports Exports of goods and services less oil exports at current prices deflated by p (sources: ETAS and ET). (F ) Real Current Transfers Abroad Net government and private transfers abroad deflated by p (source: ETAS). (I ) Volume of Imports Imports of goods and services at 1990 market prices (source: ETAS). (Kh ) Residential Fixed Capital Cumulative net real residential fixed capital formulation since the beginning of 1975 plus a constant derived from the official estimate of the capital stock for dwellings at the end of 1996 adjusted to 1990 prices. Net real residential fixed capital formulation equals gross residential fixed capital formulation at current prices deflated by p minus depreciation at current replacement for residential dwellings deflated by p (sources: ETAS and BLUE).

263


(K ) Private Non-Residential Fixed Capital Cumulative net real private non-residential fixed capital formulation since the beginning of 1975 plus a constant derived from the official estimate of the capital stock for private nonresidential fixed capital at the end of 1996 adjusted to 1990 prices. Net real private non-residential fixed capital formulation equals gross private non-residential fixed capital formulation at current prices deflated by p minus depreciation at current replacement for private non-residential investment deflated by p (sources: ETAS and BLUE). (K a ) Cumulative Net Real Investment Abroad (excluding change in official reserves) Cumulative net real foreign investment (excluding changes in official reserves) since the beginning of 1975 plus a constant derived from the official estimate of net external assets (excluding official reserves) at the end of 1996 adjusted to 1990 prices. Net real foreign investment equals net foreign investment (excluding increase in official reserves) at current prices deflated by p (sources: ETAS and UKBP). (L) Employment Employed Labour Force (source: ETAS). (P ) Real Profits, Interest and Dividends from Abroad Net interest, profits and dividends from abroad deflated by p (source: ETAS).

264


( p) Price Level Gross domestic product at market prices divided by gross domestic product at 1990 market prices (source: ETAS). (Q ) Real Net Output Net domestic product at current prices deflated by p (source: ETAS). (q) Exchange Rate (price of sterling in foreign currency) Sterling effective exchange rate average (1990 = 1.0000) (source: BOE). (r ) Interest Rate Yield on long-dated UK government securities adjusted to instantaneous rate per quarter (source: ETAS). (S) Inventories Value of real physical increase in stocks at current prices since the beginning of 1975 plus a constant derived from the official estimate of the book value of stocks and work in progress held at the end of 1996 adjusted to 1990 prices. (sources: ETAS and BLUE). (w) Wage Rate Income from employment (including pay of forces) divided by sum of employees in employment and HM forces (sources: ETAS and MD).

265


(B) Bonds Sterling debt (British government stock) held by the domestic private sector, million, stock outstanding (source: BOE). (E o ) Real Oil Exports Oil exports at current prices deflated by p (source: ET). (Gc ) Real Government Consumption Central government consumption deflated by p plus local authorities’ consumption deflated by p (source: ETAS). (K p ) Public Non-Residential Fixed Capital Cumulative net real public non-residential fixed capital formulation since the beginning of 1975 plus a constant derived from the official estimate of the capital stock for public nonresidential fixed capital at the end of 1996 adjusted to 1990 prices. Net real public non-residential fixed capital formulation equals gross public non-residential fixed capital formulation at current prices deflated by p minus depreciation at current replacement for public non-residential investment deflated by p (sources: ETAS and BLUE). ( pf ) Price Level in Leading Foreign Industrial Countries Average of implicit price deflator of US GDP and implicit price deflator of German GDP (source: IFS).

266


( pi ) Price of Imports (in foreign currency) Implicit price deflator of imports multiplied by q (source: ETAS). (M) Money Supply Money aggregate M0 (source: BOE) (r f ) Foreign Interest Rate Average of US and German long-term government bond yields adjusted to instantaneous quarterly rate (source: IFS). (T1 ) Total Taxation Policy Variable Defined so that (Q + P)/T1 = real private disposable income. Private disposable income equals net national income at market prices plus transfers less direct and indirect taxes (souces: ETAS). (T2 ) Indirect Taxation Policy Variable Gross domestic product at market prices divided by gross domestic product at factor cost (source: ETAS). (Y f ) Real Income in Leading Foreign Industrial Countries Average of index of US real GDP (1990 = 1.0000) and index of German real GDP (1990 = 1.0000) (source: IFS).

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284

Author Index

Adkins, L. C., 21 Agbeyegbe, T. D., 14, 51 A¨ıt-Sahalia, Y., 28 Andersen, T. G., 29

Courant, R., 155, 159 Cox, J. C., 19 Dahlquist, M., 21 Donaghy, K. P., 31, 115 Dothan, U. L., 19 Duffie, D., 29 Durbin, J., 3, 218 Durlauf, S. N., 46, 66, 72

Babbs, S. H., 29, 30 Bachelier, L., 2 Bailey, P. W., 31, 115 Bandi, F. M., 28 Barnett, W. A., 43, 195, 196 Bartlett, M. S., 2 Bellman, R., 35, 174, 191 Beaglehole, D. R., 29 Black, F., 2, 7, 18 Brennan, M. J., 19, 29, 31

Engle, R. F., 46, 66 Einstein, A., 2 Episcopos, A., 26 Franses, P. H., 46, 66

Chamberlin, E. H., 141 Chambers, M. J., 14, 46, 70 Chan, K. C., 19 Chen, L., 29 Chen, R. R., 29 Coddington, E. A., 35, 174, 191 Corradi, V., 46

Granger, C. W. J., 46, 66 Gandolfo, G., 7, 31, 42, 115, 164 Hall, V. B., 31, 115 Hansen, L. P., 20, 41 Harvey, A. C., 18, 30, 46, 56, 57

285

Author Index

Henry, S. G. B., 56 He, Y., 43, 195, 196 Hiraki, T., 21 Houthakker, H. S., 10 Hull, J., 29 Ingersoll, J. E., 19 Ito, K., 2 Jiang, G., 28 Johansen, S., 46, 66 Jonson, P. D., 31, 42, 115 Kan, R., 29 Karolyi, G. A., 19 Kawai, K., 27 Kirkpatrick, G., 31, 42, 115 Knight, M. D., 31, 114, 115 Knight, J., 28 Kondratieff, N. D., 239 Koopmans, T. C., 3 Krehbiel, T., 21 Kydland, F. E., 196 Langetieg, T. C., 29 Levinson, N., 35, 174, 191 Longstaff, F. A., 19, 29 Lo, K. M., 27 Lucas, R. E., 43 Lund, J., 29 Malinvaud, E., 29 Maekawa, K., 27 Mathieson, D. J., 31, 115 McCrorie, J. R., 14 McGarry J. S., 14 Mckibbin, W. J., 42

Melino, A., 20 Merton, R. C., 2, 7, 18, 19 Moses, E. R., 31, 115 Niewenhuis, H. J., 31, 115 ˜ ıguez, T. M., 27 N´ Padoan, P.C., 31, 42, 115 Park, J. Y., 46, 66, 72 Petit, M. L., 42 Peters, S., 56 Phillips, A. W., 3, 17, 42 Phillips, P. C. B., 11, 12, 16, 17, 24, 26, 27, 28, 29, 31, 35, 41, 46, 66, 72, 115 Prescott, E. C., 196 Revuz, D., 27 Ross, S., 19 ˇ Saltoglu, B., 31 Sanders, A. R., 19 Sargan, J. D., 10 Sargent, T. J., 41 Sassanpour, C., 42 Scholes, M. 2, 7, 18 Schwartz, E. S., 19, 29, 31 Scott, L., 29 Sheen, J., 42 Simos, T., 46 ¨ B., 31, 115 Sjo¨ o, Sorwar, G., 28 Stanton, R., 28 Stavrev, E., 32, 115 Stefansson, S. B., 42 Stock, J. H., 18, 46, 56, 57 Sundaresan, S. M., 7, 18, 19

286

Author Index

Takezawa, N., 21 Taylor, L. D., 10 Tenney, M. S., 29 Trevor, R. G., 42 Tse, Y. K., 21 Tullio, G., 42

Wiener, N., 2 Wren-Lewis, S., 56 Wymer, C. R., 10, 31, 32, 34, 35, 36, 41, 42, 44, 45, 48, 58, 114, 115, 121, 123, 124, 125, 126, 132, 152, 164, 173, 174, 192, 194, 195, 196, 218, 223

Van Loan C. F., 113 Vasicek, O., 19 Wandasiewicz, S., 42, 193, 194, 196 White, A., 29

Yor, M., 27 Yu, J., 26, 27, 28 Zadrozny, P. A., 18

287

Subject Index

aliasing, 41 approximate discrete model, 25, 27 asymptotic stability, 35, 174, 197, 238 autocovariance, 48, 85 Bank of England, 30, 262 Bank of Japan, 30 bifurcation, 43, 174, 192, 195, 196 bond, 2, 19, 28–29, 117, 127, 135, 144–145, 148–149, 266–267 Brownian motion, 19, 27, 30, 56, 69, 72, 74, 76, 80, 217, 221 (BRSC), 26, 28 capital, 4, 18, 33–34, 39, 58, 115–118, 128–136, 142, 149–150, 222, 224–225, 236–237, 263–264, 266

charge cards, 118, 144 cointegrated, 46 consumption, 4, 17–18, 33, 39–40, 58, 115–117, 125–129, 136, 226, 248, 263, 266 control theory, 42, 43, 193–194, 196 cycle, 17, 41, 238–240, 248–249 deep parameters, 43 discrete time, 2–6, 8–9, 20, 26, 46, 66, 80, 82, 85, 87 dividends, 39, 115, 117, 147–148, 226, 264 drift, 19–20, 27, 56, 58, 67, 71, 116, 142 dynamic, 6–7, 13–14, 44, 46–47, 79, 115, 123–124, 128–129, 144, 150, 152, 194–195

288

Subject Index

dummy variable, 40, 117, 149 eigenvalue, 174, 191–192, 195, 215, 237–238, 240 employment, 8, 37, 39, 45, 48, 56, 58, 71, 115, 117–118, 130–134, 141, 143, 222, 225, 238, 248, 265 exact discrete model, 10–12, 16, 26–27, 29–30, 36, 48, 51–52, 58, 62 exchange control, 40, 117, 149 exchange rate, 40, 43, 116–117, 141–142, 144–145, 149–151, 180, 194, 197, 234, 265 expectations, 43–44, 57, 75–77, 80, 123–124, 128–130, 140, 151, 154–155, 158, 176 exports, 4, 32, 39–40, 58, 115–117, 136, 146, 226, 263, 266 fiscal policy, 42–43, 193–194 forecasting, 5, 8, 41, 45–47, 78–79, 125, 214, 240–241, 247, 249 foreign interest, 40, 117, 145, 148–149 gaussian estimation, 14, 17, 24, 26, 28, 30–31, 46, 48–51, 66, 248 generalized vasicek, 30 government, 3, 5–6, 8, 18–19, 40, 117, 127, 263, 265, 266–267

identification, 41–42, 44 imports, 4, 34, 39–40, 58, 115–117, 136–137, 145–146, 150, 222, 232, 263, 267 income, 17–19, 28, 33, 40, 117–118, 126–127, 144–149, 248, 265, 267 indirect taxation, 40, 118, 138, 267 inflation, 37, 45, 48, 118, 127, 134–135, 141, 143, 148–149, 222 interest rate, 18–34, 40, 115, 117, 127, 131, 135–136, 144–145, 148–151, 180, 197, 225, 242 investment, 39, 116–117, 135, 147–148, 225, 264, 266 kalman filter, 18, 30 labour, 32, 37, 44–45, 118, 123, 131–134, 139–143, 194, 222, 232–233, 236, 264 Lag distributions, 13, 58, 214, 223–224, 226, 248 Lebesgue measure, 54 Lucas critique, 196 macroeconometric, 31, 34–36, 43, 46–47, 51, 71, 114, 124, 173, 196–197, 213, 247 mean reversion, 19–20, 27

289

Subject Index

mean square, 13, 55, 241–242 monetary policy, 30, 43, 174, 194 money supply, 18, 37, 39, 45, 117, 140, 143, 151, 194, 222, 233–234, 237 non-residential, 115–117, 130, 134, 224, 264, 266 official reserves, 39, 117, 150, 264 oil, 39–40, 115–117, 146, 226, 263, 266 output, 4, 17, 33, 40, 43, 56, 58, 71, 115, 117–118, 130–133, 136–140, 142, 146, 193–194, 222, 226, 237 plastic money, 118, 144, 222 price level, 34, 40, 115, 117, 130–131, 138, 140–142, 146, 237 productivity, 44, 118, 123, 132 policy, 2, 3, 6, 30, 40, 42–43, 117–118, 125–126, 193–194, 196 ´ Poincare-Liapounov-Perron, 35, 174, 185, 191

post-sample, 41, 49, 52, 66, 74, 79–80, 214, 218, 233, 240–241, 249 residential, 116, 128–129, 225, 263 sales, 6, 131, 136–137, 146, 232–233, 241 stability analysis, 49, 173, 180, 194, 215 stochastic trends, 3, 45–51, 54, 56–57, 66, 74, 79, 114, 121, 144, 152, 215, 218, 247 taxation, 40, 117, 126, 247 technical progress, 32, 37, 45, 48, 51, 56–57, 71, 138, 141–142, 222 time inconsistency, 196 transfers abroad, 39, 115–116, 147, 263 treasury bill, 26 VARMAX, 52, 63, 65–66, 68, 70, 74, 79–80, 87 wage, 34, 40, 44, 59, 71, 115, 117, 138, 141–142, 150, 194–195

290

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