Optimal Control, Expectations and Uncertainty

Optimal control, expectations and uncertainty

Optimal control, expectations and uncertainty Sean Holly Centre for Economic Forecasting London Business School

Andrew Hughes Hallett University of Newcastle

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CAMBRIDGE UNIVERSITY PRESS Cambridge New York New Rochelle Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1989 First published 1989 British Library cataloguing in publication data Holly, Sean Optimal control, expectations and uncertainty. 1. Economics. Applications of optimal control theory I. Title II. Hughes Hallett, Andrew 33O'.O1'51 Library of Congress cataloguing in publication data Holly, Sean. Optimal control, expectations and uncertainty. Bibliography. 1. Economics - Mathematical models. 2. Control theory. 3. Rational expectations (Economic theory) 4. Uncertainty. I. Hughes Hallett, Andrew. II. Title. HB141.H62 1989 33O'.O1'51 88-9523 ISBNO 521 26444 8 Transferred to digital printing 2004

MC

Contents Preface 1 Introduction

page xi 1

2 The theory of economic policy and the linear model 2.1 The linear model 2.2 State-space forms 2.2.1 Dynamic equivalences 2.3 The final form 2.4 Controllability 2.4.1 Static controllability 2.4.2 Dynamic (state) controllability 2.4.3 Dynamic (target) controllability 2.4.4 Stochastic controllability 2.4.5 Path controllability 2.4.6 Local path controllability 2.5 An economic example

12 12 13 18 19 21 21 23 23 25 25 28 28

3 Optimal-policy design 3.1 Classical control 3.1.1 Transfer functions 3.1.2 Multivariate feedback control 3.1.3 Stabilisability 3.1.4 The distinction between stabilisation and control 3.1.5 An economic example 3.2 Deterministic optimal control by dynamic programming 3.2.1 A scalar example 3.3 The minimum principle 3.4 Sequential open-loop optimisation 3.5 Uncertainty and policy revisions 3.5.1 First-period certainty equivalence 3.5.2 Certainty equivalence in dynamic programming

31 31 31 33 34 36 37 39 43 45 48 50 50 51

VI

CONTENTS

3.6 3.7

3.5.3 Restrictions on the use of certainty equivalence 3.5.4 Optimal-policy revisions 3.5.5 Feedback-policy revisions Sequential optimisation and feedback-policy rules Open-loop, closed-loop and feedback-policy rules

4 4.1 4.2

Uncertainty and risk Introduction Policy design with information and model uncertainty 4.2.1 Additive uncertainty (information uncertainty) 4.2.2 Multiplicative uncertainty (model uncertainty) 4.2.3 Optimal decisions under multiplicative uncertainty 4.2.4 A dynamic-programming solution under multiplicative uncertainty 4.3 The effects of model uncertainty on policy design 4.3.1 Parameter variances 4.3.2 Parameter covariances 4.3.3 Increasing parameter uncertainty 4.4 Policy specification under uncertainty 4.5 Risk-averse (variance-minimising) policies 4.6 Risk-sensitive decisions 4.6.1 Optimal mean-variance decisions 4.6.2 Trading security for performance level 4.6.3 The certainty-equivalent reparameterisation 4.6.4 Risk-sensitive policy revisions 4.6.5 The stochastic inseparability of dynamic risk-sensitive decisions 4.7 The Kalman Filter and observation errors 5 Risk aversion, priorities and achievements 5.1 Introduction 5.2 Approximate priorities and approximate-decision rules 5.3 Respecifying the preference function 5.4 Risk-sensitive priorities: computable von Neumann— Morgenstern utility functions 5.5 Risk-bearing in economic theory 5.6 The utility-association approach to risk-bearing 5.7 Risk management in statistical decisions 5.7.1 The regression analogy 5.7.2 The optimal risk-sensitivity matrix 5.7.3 A simplification 5.8 Achievement indices 5.9 Conclusions: fixed or flexible decision rules?

52 52 55 56 58 61 61 64 64 65 66 67 68 69 71 72 73 75 77 77 78 78 79 80 82 87 87 88 90 92 95 97 99 99 101 102 102 103

CONTENTS

vii

6 Non-linear optimal control 6.1 Introduction 6.2 Non-linearities and certainty equivalence 6.3 Non-linear control by linearisation 6.3.1 Linearisation by stochastic perturbation 6.3.2 A linearisation in stacked form 6.4 Non-linear control as a non-linear programming problem 6.4.1 A modified gradient algorithm 6.4.2 Repeated linearisation

105 105 106 108 109 110 111 114 115

7 7.1 7.2 7.3 7.4

117 117 118 120 121 122 123 124 126

The linear rational-expectations model Introduction The rational-expectations hypothesis The general linear rational-expectations model The analytic solution of rational-expectations models 7.4.1 A forward-looking solution procedure 7.4.2 The final-form rational-expectations model 7.4.3 A penalty-function solution 7.4.4 Multiperiod solutions 7.4.5 The stochastic properties of rational-expectations solutions 7.5 Terminal conditions 7.6 The numerical solution of rational-expectations models 7.6.1 Jacobian methods 7.6.2 General first-order iterative methods 7.6.3 Non-linearities and the evaluation of multipliers 7.6.4 Shooting methods 7.7 Stochastic solutions of non-linear rational-expectations models 7.8 Controllability and the non-neutrality of policy under rational expectations 7.8.1 Dynamic controllability 7.8.2 An example: the non-neutrality of money 7.8.3 Conclusion: the stochastic neutrality of policy

141 142 144 147

8 Policy design for rational-expectations models 8.1 Introduction 8.2 The dynamic-programming solution 8.3 Non-recursive methods 8.3.1 A direct method 8.3.2 A penalty-function approach

148 148 148 150 150 151

127 128 131 132 133 136 138 139

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CONTENTS

8.4

Time inconsistency 8.4.1 A scalar demonstration 8.4.2 Suboptimal versus time-inconsistent policies 8.4.3 A simple example 8.4.4 The case of zero policy-adjustment costs 8.5 Microeconomic instances of inconsistency 8.6 Precommitment, consistent planning and contracts 8.7 Rules versus discretion 8.8 Sequential optimisation and closed-loop policies 8.8.1 Open-loop decisions 8.8.2 Optimal revisions 8.9 Sources of suboptimal decisions 8.10 An interpretation of time consistency

152 152 153 153 156 156 159 161 163 164 165 167 168

9 Non-cooperative, full-information dynamic games 9.1 Introduction 9.2 Types of non-cooperative strategy 9.2.1 The Nash solution 9.2.2 Simplified non-cooperative solutions 9.2.3 Feedback-policy rules 9.3 Nash and Stackelberg feedback solutions 9.3.1 The Nash case 9.3.2 The Stackelberg case 9.4 The optimal open-loop Nash solution 9.5 Conjectural variations 9.5.1 The conjectural-variations equilibrium 9.6 Uncertainty and expectations 9.7 Anticipations and the Lucas critique 9.7.1 A hierarchy of solutions 9.8 Stackelberg games: the full-information case 9.9 Dynamic games with rational observers 9.9.1 A direct method 9.9.2 A penalty-function method

169 169 171 171 173 174 175 176 179 180 181 182 184 187 187 189 192 192 194

10 10.1 10.2 10.3

197 197 198 202 203 204 206

Incomplete information, bargaining and social optima Introduction Incomplete information, time inconsistency and cheating Cooperation and policy coordination 10.3.1 Cooperative decision-making 10.3.2 Socially optimal decisions 10.3.3 Some examples

CONTENTS

10.4 Bargaining strategies 10.4.1 Feasible policy bargains 10.4.2 Rational policy bargains 10.4.3 Relations between bargaining solutions 10.5 Sustaining cooperation 10.5.1 An example 10.5.2 Cheating solutions 10.5.3 The risk of cheating 10.5.4 Punishment schemes 10.6 Reputational equilibria and time inconsistency 10.6.1 Asymmetric information and uncertainty 10.6.2 Generalisations of reputational effects 10.6.3 Conclusions

IX

209 212 212 214 215 215 216 217 218 219 221 224 225

Notes

227

References

231

Index

241

Preface This book was started while the first author was at Imperial College, London, and the second author at the University of Rotterdam. It began with the objective of first covering the standard optimal-control theory - largely imported from mathematical control theory - and parallel developments in economics associated with Timbergen and Theil; then secondly to relate this work to recent developments in rational-expectations modelling. However, the rate of progress has been such that the original objectives have been revised in the light of many interesting developments in the theory of policy formulation under rational expectations, particularly those concerned with the issues of time inconsistency and international policy coordination. The first six chapters are devoted to various aspects of conventional control theory but with an emphasis on uncertainty. The remaining four chapters are devoted to the implications of rational, or forward-looking, expectations for optimal-control theory in economics. We also examine the increasing use which has been made of game-theoretic concepts in the policy-formulation literature and how this is relevant to understanding how expectations play a role in policy. We owe many debts to colleagues past and present. A number of people have read and commented upon previous drafts of this book. In particular we are grateful to Andreas Bransma, Bob Corker, Berc Rustem, John Driffill, Sheri Markose, Mark Salmon and Paul Levine. We also owe a particular debt of gratitude to Francis Brooke, whose editorial patience was inexhaustible. We are also indebted to first Bunty King and then Sheila Klein, who shouldered the enormous burden of typing, retyping and typing again many, sometimes incomprehensible, drafts.

1

Introduction

Control methods have been used in economics - at least at the academic level - for about 35 years. Tustin (1953) was the first to spot a possible analogy between the control of industrial and engineering processes and post-war macroeconomic policy-making. Phillips (1954, 1957) took up the ideas of Tustin and attempted to make them more accessible to the economics profession by developing the use of feedback in the stabilisation of economic models. However, despite the interest of economists in the contribution of Phillips, the use of dynamics in applied and theoretical work was still relatively rudimentary. Developments in econometric modelling and the use of computer technology were still in their very early stages, so that there were practically no applications in the 1950s. However, matters began to change in the 1960s. Large-scale macroeconometric models were being developed and widely used for forecasting and policy analysis. Powerful computers were also becoming available. These developments plus important theoretical developments in control theory began to make the application of control concepts to economic systems a much more attractive proposition. Economists with the help of a number of control engineers began to encourage the use of control techniques. Independent of the development of control techniques in engineering and attempts by Tustin and Phillips to find applications in economics, there were also a number of significant developments in economics and in particular in the theory of economic policy. Tinbergen, Meade and Theil developed, quite separately from control theory, the theory of the relationship between the instruments that a government has at its disposal and the policy targets at which it wishes to aim. Theil (1964), in particular, developed an independent line of analysis which could be used to calculate optimal economic policies which was quite distinct from the methods emerging in optimal-control theory. Initially the work of Theil was overshadowed by modern control theory but as will be shown in this book his methods are capable of providing solutions to problems that are not directly amenable to the methods of modern control theory.

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1 INTRODUCTION

The main development on the engineering-control side was the emergence of modern control theory, which had a much firmer basis in applied mathematics than traditional concepts, many of which had developed as rules of thumb, which could be used by engineers to design control processes for engineering systems. Modern methods provided the tools needed to design guidance systems to send rockets to the moon and seemed ripe for application to economic models. They also provided a precise criterion for the performance of a system in contrast to the more ad hoc classical methods. A further attraction of modern control methods was that they offered ways of analysing economic policy under different kinds of uncertainty. In principle, optimal-policy rules could be designed which took account of both additive and multiplicative uncertainty. Furthermore, the insights provided by classical control theory concerning the importance of feedback were regarded as particularly relevant to the way in which economic policy had been conducted since the Second World War. If proper allowance was not made for the dynamic behaviour of the economy, stabilisation policy could actually make matters worse compared with a policy of non-intervention. While this observation was not new to economists, since a similar point was made by Dow (1964) in a study of the post-war performance of the UK economy and by Baumol (1961) in the context of the simple multiplier-accelerator model, optimal-control theory offered a practical way of designing policy which would prevent policy-makers from destabilising the economy and indeed would make a useful contribution to the overall stability of economic activity. According to this view economic policymaking was not being designed properly. This was not to deny that there were not a number of difficulties associated with the detailed application of control methods to economic policy-making. 1 However, the feeling was that these difficulties were amenable to technical solution and that the piecemeal introduction of control methods which could prove themselves against existing methods of policy design was the best way to proceed. This optimism proved to be misplaced. This was not because policymakers failed to recognise the benefits but because a major revolution was occurring in mainstream macroeconomics which was to pull the rug out from under optimal control. The way it was being applied to economic systems was inappropriate. The introduction of the rational-expectations hypothesis into macroeconomics produced some radical changes in the way in which economic policy was perceived. The orthodox theory of economic policy was turned on its head. In the most extreme and arresting formulation of the hypothesis there was no role for macroeconomic policy, indeed there was no need for stabilisation policy. Although it was recognised later that this analytical result also required perfectly flexible prices as well as rational or forward-looking

1 Introduction

3

expectations, it was of considerable political importance since it provided some intellectual justification for widespread disillusionment with conventional economic policies in a world suffering from rising inflation and rising unemployment. With the benefit of hindsight the effect that the rationalexpectations revolution has had on optimal-control theory - as it was being applied to economics - was salutary. It helped to highlight the basic fallacy of treating economic systems as closely analogous to physical systems. Human agents do not respond mechanically to stimuli but anticipate and act in response to expectations of what is going to happen. For a long time economists - while recognising this point in theoretical analysis - usually fell back in empirical work upon various devices which enabled them to make expectations a function only of the past behaviour of the economy. But there are other circumstances where the analogy between economic and physical systems breaks down. Decision-making by economic agents is a very complex affair, especially when a number of different groups or individuals are involved, who all may want different or competing things. They may also have insufficient information both about how the economy works and about what other individuals and groups are up to. They may also have great difficulty in giving a precise expression to their preferences and objectives in a form which can be easily characterised by an objective, utility or loss function. These difficulties may extend as far as specifying how they would respond to uncertainty and how they would rank uncertain outcomes. In this book we try to take greater account of the difficulties and special characteristics of using optimal-control techniques in economics. The area has developed sufficiently to allow us to address the particular problems found in economic systems whether they concern forward-looking expectations, risk, game theory or the specification of objective functions. This does not mean that we are able to provide complete answers to all the problems that could be examined. Many difficulties remain. Nevertheless we can be a lot more confident that, in comparison with the position that optimal control found itself in straight after the rational-expectations revolution, it is now possible to analyse models in which expectations are forward-looking under a variety of assumptions about the amount of information which is available both to policy-makers and to economic agents. The book is divided into two main parts. The orthodox optimal-control framework is covered in chapters 2 to 6. Different types of policy under different degrees of uncertainty are examined. At the same time we try to extend the standard analysis and to bring out some of the elements which we feel to be of particular relevance to economics. We also examine the effect of multiplicative uncertainty, the question of risk, and non-linearities in the econometric model. In the second part of the book we take up the question of expectations and examine how the standard optimal-control problem is

4

1 INTRODUCTION

transformed by forward-looking expectations. We provide methods by which the way in which forward-looking expectations are formed can be treated jointly with the determination of the optimal policy. However, even this task is considerably complicated by the possibility that the principle of optimality may break down. This raises the possibility that there may be incentive for policy-makers to renege on their commitments and to indulge in reversals of policy. There is a temptation to seek a solution to this problem of time inconsistency by recasting the problem of economic-policy formulation in terms of a dynamic game between the government and the private sector. However, we have to be careful because there may be no strategic interdependencies between the government and the private sector if the private sector is comprised of atomistic agents. However, there are similarities between some of the problems of game theory and those of rational expectations which we consider. Moreover, policy formulation in an open economy does raise gametheoretic considerations. In chapter 2 the basic framework and notation are set out. Since we want to remain as flexible as possible we employ the standard economic linear model, the state-space model commonly used in control theory and also the stackedvector notation introduced by Theil, in which a set of dynamic equations for T periods is rewritten in a static form. We also show how they are related to each other and how both minimal and non-minimal realisations of state-space models are related to the standard form of the econometric model. Once the basic linear framework has been set out we then examine the nature of the policy problem as one between a set of policy instruments and a set of targets or objectives at which the policy-maker is assumed to aim. The question of 'controllability', or the existence of a policy, is something which comes before the question of what is feasible and desirable and simply establishes the limits within which a policy choice should be made. In dynamic models the original assignment problem solved by Timbergen has to be modified to allow for the possibility, if not the requirement, that a policy-maker will have objectives which refer not just to some future date or 'state' but also to the path by which the future state is arrived at. In comparison with static controllability, dynamic controllability is a more rigorous requirement that a policy should fulfil, especially if the policy-maker is constrained by the frequency with which instruments can be changed. In chapter 3 the question of how optimal policies can be designed is taken up. A policy is described as optimal if it is the minimum of an objective function which, mainly for reasons of tractability, is assumed to be quadratic in deviations of the set of targets and instruments from desired or 'ideal' values. The objective function is also intertemporal in that it involves a number of time periods. This objective function is minimised subject to a dynamic linear model (the subject of non-linear constraints is taken up in

1 Introduction

5

chapter 6). Three main kinds of method are described. The first is the classical method of control theory which does not necessarily provide an 'optimal' solution in the sense of minimising an explicit objective function. Instead there is greater emphasis placed upon the dynamic properties of the system under control and upon the need to obtain, often by means of trial and error, a controller or feedback rule which is robust when things go wrong or when the model of the system is poorly specified or the system is itself subject to change over time. The second method we consider - that of dynamic programming - exploits the additively recursive structure of the objective function and breaks the problem down into a number of subproblems. The optimal policy is then obtained by solving backwards from the terminal period to the initial period, so that for each subperiod the optimal solution is obtained subject to the asyet-to-be-solved separable part of the whole problem. In the third method, which uses the stacked approach of Theil, the dynamic problem is recast as a large static problem. The optimal policy is then obtained in one go rather than by backwards substitution. All three methods have their advantages and disadvantages and, although classical techniques have a number of things to be said for them, especially when, as is often the case in economics, the system under control is poorly understood or subject to 'structural' change, we devote most of the book to the study and use of the last two methods. It is worth highlighting some of the differences between the solutions that are provided by dynamic programming and the stacked approach. The most noticeable feature is that the recursive method, at least for the case we consider in chapter 3 of a quadratic cost, a linear model and additive disturbances, produces an explicit feedback solution. It is easy to see why, for a wide variety of physical, biological and engineering systems, the dynamic-programming approach should be preferred to a stacked approach. As the number of time intervals rises the dimension of the stacked form becomes impossible to handle. The stacked form is also only suitable for discrete time models. But more significantly the stacked form only provides a solution in the shape of an open-loop policy. By open-loop is meant a policy which gives precise values for the policy instruments or control variables over the period of the plan. On the other hand a feedback solution does not provide explicit values 2 but instead it provides a function which makes the control or policy instrument dependent upon the performance of the system under control. Among other things this means that the feedback controller does not have to carry a large amount of information around. All it needs apart from the feedback rule is a solution since for a convex objective function and linear constraints there is only one policy which is optimal. However, the failure of the method of Theil to provide a policy which is responsive to the evolving behaviour of the economy is more apparent than real. It has to be conceded that the stacked approach is inappropriate when the need is to design simple, cheap and

6

1 INTRODUCTION

economical feedback mechanisms which have to function without human intervention. But when, as is normally the case with macroeconomic policy, the period of gestation between one policy action and another is quite long measured in months rather than microseconds - it is perfectly straightforward to update the stacked 'open-loop' policy in each time period as new information becomes available so that this sequential multiperiod approach actually mimics the feedback solution. Even when economic policy is conducted over quite short intervals - as with money-market or exchangemarket day-to-day intervention by a monetary authority - a formal econometric model is rarely available, so that automatic intervention in the form of a feedback rule is neither possible nor, probably, desirable. More significantly, it is useful to draw a distinction between a feedback rule and a 'closed-loop' rule. Although these are treated as synonymous in much of the literature 3 we shall use them in distinct senses. In the usual econometric model policy instruments and exogenous variables are treated as the same. But we need to distinguish between those variables which are at the discretion of the policy-maker and those - such as events external to a single country which are not. The values that exogenous variables take are an important input to most econometric models, especially when forecasts are being made. Values for the exogenous variables are produced by a variety of methods. Some of these may involve submodels such as time-series models, while others will be produced by much less formal methods. This implies that assumptions which are made about exogenous variables could change in subsequent periods. This will require the recalculation of at least the intercept term in the feedback law so that there are not necessarily any computational benefits to be gained from using a feedback rule rather than a sequential updating of the stacked solution. Henceforth we will use the term 'closed-loop' to denote a full sequential policy in which current policy is revised in the light of not only stochastic and unanticipated disturbances but also anticipated shifts in expected outcomes for the exogenous variables. It has sometimes been argued that an explicit feedback rule has the advantage of being simple and easy for economic agents to understand. In practice this may not prove to be so since the optimal-feedback rule in the most general intertemporal case is a very complex, time-varying function of the structure of preferences and the parameters of the model, though we do examine some simple cases in chapter 3 where the feedback rule does become reasonably simple. It is important not to confuse what is after all only a question about the computational advantages of one method over another with matters of real economic substance. All that matters for the policy-maker in the end is what kind of economic policy is pursued, not how it is arrived at. As long as both approaches yield the same outcome it does not really matter which is used. Where it does matter is when there is something that one method can produce

1 Introduction

1

that the other method cannot and we will examine a number of situations in this book where one method is decidedly superior to the other. Nevertheless, the preoccupation with feedback rules and with a supposed desire for an endogenous response to random disturbances masks an important aspect of economic policy-making concerned with how the policymaker responds to changes in assumptions about the future behaviour of exogenous variables. Feedback rules are essentially backward-looking and, while very sensible for engineering systems, they are not sufficiently general for economic policy to encompass situations in which policy-makers want to respond in anticipation of what they expect to happen in the future. If policymakers confined themselves to responding only to unanticipated disturbances coming through the error term in the econometric model they would be introducing an inefficient restriction upon the freedom with which policy could be carried out. In essence policy-makers must also have regard to the feedforward aspect of policy. Feedforward refers to the forward-looking, anticipatory aspect of economic policy whereby policy-makers take account of expected events and revise current policy. A combination of feedback and feedforward is the closed-loop policy referred to above. In linear models with quadratic cost functions an important simplification results from what is known as the certainty-equivalence principle. What this means is that the problem of solving a stochastic control problem can be solved by ignoring the disturbances and just solving the deterministic problem. Although feedback is not very meaningful in the deterministic case, since there is nothing on which to feed back, this separation result does mean that the task of obtaining an explicit solution is much easier. Put another way, certainty equivalence means that in the stochastic problem we only need to know the first moment of the probability distribution of stochastic disturbances. However large the uncertainty of the outcome - in the sense that there is a large variance attached to the forecast error - the optimal-feedback rule will be the same. It is important to stress that this does not mean that the actual policy which is implemented will be the same, only that the structure of the optimal-feedback rule is the same. If the variance of stochastic disturbances alters exogenously then the closed-loop response will also alter. In chapter 4 we take up a number of cases for which certainty equivalence does not hold. For example, if there are multiplicative disturbances so that the parameters of the econometric model now have a probability distribution, we find that the structure of the optimal-feedback and feedforward rule will depend upon the covariance matrix of the parameters, but that the second moment of the additive disturbances will have an effect only in so far as they co-vary with the model parameters. The effect on economic policy of parameter uncertainty was first established by Brainard, who was able to show, at least in a static model, that, as the variance of the parameter on the policy variable increased, the intensity with which the policy was pursued

8

1 INTRODUCTION

tended to diminish, that is, the strength of the feedback response to random disturbances tended to weaken. One consequence of this is that the greater the degree of uncertainty attached to parameters the less interventionary a policy will tend to be. The generality of this result to dynamic models and intertemporal objective functions is not quite so clear-cut, as we will show in chapter 4. Another important issue taken up in both chapter 4 and chapter 5 is the role of risk in economic policy. Certainty equivalence seems to permit no role for risk aversion in the formulation of economic policy if we mean by this that there is no obvious way in which policymakers' dislike of high variance can be taken account of. Risk has long been recognised as an important influence on economic decision-making at the microeconomic level though the effect that it can have at the level of macroeconomic policy has only been considered more recently. There are a number of problems with the incorporation of risk considerations into the optimal-control framework. The quadratic objective function, though it has many attractions because of its tractability, is not ideally sited to the treatment of risk since the curvature of the preference function does not have the properties normally associated with the analysis of risk in economic theory. One way of overcoming this problem is to reconsider the quadratic function as some local approximation to a more general preference function which does have the desired properties. We show that approaching the problem from this perspective permits us to reparameterise the problem as one in which the policy-maker, when specifying this objective function, must have regard for the variability of the targets and instruments about which he is risk-averse. This means that in practice the prior or interactive specification of the objective function cannot be achieved without some knowledge of the stochastic behaviour of the model of the economy and in particular of how the model behaves stochastically under closed-loop control. What this point also highlights is that there is already a strong element of sensitivity to risk implicit in a quadratic formulation. The risk premium can be interpreted as approximately equal to the variance times the coefficient of risk aversion. With a quadratic objective function we are minimising the squared deviation of both targets and instruments about a desired path; so the closedloop policy can be seen as variance-minimising behaviour on the part of the economic policy-maker. We examine in chapter 4 one further question concerned with uncertainty and that is how to handle the very common problem of what form policy should take when there is uncertainty about the initial state; or in other words when the policy-maker is uncertainty where the economy actually is when the appropriate policy decision is made. There may be incomplete information available because of delays in collecting data on the economy or initial

1 Introduction

9

measurements may be subject to subsequent revision. What is required is a filter which generates optimal estimates of the inital state and this is provided by what is known as the Kalman Filter, which uses information on both the expected value of the estimate and the variance of the measurement or data revision error to calculate the optimal estimate of the initial state. The certainty-equivalence principle still applies but it is worth noting that while the structure of the optimal-decision rule is independent of the variance of the measurement error the actual policy change which occurs in response to some disturbance to the economy will be affected by the reliability, i.e. the variance of measurements of economic activity. In chapter 6 we examine the effects of having non-linear models. The vast majority of this book is concerned with linear models. This is because of the desire to obtain as far as possible explicit analytical results. Yet practically all the econometric models which are used for forecasting and policy-making are non-linear. We consider two main approaches to the problem of non-linear models. The first adapts the strategy of linearising the model about some reference path and then applying the linear theory to the linear approximation. The second approach treats the problem of minimising a quadratic function subject to a non-linear model as a non-linear optimisation problem and applies a number of algorithms which have emerged in the last few years. Whichever approach is used the difficulties attached to linearisation and problems of approximation, as well as the well-known problems of possible multiple solutions and convergence difficulties in non-linear optimisation, have to be recognised. The second part of this book starts with chapter 7. Here we switch from the standard linear econometric model of chapter 2 to the linear rationalexpectations model in which expectations of private agents are assumed to be forward-looking. This change has major effects both upon the kind of techniques which can be used to solve models of this type and upon the whole conduct of economic policy in a world in which private agents cannot be assumed to respond passively to changes in current and future expected government actions. Chapter 7 is mainly devoted to aspects of the analytical and numerical solutions of linear and non-linear rational-expectations models, and the role of transversality or terminal conditions. Casting the rational-expectations model into a version of the stacked form of Theil proves to be a particularly useful approach and we then use this form to examine the controllability conditions for rational-expectations models. Controllability is of special importance because one of the factors which gave a boost to the interest in rational-expectations models was the result of Sargent and Wallace (1975) that in a special class of model there was no role for economic policy. This controllability result led to a large amount of literature which provided counter-examples or extensions to more elaborate models. In chapter 7 we

10

1 INTRODUCTION

provide straightforward ways of establishing the degree of controllability of any linear rational-expectations model and also apply them to the original Sargent and Wallace model. In chapter 8 we take up the question of the design of optimal policies in rational-expectations models. In comparison with the standard model this is still a relatively unexplored area and it produces a number of intriguing problems. We first show that the recursive method of dynamic programming is not capable of generating an optimal policy for the initial time period, let alone for subsequent periods. The reason for this lies in the breakdown of the additive separability of the objective function in rational-expectations models. We show that this difficulty can be easily overcome if the stacked form is adopted so that an open-loop optimal policy can be obtained for the current period. A sequential approach can then be used to update policy as new information arrives. However, there is still a serious difficulty with the implementation of the optimal policy because of the problem that as the policy-maker moves forward in time it may not be optimal for him to implement the previously optimal policy, even when there are no disturbances. This difficulty, referred to as 'time inconsistency', was first identified by Kydland and Prescott in the context of rational-expectation models, though Strotz had much earlier demonstrated the nature of the difficulty in intertemporal consumption theory. Strotz proposed two ways in which the consumer could respond. He could enter into binding commitments so that subsequently he would be committed to a particular strategy. Or he might resign himself to the fact that the optimal plan he formulates today cannot be implemented so he might as well select the current decision which is best given that he will go back on his plans in the future. He will seek the best plan among those he will follow. One such consistent plan is that provided by dynamic programming. For the macroeconomic policy-maker the problem is not quite the same since the cost of reneging will be borne by the public. Moreover, if the government can enter into binding precommitments - perhaps in the form of a constitutional amendment - it would be better if the commitment were made to the ex-ante, open-loop policy rather than the suboptimal dynamicprogramming solution - with suitable sequential revisions to policy in a stochastic environment. In practice binding constraints on governments are difficult to implement, to police and even to design given the problems attached to reversing the rule if circumstances change. One other area in which dynamic inconsistency has also been observed to arise is in game theory, and this is the subject of the remaining two chapters. We show that the question of optimal-policy design in rational-expectations models can be given a much clearer motivation if the implicit game aspects of rational-expectations models are brought to the fore. This is despite the absence of strategic interdependencies. Two questions are then important:

1 Introduction

11

first, how much information each player in a 'game' has about the intentions and constraints of the other players and, secondly, whether players cooperate or collude. Typically the government is in the position of facing a large number of firms, organisations and individuals about whom it would be next to impossible to have full information. On the other hand the public is likely to have a lot of information about what the government is up to and what its preferences are, although the public will not know everything. Noncooperative solutions have the characteristic of not being Pareto-optimal in the sense that if cooperation were possible an outcome beneficial to everyone could be found. The fundamental problem is how to calculate policies which are Pareto-efficient, how to choose among the efficient set - perhaps by bargaining - and then how to ensure that players keep to their side of the agreement. The reason that methods of enforcement can be important is that there may well be incentives for players to renege on the cooperative solution. Even in non-cooperative games cheating can be a problem if there is incomplete information about the intentions of, for example, the government so that a policy announcement is made to elicit a favourable response on the part of economic agents but a different policy is actually pursued. The question of dynamic inconsistency, cheating and reneging are, then, interwoven aspects of the same problem. We consider one way in which the possible indeterminacy of macroeconomic - and presumably microeconomic - policy can be resolved and that is in terms of reputation. One of the costs a government might bear if it reneged is a loss of its reputation so that in subsequent periods the public might attach no credence whatsoever to its announcements. This might result in a suboptimal outcome for the government and make it desirable for the government to balance the advantages at the margin of reneging or pursuing the time-inconsistent policy against the costs of a loss of reputation in the future. The theory of economic policy has come a long way from the simple targetinstrument assignment framework of Tinbergen, and from the early applications of optimal control to economics. If anything, what the introduction of expectations and game theory into the theory of economic policy does is to re-establish the political-economy features of policy-making. The design of an economic policy is not simply a matter of estimating the appropriate econometric model and of applying an algorithm to minimise some objective function - even when there are complications with uncertainty in the model and in the specification of the objective function. We need a clearer understanding of how people's views of the political climate interact with the formation of expectations and how policies can be designed in such a way as to ensure that there are not serious inconsistencies in the announcement and implementation of policies.

2

2.1

The theory of economic policy and the linear model THE LINEAR MODEL

The general linear constant-coefficient dynamic stochastic model common in the econometric literature is written in structural form: A(L)yt = B(L)xt + C(L)et + et (2.1.1) where yt is a vector of g endogenous variables, xt a vector of n policy instruments and et a vector of k exogenous variables. A(L), B(L) and C(L) are g x g, g x n and g x k matrices of polynomials in the lag operator. Thus:

r, 5 and q determine the maximum lag at which endogenous variables, policy instruments and exogenous variables enter into the structural form. Ao is subject to the normalisation rule that each of the diagonal elements is unity. The vector et of random disturbances follows a fixed distribution with zero mean and known covariances. Contrary to the custom in econometrics, we do not subsume the set of policy instruments in the exogenous variables since our primary interest is in how the values of these policy instruments are to be determined. The exogenous variables are assumed to be completely outside the influence of policy-makers. It is a characteristic of the structural form that the matrix Ao is not diagonal (i.e. Ao = I). Assuming that Ao is invertible, the instantaneous relationship between the endogenous variables is removed in the reduced form, so that each endogenous variable is expressed in terms of lagged endogenous variables and current and lagged policy instruments and exogenous variables: yt = -AoHA,^-,

+ ... + Aryt_r) + A^{B{L)xt

+ C{L)e^ + A;ht

(2.1.2)

2.2 State-space

13

forms

Since a change in both policy instruments and exogenous variables can influence the endogenous variables immediately (AQ 1BO # 0, AQ X CO ^ 0), the system represented by (2.1.1) would be described by control theorists as 'improper'. Most engineering systems, either naturally or by design, have controls (or policy instruments) which can affect the system only after a lag of one or more periods. Provided that the reduced form is stable in the sense that the roots of the polynomial matrix A(L) in (2.1.1) lie within the unit circle, A(L) can be inverted to give the final form: l

B{L)xt

yt =

* C(L)et

(2.1.3)

The final form is identical to the transfer-function or input-output (blackbox) systems developed in engineering. The rational matrix [>l(L)j - 1 [B(L)], each element of which is the ratio of two polynomials (but with identical denominator polynomials, unless common factors cancel), expresses each endogenous variable as an infinite distributed lag function of the policy instruments, exogenous variables and disturbance terms. An alternative way of writing the reduced form, and one which will be of use later, is to stack the equations as: (2.1.4) where the vectors and matrices are redefined as follows:

Ao'A

Ao

/

A=

0

0

0 / 0

A01B0

Ar

I

B =

...

Ao

1

0

0

0 0

0

C=

2.2

STATE-SPACE FORMS

In the previous section we described the structural-, reduced- and final-form representations of the linear econometric model. However, a major part of modern control theory deals with models of systems in state-space form. An essential feature (because of its mathematical tractability) is that the dynamics

14

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

of the system are described by a set of first-order difference equations. The state of a dynamic system is essentially a set of numbers such that knowledge of these numbers (together with a knowledge of the equations describing the system), plus the future sequence of policy instruments and exogenous variables, is sufficient to determine the future evolution of the system. The n~ dimensional space containing the state variables is the state space - hence the term state-space form to denote models represented in this way. The problem of translating (or realising) the input-output- or final-form representations of equation (2.1.3) into a representation involving first-order difference equations was first tackled by Kalman (1961, 1963). There is nothing, however, which makes the final form the only stage at which realisation occurs. The two best-known approaches to optimal control in economics, those of Pindyck (1973) and Chow (1975), used the reduced form as the basis for realisation; realisations of the structural form have also been explored (Preston and Wall, 1973). Indeed Chow has managed to develop his approach with hardly a mention of state space and its associated concepts (see also Aoki, 1976). The state-space form of a system of linear equations is written: zt = Fzt_x + Gxxt_x + Geet_x + GEet^

(2.2.1)

yt = Hzt + Dxxt + Deet + DEet

(2.2.2)

The vector zt is the state vector with state variables as elements. The dimension, / , of the state vector is the dimension of the state space. The task of getting from the structural, reduced or final form is denoted a realisation, and amounts to determining the set of constant matrices (F, Gx, Ge, Ge, H, D x , D e , DE) such that, for example in the final form (2.1.3), the mapping (xn et9 st) -+" yt is determined by equations (2.2.1) and (2.2.2). Equation (2.2.1) is the state equation and (2.2.2) the observation or measurement equation. State-space realisations are not unique since there are a number of ways of factorising the final or reduced forms. Kalman (1966) has proved that all minimal state-space realisations are those which have the minimum number of states. For the reduced form (2.1.2) the minimum state dimension is g, the number of endogenous variables times the maximum lag given by r, s or q. Thus if policy or exogenous variables appear with a lag greater than r there will be explicit state variables associated with policy or exogenous variables. One particular reduced-form realisation (Aoki, 1976) defines the state vector (assuming r> s,q and no noise term) as: z

t

=

z

t

AQ

^0xt

^0

^O^t

and zt_l=yt—A

0

A1yt_1—A

0

B1xt_1—A

0

C 1e t _ 1

2.2 State-space forms

15

Lagging z\ and substituting into the equation for zf2 and reorganising give: Repeating this procedure with z 3 we obtain an equivalent expression for z 3 and so on. The state-space form is then:

"A^A,

I

i^A2

o ... o"

0 /

... 0

z^i+G^^+G^.,

i

t-'A,

0

...

(2.2.3a)

0_

yt = (Ig:0)zt + Dxxt + Deet

(2.2.3b)

where

[

and

A~*

A

R 4- /? ~1

^ l 0 -^-1 A~l A ^0 ^ 2

^0 ' ^1 R 4 - R ^ 0 ' r > 2J

^ e

f~ A~ 1 A

C 4- C ~1

^ 0 ^1 \ A~ 1 A L 0 ^r

^O^^l C ~i- C ^0 ' W J

Each state is a linear combination of endogenous and exogenous variables and policy instruments and therefore has no direct economy interpretation. The convoluted structure of the Gx and Ge matrices is due entirely to the improper nature of the model. If the model is strictly proper (Bo = C o = 0) the companion matrix F retains its structure but Gx and Ge can be interpreted more easily. Moreover, the first g elements of the state vector zt refer directly to endogenous variables since the observation matrix takes on a trivial form. For sor^f greater than r we simply define extra states: — 7r+J+1 —Zt_1

-L R v A- C Y -f- £>r + jXt _ ! •+- 1^r + jXt _ !

for j = 1, max(s, q). The treatment of the noise term is exactly parallel to that for exogenous variables. If the error process is actually more general than (2.1.1) with autoregressive or moving-average disturbances, then the procedure is straightforward. If the error process is:

where now vt is a sequence of independent random processes with zero mean and known co variances, we can transform yt9 xt and et by multiplying through by V(L) and then treating the moving-average process D(L) as an extra set of exogenous variables. The main difficulty is not at the realisation stage but in obtaining estimates of the state, given that states will now be defined as random variables. To initiate control at time t we require an estimate of the

16

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

state at time t — 1, and usually this will be provided by the Kalman Filter, which generates recursive estimates of the state. But because of the particular characteristics of econometric models the noise terms in (2.2.1) and (2.2.2) are usually identical (a case when this is not so is examined in chapter 4). Thus we can solve (2.2.2) for et and substitute into (2.2.1) to give: zt = Fzt_x + G ^ . j + Geet_, + Kyt.t

(2.2.4)

where F = F-KH,

GX = GX-KDX,

Ge = Ge-KDe,

K = D~1GE

As long as F is asymptotically stable then, as Zarrop et al. (1979a) argue, the initial state zt_x can be calculated to a high degree of accuracy by running (2.2.4) from some date, /c, in the past (using historical values on y, x and e) and assuming, for example, that zt_k is zero or equal to zt _k + 1 so that for the first term period we use:

Structural realisations Some of the loss of transparency of minimal state realisations can be removed if we adopt the structural-form realisation proposed by Preston and Wall (1973). Rewriting (2.1.2) as: yt=-t

AjA^yt-j* j=l

£ Bjxt-j+ £ C^_,

j=0

j=0

where the endogenous variables have been redefined as yt = Aoyn we can then treat this as a reduced form and obtain an equivalent realisation to (2.2.3), but where now the observation equation is: This observable canonical structural-form realisation also provides explicit information about the contemporaneous matrix Ao as well as the impact multipliers Dx and De. So far we have described realisations of both the reduced and the structural forms with little attention paid to final-form or transfer-function realisations. But there may be circumstances in which a final form is worth while. For example, if the structural model is very large - which most econometric models used for policy analysis are (the question of non-linearity is taken up in chapter 6) - then the target variables may be quite a small subset of the total number of endogenous variables. Rather than provide a full realisation for all endogenous variables, which we will have to do for the reduced form, the final form can be used to strip out all but the target variables, with possibly a

2.2 State-space forms

17

considerable saving in the dimension of the state. The number of exogenous variables might still be large but this could be reduced by defining new 'exogenous' variables as linear combinations of existing exogenous variables with the weights provided by the elements in the final-form matrices, i.e. if the exogenous-variables part of the final form is: yt = A(L)et where yt is a vector of m(^g) target variables, and A(L) is a matrix of polynomials, A(L) = A o + KXL + . . . , then for the ith target in period t = 0 we can define a new 'exogenous' variable as:

where ej is the 7th exogenous variable. A scalar example of a state-space model A scalar example will help to highlight some of the basic features of state-space representations. Consider the second-order, constant-coefficient, scalardifference equation: 2 ~T~ OQX*

—| u-tXf

i "T~ O2X*

2

(2.2.5)

The realisation employed by Chow (1975) would be: zt =

(2.2.6)

Fzt_x+Gxxt

where ax F=

a2

1 0 0

&! b2 0

0

0

0

0

0

0

0

b0 _

si

0

(2.2.7)

0.

with each element of the state vector:

A = yt>

zf = xt K

t~i

Each state has a direct interpretation in terms of current and lagged values of the endogenous variables and the policy instrument. Note that x,_! appears in the state equation. An equivalent minimal realisation can be written: a2 y, =

(2.2.8) (2.2.9)

18

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

with

2.2.1

Dynamic equivalences

The equivalence between the various forms of econometric model representation can be seen from the dynamic structure. If we rewrite the state equation (2.2.1) using the matrix lag operator L as: (/ - FL)zt = GxLxt + GeLet + GELet

(2.2.10)

and then substitute into (2.2.2): + [H(I-FL)-1GeL

+ De]et

l

+ [H(I - FL)~ GEL + £D,]e

(2.2.11)

Equating terms between (2.1.2) and (2.2.11), we have: [A(Lj] ~x B(L) = U(L) = H(I-FLy1 1

GXL + Dx

1

[A(L)\ - C{L) = A(L) = H(I-FL)[A(LY\ ~*

GeL + De

= T(L) = H(I -FL)-

1

GEL + De

Taking power-series expansions: (2.2.12) (2.2.13)

i=0

(2-2.14) we have: Gx Ge o

(2.2.15) (2.2.16) (2.2.17)

The final-form matrices, IT,, A,, Ti9 i = 1,2,..., are the matrix multipliers of the econometric model. Thus n 0 = Dx, A o = De are the impact multipliers of the policy instruments and exogenous variables respectively. Similarly the

23 The final form

19

matrices Tlh Ah Ti9 i = 2 , 3 , . . . are the dynamic matrix multipliers at lag i. So each element describes the impulse response of the endogenous variables to a one-off shock to the policy instruments, exogenous variables and random disturbances. On the assumption that F is stable (all roots lie within the unit circle) the steady-state matrix multipliers (step responses) representing the equilibrium effects on the endogenous variables of a sustained unit-change in the policy instruments or exogenous variables are given by: n°° = Dx + H(I -F)~1GX

(2.2.18)

A00 = De + # ( / - F)" x G e

2.3

(2.2.19)

THE FINAL FORM

Another way to represent the linear model, in a form which will prove useful for a number of policy-optimisation problems, is to use the stacked final forms ofTheil (1964). Consider the linear reduced-form model (2.1.4). By backward substitution as far as some known initial condition y0, we can write: l

jZ

j=0

^+\ j=0

Aet.j

(2.3.1)

j=0

Consequently all the policy-impact multipliers in this system and the dynamic multipliers: dyjdxt

and

dyt _/dx f for T - t > j ^ 1

can be evaluated directly from the coefficient matrices of the reduced form: dyjdxt = B

and

dyt +j/dxt = AjB

The same holds for those multipliers of the exogenous (uncontrollable) variables, et, and those of the random disturbances. Equation (2.3.1) expresses the model in its 'final' form for given initial values, y 0 . For the planning interval t= 1 , . . . , T as a whole we can now describe the behaviour of the endogenous variables in terms of the realisations of the instruments, the uncontrollable exogenous variables and random errors as follows. If we define the stacked vectors:

then the final form model can be written (Theil, 1964): Y = RX + s

(2.3.2)

20

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

where R2

R=

_RT

...

R, _

and A

c.

I

2

AC

A

A 5 =

l

C

sT

4T-1

C

and of course Rt = A1 XB. Thus R contains all the impact and dynamic multipliers relevant to the period of study, while s collects together all the external influences on the system. X, and through it 7, is subject to the policymaker's choice. The vector s is not subject to his choice and so the value of X which is selected must be conditional on some assumed values for s. Once these are fixed the policy-maker is free to select the whole of X in one operation, or alternatively to select each component x, separately. A difficulty which arises is that, since 5 is stochastic, it will be unknown to the policymaker in advance. In this case it does matter whether X is actually calculated as a whole, or whether each component xt is calculated sequentially and therefore revised at each period. If there are sequential revisions then it will be possible to exploit any information which becomes available during the period in which policy is being implemented on the actual realisations st-j9 j ^ 1, and/or on revisions to the assumed values st+j, j ^ 0. The relationship between the stacked final forms (2.3.2) and the first-order difference equation (2.1.4) can be seen more clearly if we take the yt _ x term to the left-hand side and then stack the difference equations as: 0

/ -A

/

B

0

0

B

-A -A C

I

yT

x1 x2 XT

0 (2.3.3)

0

C

2.4 Controllability

21

and then premultiplying by: / -A

0 I -A

0

I

-l

A

/

2

A

A -A

I

A7 - 1

AT-2

gives us equation (2.3.2). It will prove useful analytically to be able to solve for the endogenous variables in this manner. Computationally, however, there is no advantage. 2.4

CONTROLLABILITY

Before we consider the relative merits of different policy-design techniques, it is important to check that sufficient intervention instruments actually exist. Given a model of an economy, in any one of the forms examined in previous sections, the fundamental question is whether, given the range of policy instruments at the disposal of the policy-maker, it is possible to reach any desired policy objectives. If this is possible, the model is said to be controllable. Thus controllability is an essential property guaranteeing the existence of a sequence of decisions which can achieve arbitrary pre-assigned values for the target variables within some finite planning interval (Preston, 1974; Preston and Sieper, 1977). And, as we shall see later in this section, and in subsequent chapters, the concept lies at the heart of most discussions of economic policy. Indeed many of the conflicts which can be observed in discussions of economic policy arise because of differing views of the degree of controllability of the economy, the effectiveness of particular instruments, and the desire to invent new instruments to improve the controllability of the economy, as well as differing views of the short-term and long-term effects of policy. Controllability, however, only establishes the necessary conditions for the existence of policy. Sufficient conditions involve much broader considerations than just the technical properties of a given model. There may, for example, be severe institutional and political constraints (apart from technical constraints such as non-negative interest rates) on the range over which policy instruments can vary, as well as on their rates of change. Moreover, there is a class of macroeconomic models which, under certain assumptions about the formation of expectations by private agents, is not always controllable. This class of model is examined later, in chapter 7. 2.4.1

Static controllability

Suppose there are m targets yt (a subset of all the endogenous variables in yt), and n instruments xt9 in each decision period t = 1 , . . . , T. A naive strategy is

22

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

to pick xt to reach the desired yt values period by period, each time taking all lagged variables and expected values of other exogenous variables as temporarily fixed. To illustrate this, suppose the targets and instruments are linked through the model (2.1.4), so that: yt = Ao M I

AjK-j + Bo*t + s + X Bjxt-j) + ut

(2.4.1)

where ut contains all the non-controllable (random) variables. Let yf, for f = l , . . . , r , be the ideal target values. Then, according to this myopic strategy, the interventions required are: xf = D~l{~/t - at) for t = 1 , . . . , T

(2.4.2)

where D contains just the rows corresponding to the target variables from A^BQ, and where dt contains the same rows from:

The vector at will be a known quantity at the start of period t if Et(ut) represents the expected value of the non-controllable elements. Obviously xf only exists if n = m and D is non-singular. (If n > m, multiple solutions exist; but (2.4.2) can still be used by transferring n — m surplus instruments into s.) Under these conditions yf will be achieved exactly in every period, provided the realised values of ut actually coincide with their expected values. The model is then said to be statically controllable. This proposition is due to Tinbergen (1952). It is often convenient to distinguish the simple necessary condition, that there should be at least as many instruments as targets (n > m), from the more complicated necessary and sufficient condition that those instruments should also be linearly independent (D non-singular). There is nothing in this framework to ensure that the interventions xr* are 'realistic', or politically and administratively feasible; the static-controllability conditions do no more than guarantee the existence of a policy package to achieve arbitrary target values. There are a number of other difficulties: (i) There is no indication of how to pick x* if n < m. (ii) It is obviously myopic because the consequences of xf will determine, in part, the values which will be necessary for x f * +1 ,.. .,x£. Policymakers are seldom indifferent to the size of the interventions, and a strategy which takes account of the implications of xf for the values needed for x* + 1 ,.. .,x% will almost certainly reduce the size of the interventions required. (iii) On what grounds is the use of Et(ut) justified? Given that we can deal with no more than expected target values here, a risk-averse strategy to reduce potential variability in the actual yt values might be preferable.

2.4 Controllability 2.4.2

23

Dynamic {state) controllability

It is possible to avoid some of these difficulties if we try to reach the ideal target values at some pre-assigned date in the not-too-distant future rather than in every decision period. The targets are allowed to take whatever values they need along the way. In the control-theory literature, controllability describes the general property of being able to transfer a dynamic system from any given state to any other by means of a suitable choice of values for the policy instruments. Ignoring the exogenous variables and the random terms in equation (2.2.1), the difference equation: z^Fz^.+G^.,

(2.4.3)

can by backward substitution be written solely in terms of the initial state and past policy-instrument values: zt = Fz0 + PtX0

(2.4.4)

where Xo is a m-dimensional stacked vector of past policy instruments X'o = (x' 0 ,xi,.. .,x,'_i)

(2.4.5)

and where Pt = (G'-1Gx9F'-2Gx9...9FGx9Gx)

(2.4.6)

is of dimension / x tn. Thus to transfer any given state to some alternative state, z°, requires the solution of the equation: z*-Ftzo-PtXo = 0

(2.4.7)

Thus the dynamic system (2.4.3) is said to be completely state-controllable if the matrix Pt in (2.4.7) is of rank / , denoted r(P) = f.

2.4.3

Dynamic (target) controllability

The property of state controllability is very important, but, as we have seen in section (2.2), a state vector in the minimal-form realisation is some combination of economic variables. This is not always easily interpreted and it is not always possible to relate the necessary conditions for controllability directly to the characteristics of an economic model. A more tractable formulation can be derived if we consider the ancillary problem of transferring the value of an endogenous variable from any initial point to any other point. For the case of no exogenous variables or random disturbances the observation equation is: yt = Hzt + Dxxt

(2.4.8)

24

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

Substituting (2.4.2) into (2.4.6): yt = HFz0 + HP*X0 + Dxxt

(2.4.9)

the system is said to be completely target-controllable if the matrix: P* = (HFt~1Gx,...,HGFx,Dx)

(2.4.10)

is of rank t (the number of endogenous variables) where t>g. But rank(F) < g, so (2.4.10) reduces to rank(P*) = g. This condition for controllability can be more readily understood since each of the terms in (2.4.10) can be seen to be constructed from the impact and dynamic multipliers of the econometric model (recall section 2.3). Indeed this controllability condition for the non-minimal realisation advocated by Chow (1975) makes explicit exactly this point. By backward substitution, (2.1.4) can be written as: PtXo

(2.4.11)

Pt = {At~lB,At-2B,...,AB,B)

(2.4.12)

where now and

and (without loss of generality) all uncontrollables have been set to zero again. The general economic system represented by (2.1.4) is therefore dynamically controllable if and only if the matrix Pt has rank g, where t > g (for a direct proof, see Turnovsky, 1977, p. 333). Hence there must be as many independent policy choices which can be made between now and t as there are targets in yr If we have just one instrument, we could in principle assign it to a different target in each period and give that target just the impulse required for the dynamics of the system to carry it along to its ideal value at t (t>g). If we had more instruments, we could reduce the lead time needed to get all targets to their ideal states because multiple assignments are possible in each period. Ultimately, if n > g, it could all be done in a single period. There is therefore a trade-off between using more instruments to reach the desired target values earlier, and allowing a longer lead time to compensate for a lack of, or a smaller number of, independent instrument choices. The lead time (or degree of policy anticipation) required to reach ydt from an arbitrary state t periods earlier, y0, is of course t — 1 periods. On the other hand there is no point in considering t>g since the final policy impulse (AgB) will, by the CayleyHamilton theorem, be a combination of the earlier impulses (AlB, for 0 < / < g — 1) so no independent instrument choices are added by extending the lead time (Preston and Pagan, 1982). Thus the necessary and sufficient

2.4 Controllability

25

condition for dynamic controllability over t periods is r a n k ^ " 1 ^ , . . .,AB,B) = g. The dimensions of this matrix show nt^g is necessary for dynamic controllability, and that implies that the choice of lead time is bounded by (g — n)/n m, which implies that the choice of t will now be bounded by (m — n)/n g)9 is assumed if and only if B

...

A2g~2B

B

9 l

(2.4.13) 0

...

A ~B

has rank at least g2 (see Preston and Pagan, 1982). However, we are hardly very interested in controlling all the endogenous variables at once. Once again this condition simplifies for the path controllability of just the target variables after premultiplying the system by the selection matrix in order to pick out the targets. In this case, path controllability implies that (2.4.14) 0 must have rank at least m2 (where t > m). In large systems, the path controllability condition - especially that for the complete system - rapidly becomes unwieldly and difficult to compute in practice. Fortunately Aoki and Canzoneri (1979) have provided a set of sufficient conditions which are easier to verify. These conditions, applied to the complete system, are formulated in terms of the g x g0 matrix E = (Ig: 0); A from (2.1.4); and D' = (D'u.. .D'r) where Dt = AtB0 + Bt} Then (2.1.4) is path-controllable if one of the following is true: (a) The first non-zero matrix in the sequence B0,EAD9.. .9EA9~1D has rank g. (b) The matrix I

has rank 2g.

2.4 Controllability

27

(c) A is non-singular and (Bo — EA~1D) has rank g. (d) Bo has rank p > g, and a permutation matrix P exists such that P(E: Eo) = r R i L say, where B% is p x # and ° has rank IE2DJ Bo and Bo differ in rank by g. (e) 0 Note that the only function of £ is as a selection matrix to pick out the leading g rows from A or D. These sufficient conditions may obviously be reformulated for the path controllability of just the target variables by defining E to annihilate the nontarget rows of Yt9 A and D, in addition to their lower g(r — 1) rows which correspond to the lags in Yt of (2.1.4). That is done by setting the unit elements in non-target rows of Ig to zero and then replacing g by m throughout the rank conditions (a) to (e) above. Thus in performing these calculations E will actually be replaced by the appropriate target-row selection matrix SE, Bo will be replaced by SB0, and the rank conditions will involve the number of target variables m. To these simple sufficient conditions, we should add one simple necessary condition derived directly from (2.4.14); namely, n^m or static controllability.2 Together these various conditions provide a practical (if approximate) way of testing for path controllability. The path-controllability conditions have been presented here directly in terms of the parameters of the econometric model. To do this we have used the discrete-time version of the Aoki-Canzoneri analysis due to Wohltmann and Kromer (1984), and have further translated their conditions from a statespace model system to the parameters of the underlying econometric model using the relationships between the two systems set out in Hughes Hallett and Rees (1983, p. 178). This has allowed the conditions to be condensed on just the target variables from a large econometric system. Further discussion of the conditions for path controllability, including corrections to several of the early statements of those conditions, will be found in Preston and Pagan (1982), Wohltmann (1984), Wohltmann and Kromer (1984) and Buiter and Gersovitz (1981, 1984). Finally, path controllability conditions may be expressed also in terms of the state-space model parameters of (2.2.1) and (2.2.2). The matrix in (2.4.13) becomes: D

HG

...

HE2f-xG (2.4.15)

0

D

with rank at least ( / + \)g. Similarly the sufficient conditions (a) to (e) are as stated above, but with D of (2.2.2) for B o , H for £, F for A, and G for D. Then / replaces the rank value of g throughout, except in (b) where 2g becomes / + g.

28

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

2.4.6 Local path controllability A less stringent requirement would be local path controllability. In this case we seek a policy sequence which drives the (expected) targets along a preassigned path from yd to yd+t over the interval (t, T+t) inclusive but not necessarily beyond. Given an arbitrary initial state y0, the necessary condition for local path controllability is (Preston and Pagan, 1982): m(T+l) (m — n)(T+ l)/n, and this reduces to zero only under static controllability. As before, the maximum lead time is m periods. Similarly the period of local controllability is given by: T + l ^ - ^ — i f m>n m-n

(2.4.17)

Hence the controlled period can be no more than mn/(m — n) if m > n, but it may be unbounded if m = n (or m < n). One could therefore regard dynamic controllability as a special case of path controllability where the control period reduces to a point (T = 0); dynamic controllability is therefore also necessary for path controllability. 2.5

AN ECONOMIC EXAMPLE

Some of the concepts of the previous sections can be reinforced with an economic example. Black (1975) has developed a simple continuous time version of the quantity theory with a Phillips curve in a closed economy. The model is written in logarithms. The log of the ratio of actual real output at its normal level (y) changes in response to deviations of actual (ms) from desired money balances (md) : y = a(ms-md)

a>0

(2.5.1)

Normal output grows at a constant, exogenously determined, rate h. The demand for money (md) is a function of real incomes, prices and the expected rate of inflation (pe): m

= k + ht + y + p-cpe

c>0

(2.5.2)

The actual rate of inflation (p) is described by the expectations-augmented Phillips curve: p = p* + vy

v>0

(2.5.3)

Expectations of the rate of inflation are determined adaptively: pe = b(p-pe)

b>0

(2.5.4)

2.5 An economic example

29

By substitution and differentiation we finally obtain the third-order differential equation for y (i.e. deviations of output from normal): y + ay + av(l — bc)y 4- abvy = am

(2.5.5)

Although this is a differential equation, conversion into state-space form is precisely analogous to that for difference equations. Given the nth-order differential equation: y ° + ai/""" 1 * + ... + ocny = poxin) + f$x{n~l) + ... + finx

(2.5.6)

(3)

i.e. y = y, etc., it can be written in first-order matrix form by defining zt = y, z2 = y 2 , . . . , z n = y", and since zt- = zi + 1 , we have: (2.5.7)

z=Az + bx y=(l

0

...

(2.5.8)

0)

where since fio = 0 a

A=

av(l-bc) abv

1

0

0

1

a b=

0 0

0 0

and each state is defined as Zi=y

z2 = y — abvy z

= y — abvy — av(l — bc)y — am

The stability properties (in the absence of feedback) of the system (2.5.7) can be determined from the state-transition matrix A. The system (2.5.7) is asymptotically stable if and only if A is a stability matrix, i.e. all the characteristic roots of kk oiA have negative real parts. The characteristic roots of A are the roots of the characteristic polynomial (Gantmacher, 1959): det(2/ - A) = P + aP + av(l - bc)X + abv

(2.5.9)

The Routh-Hurwiz criterion can be employed to test for the stability of A using the coefficients of the characteristic polynomial without evaluating the roots explicitly. The criterion can be expressed with reference to determinants formed from the coefficients of (2.5.9): 1

0

abv av(l—bc) 0

0

a ab

= abva2v(l — be) — abv

30

2 THEORY OF ECONOMIC POLICY AND THE LINEAR MODEL

'2 = 1 , ,« . \ = 1 abv av(l —be)

a2v(l-bc)-abv

The roots of the characteristic polynomial all have negative real parts if and only if the determinants V1, V2, V3 are greater than zero. Since a, b, c and v are all positive, the necessary and sufficient conditions reduce to: a2v(l —be) > abv bc g and B non-singular). (But if we are concerned with only the m targets in yt, premultiplying (2.1.19) by the selection matrix S implies we would actually

36

3 OPTIMAL-POLICY DESIGN

need n> m and static controllability again.) However, given dynamic controllability, we can pick Kt so that (A + BKt) has an arbitrary set of eigenvalues. This follows from the pole-assignment theorem (Wonham, 1974): for any real matrices C (of order g) and D (of order g x n) and a set of arbitrary complex numbers \i^ j=l,...9g, then, provided r(D9CD,.. .,C9~x) = g where r(.) denotes rank, a real matrix F exists such that (C + DF) has eigenvalues /i;,; = 1 , . . . , # . In this case we can impose any desired cyclical and growth characteristics on the model, but not an arbitrary dynamic structure. This result is independent of the values yf, and of the numbers of targets and instruments. In particular we can choose Kt such that the spectral radius p(A + BKt) < 1 for all t. That means we can always impose stability on the economy. 3.1.4

The distinction between stabilisation and control

It is perhaps useful to emphasise the difference between stabilisability and the imposition of an arbitrary dynamic structure. The former implies only that the eigenvalues of A + BKt can be selected at will, whereas the latter implies that every element in that matrix may be picked as well. Now the canonical form of that matrix is: (A + BKt) = WM

W~1=YJ

iijWjWj

(3.1.20)

where W is a matrix of eigenvectors and M one of eigenvalues /i. We have written the ;th column of W as Wj9 and thejth row of W'1 as Wj. The amplitude, cycle length and phase relations (i.e. the leads or lags) between the component cycles of (3.1.20) are all determined by the real and complex parts of fij (Theil and Boot, 1962). But the dynamic characteristics of the realisations of each y, element are dictated by how those component cycles combine; and the way in which the cycles combine is determined by the real and complex parts of the elements of W. As we have just seen, stabilisability allows us the pick of the component cycles via M. Consequently these component cycles combine to affect any given target in a way which depends on the elements of W as well as M. Hence, through stabilisability, we can arbitrarily control the amplitude, length and stability of economic cycles. But without control over Was well we cannot complete the choice of dynamics by determining how the component cycles combine within any particular target. That means we cannot force targets to hit pre-arranged values in each period; but we can ensure that they necessarily tend to such values over a longer horizon. Moreover, it is not possible to extend the pole-assignment theorem, and hence stabilisability, to a 'decoupled'-policy regime in an arbitrary economy. A 'decoupled'-policy regime is one in which one policy instrument is assigned exclusively to the achievement of each target (and vice versa) so that K

3.1 Classical control

37

becomes a diagonal matrix. We have used feedback-policy rules throughout this section since the optimal interventions can usually be written in this form (see below). Stabilisability is therefore a stochastic extension of dynamic controllability. But, because the complete dynamic structure cannot be selected at will, the stabilisation of a stochastic system is effectively the same thing as steering it. For example, we might pick policies (and Kt) by specifying suitably smooth, desired target and instrument values, and getting as close to them as possible. This is the standard control-theory approach. But it fails to guarantee that the consequences of a stochastic shock are damped out as rapidly as possible, or that the variance of each target is reduced, or that the way that cycles combine in the controlled system is necessarily satisfactory. For example, altering the cycles in the system's dynamics may lead to shocks setting up cycles with low amplitude but high frequency when the targets were intended to follow a smooth trajectory. If the short cycles become relatively more important for some targets as a result of the control rule, then the variance may be increased. There are well-known examples in Baumol (1961) and Howrey (1967,1971) where 'stabilisation', despite reducing the modulus of the roots of the system, actually increases the amplitude of the cycles for certain variables and also increases the variances of the endogenous variables. Policy design can hardly be said to be successful if the target variables are the ones with the increased variances or amplitudes.

3.1.5

An economic example

The economic example of section 2.5 can be extended to examine the role of feedback. In a discussion of the policy aspects of his model, Black (1975) considered various kinds of feedback rules for the monetary instrument, ma, designed to stabilise the economy. One possibility is to have a feedback rule for the money supply which depends on output deviations from trend due to the actual endogeneity of the money stock resulting from cyclical and autonomous movements in the fiscal stance of the public sector: ms = g-wy

(3.1.21)

Substituting (3.1.21) into (2.5.1) and re-deriving the differential equation for y we have: y + ay + a[w + v(l - bc)~]y + abvy = 0

(3.1.22)

for which the necessary and sufficient conditions for stability are then: a2[w + i;(l -be)] >abv w/v + 1 > be

38

3 OPTIMAL-POLICY DESIGN

Thus some sufficiently large w can ensure that both these conditions are satisfied if the original open-loop system is unstable, i.e. if the roots of the characteristic polynomial of equation (2.5.5) are in the left half of the complex plane. Questions of stabilisation policy, however, are not concerned only with whether an economic system is stable in this sense. An economy may be asymptotically stable in the sense that for the model described above the response of y to an exogenous shock eventually dies away, so that output returns to its normal level. If, however, the roots of the open-loop system are close to unity the amount of time which elapses before convergence may be considerable. Complex roots, moreover, will mean that the economy will oscillate about its equilibrium path. The feedback rule (3.1.21), although it can be used to ensure the asymptotic stability of the closed-loop system, is too simple to give adequate influence over dynamic behaviour. Consider instead the state-feedback rule for the money supply: (3.1.23)

rrft = kzt

where k is a 3 x 1 vector. Substituting (3.1.23) into (2.5.7) we have the closedloop system: (3.1.24)

z=(A + bk)z where a (A + bk) =

av{\ — be) abv

1 0

— akx

1

0

0 0

0

0

+

— ak2

— ak3

0

0

0

0

(3.1.25)

So the money supply is a linear function of all three state variables. In contrast the feedback rule (3.1.21) describes the money supply as a linear function of a single state only (the first state, zx). The problem now is to choose the elements of the vector k so that the coefficients of the closed-loop characteristic polynomial take whatever values are desired. The closed-loop system matrix (3.1.24) has the characteristic polynomial: (3.1.26) with Sj = a(akx - 1)

(3.1.27)

s2 = a2bvk3 + av(l — bc)(ak2 —

(3.1.28)

5 3 = abv(ak2 — 1)

(3.1.29)

Suppose the decision was to set s1 = s2 = s 3 = 0, equivalent to locating the closed-loop poles at the origin in the complex plane. This would ensure, due to the feedback rule for the money supply, that output - even when there is an

3.2 Deterministic optimal control

39

exogenous disturbance - will never deviate from its normal level. Solving for kx from (3.1.27), k2 from (3.1.29) and then k3 from (3.1.28) we have the feedback coefficients:

/c1 = l/tf;

k2=l/a;

k3=l/(a2bv)

Thus the first two feedback coefficients are equal to the reciprocal of the responsiveness of output to excess money balances while the third coefficient depends also on the elasticity of inflation with respect to excess demand (the slope of the Phillips curve) and the speed of adjustment of inflationary expectations. The comparatively straightforward way in which we were able to choose the feedback coefficients is entirely a consequence of the fact that there is only one policy instrument which enters the state-space form in a simple way. With more complex systems matters are less easy (Elgerd, 1967; Barnett, 1975). The application of the state-feedback rule (3.1.23) to (2.5.7) means that, with the closed-loop poles located at the origin, the ability of output to recover from some initial disturbance is considerably enhanced. This is only achieved, however, by requiring that the money supply adjust very rapidly to any disturbance. The further to the left in the complex plane the closed-loop poles are set, the faster convergence to the zero state can be achieved. To make the system move fast, however, may require the amplitude of instrument changes to increase considerably. Since, in practice, the instrument amplitude will be bounded, if only by institutional and political constraints, there is an implicit trade-off between the speed with which an economy can be moved to a desired state and the freedom with which policy instruments can be varied. It may well be realistic to bound the permissible fluctuations in the instruments (or at least penalise them) in order that they shall remain within what is 'acceptable' even if this entails some loss in the degree of effective control. One response to these perceived shortcomings of classical control methods is to devise explicit criteria by which the performance of a system can be judged, and for this we turn to optimal-control theory. 3.2

DETERMINISTIC OPTIMAL CONTROL BY DYNAMIC PROGRAMMING

An 'optimal' economic policy means one which is 'best' in some sense. But before the term can be given a more precise meaning we need to be able to define what is preferred in order to provide a measure of what 'best' means, and to be able to spell out the range of possible alternatives among which a choice can be made. A mathematical statement of the optimal deterministic economic-policy problem consists, first, of a description of the economy provided by equation (2.1.4): y^Ay^^Bx^Ce,

(3.2.1)

40

3 OPTIMAL-POLICY DESIGN

Secondly, we need a description of the task which is to be accomplished in terms of the targets of economic policy and the possible constraints and, thirdly, we need a statement of the criterion by which performance is to be judged. The criterion most commonly used to embody the structure of preferences is an additively separable quadratic function: J = 1/2 £ (dy'tQtSyt + dx'tNt6xt)

(3.2.2)

t= i

The matrices Nn t = 1 , . . . , T are assumed to be symmetric and positive definite while the matrices Qt, t = 1 , . . . , T are assumed to be symmetric and semidefinite. The deviations 5y and dx are defined by:

Sxt =

xt—xdt

where the superscript d denotes the desired (or preferred, or ideal) value for the target or instrument. The diagonal elements of Q and N penalise these deviations. The problem is then to determine the vector of instruments (xt: 1 ^ t ^ T) from the first time period to some (unspecified) horizon at time period T, such that the objective function (3.2.2) is minimised subject to the constraints of the econometric model described by (3.2.1). The basic characteristic exhibited by dynamic programming is that a Tperiod decision process is simplified to a sequence of T single-period decision processes. This reduction is made possible by what is called the principle of optimality (Bellman, 1957): 'An optimal policy has the property that, whatever the initial state and the initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.' If an economic policy (x*: 1 ^ t ^ T) is optimal over the interval t = 1 to t = 7", then it must also be optimal over any subinterval t = T to t = T, where 1 < T < T. Take a simple example. Suppose someone wanted to travel from London to Edinburgh by the shortest and least expensive route, and this was found to be to go via York and Newcastle upon Tyne. Now suppose the traveller had reached York and nothing had changed in the sense that no new information had been obtained about some alternative route, then it must be the case that the optimal route from York to Edinburgh is still to go via Newcastle. For the principle of optimality to hold, the minimum of (3.2.2) must be decomposable in the following way: min[J(r, T)] = min{J(f, T- 1) 4- min[J(T, 71)]} (3.2.3) where the more explicit notation is designed to make it clear to which period of time the cost applies. Thus the cost of an optimal policy from periods t = 1

3.2 Deterministic optimal control

41

to T must be equal to the sum of the cost of an optimal policy from period t = 1 to T— 1 and the cost of the optimal policy for the remaining period T. Formally, the necessary condition for the principle of optimality to hold is that the constrained-obJQdive function must satisfy the Markov property: 'After any number of decisions, t, we wish the effect of the remaining T—t stages of the decision process upon the total return (the constrained objective function value) to depend only on the state of the system at the end of the rth decision and the subsequent decisions' (Bellman, 1961). As can be seen from (3.2.3) - or from general considerations - this amounts to requiring the constrained-objective function to be additively recursive with respect to time so that the decision variables themselves are nested with respect to time intervals. If that is not the case, obviously it will not be possible to optimise the sum of objective-function components by optimising component by component sequentially backwards (as in (3.2.3), or in (3.2.5), (3.2.8) and (3.2.12) below). Thus if the unconstrained objective can be written J = lI=iJt(yi>xt)> t n e n the constrained objective must be of the form J = Yl=i(xt>--->xi)- This necessary condition for the optimality of dynamic programming is important because dynamic games and models with rational expectations introduce mutual dependence between time periods, rather than the recursive dependence found in conventional dynamic models. In those cases this necessary condition will be violated. Consider the optimisation problem for period T, the last period of the planning horizon, where xT has to be chosen to minimise: J(T, T) = l/lidy'jQjdyr + Sx'TNT5xT)

(3.2.4)

Expanding (3.2.4) we have: J(T, T) = l/2(y'TQTyT

+ x'TNTxT

- 2y'TQTydT - 2x'TNTxdT + ydTQTydT + xdTNTxdT) (3.2.5)

After substituting (3.2.1) into (3.2.5) and differentiating with respect to xT we have the optimal-feedback rule: x*T = KTyT_1+kT KT=-(NT

+

(3.2.6) B'QTB)-1B'QTA

Equation (3.2.6) is arranged in the form of a feedback rule or law. The first term on the right-hand side is referred to as the feedback matrix. The second term is the tracking gain comprised of desired values for the policy instruments, and terminal weights on the targets and instruments. The tracking gain is thus a constant independent of the state of the system.

42

3 OPTIMAL-POLICY DESIGN

Substituting (3.2.1) and (3.2.6) into (3.2.5) we have: J(T,T)= l/Ky'j^Pryr. 1 - 2y'T_xhT + aT) where pT = (A + BKT)'QT(A + #K T ) + K'TNTKT d

fcT = (.4 4- BKT)'QTy

T

(3.2.7) (3.2.7a)

- K'rNTx*

d

a T = k'TNTkT - 2QTy T(BkT d

+ CeT) -

(3.2.7b) d

2kTNTx T

d

+ yrGryr + x TNTx T + (BfcT + CeT)'QT(BkT + Ce r )

(3.2.7c)

The quadratic form of (3.2.7) provides an expression for the minimum cost at the terminal period. For some intermediate time period where T can be period T — 1 we can proceed by assuming that the solution will also be of quadratic form like (3.2.7). Thus: J(t, T) = min 1/2(5^6,^ + Sx'tNtdxt) + J(t + 1, T)

(3.2.8)

This is a difference equation with the boundary condition provided by (3.2.7). The differential of (3.2.8) with respect to xt is: a/(T,7) dxx

=

a/(T,T) dxx

t

dxz

dyxdJ(x+l,T) dyx

where the second term on the right-hand side follows because J{x + 1, T) is an implicit function of xT through the effect of x t on yx and yx on J(T -f 1, T). Rewriting (3.2.8) in the expanded form we have: J(7\ T) = l/2(>;er>'t + x;NTxt - 2yfxQxyd - 2x'xNxxd + >f &>? + xf Nrx?) + J(T + 1, T)

(3.2.10)

which, when a quadratic expression for J(T + 1, T) like (3.2.7) is substituted into (3.2.10), gives: - 2x'xNxxdx + /x'Qxyd + xd'Nxxd-] +ax + 1

(3.2.11)

where note that time period x + 1 is assumed to be the terminal period. The structure of (3.2.11) is very similar to that of (3.2.5). The expression only contains terms in yT and xT and current and future constant terms embodying system parameters, desired values and exogenous assumptions and preferences. So if we substitute y\ out, using (3.2.1), and differentiate with respect to xr we have: x? = K r y T _ 1 +fc T

(3.2.12)

3.2 Deterministic optimal control

43

We now find that the optimal feedback and tracking gains depend upon future as well as current system parameters, exogenous assumptions and preferences. Substituting (3.2.1) and (3.2.12) into (3.2.11) and rearranging again we have: J(T,T) = y't_1PTyt_1-2yx_1hr where now

+ a.

(3.2.13)

PX = (A + BKx)'(Qr + *\ + i)(A + BKX) + K'XNXKX

(3.2.13a)

hx = (A + BKxy(Qx + Px + 1/x-hx_1)-K'xNxxdx

(3.2.13b)

ax = ax_t + k'xNxkx ~ 2(Qzyf + K + l){Bkz + Cex) - 2kfxNxkx + /X'QX/X + * M

(3.2.13c)

We thus have the optimal-feedback solution for two time periods. The solution for further periods can now be obtained by using repetitions of the backward recursion described by equations (3.2.3) to (3.2.13). The recurrence equations (3.2.13a) to (3.2.13c) are matrix difference equations, also known as Riccati equations in continuous-time problems. They can be solved from the boundary conditions provided by the terminal-period equations (3.2.7a) to (3.2.7c). The optimal-feedback rule for time period T is for the deterministic case dependent on all future feedback rules, desired values and therefore exogenous variables up to the terminal period. Note also that the feedback matrix KT, although time-varying, is only a function of system parameters and the weights embodied in the preference structure of the objective function. On the other hand the tracking gains take account of the desired trajectories as well as of current and future exogenous variables. In the stochastic case future exogenous variables are only known in expectation. If assumptions about future exogenous influences were to change, this would only shift the timevarying optimal tracking gain but would not alter the feedback response. The feedback matrix is essentially backward-looking, since, even though it is affected by future preferences, it responds only to unanticipated disturbances after the event. The tracking gain is forward-looking and takes account of anticipated exogenous influences. For the stochastic case these tracking gains would therefore have to be 'kept up to date' by re-evaluating all kx for t^x^T, if one current expectation of a future exogenous variable, say Et(eT), is revised at t. In principle these re-evaluations will be repeated at each t=l,...,T. 3.2.1 A scalar example Some aspects of the use of a quadratic objective function in the optimalcontrol solution can be clarified with a simple scalar example: St

(3.2.14)

44

3 OPTIMAL-POLICY DESIGN

where exogenous influences are excluded. The objective function for a twoperiod problem is:

J=l/2 iiqjrf + Sx?) t= 1

where the weight on the instrument is assumed to be unity and the same for both periods. The feedback rule for period t + 1 is: xt + l=Kt + iyt + kt + l

(3.2.15)

with

There are a number of features of this (single-period) optimal result worth noting. If we take the limiting case where there are no costs of adjustment attached to variations in the policy instrument, then the optimal-feedback rule for the single-period problem is: -^r-i)

(3.2.16)

which when substituted into (3.2.15) gives: >>, = >>? + £, So with zero adjustment costs the autoregressive response of yt to an innovation in ef is completely eliminated. The target fluctuates randomly about its desired value with variance of as compared with a larger variance of (1 + a2)o2 in the open-loop case. With policy-adjustment costs the autoregressive coefficient on yt_x is no longer zero but is: X = a - qab2/(qb2 + n) and given n ^ 0, |A| is always less than | n (there are more targets than instruments) it is possible to solve problems with no penalties on the instruments (which is not possible with the dynamic-programming approach) 2 , or no penalties on the targets, as well as any combination in between. It is not necessary, therefore, to penalise the use of all instruments in every period or of all target variables in every period (or, indeed, to provide penalties for every period from t= 1 to T). In this sense the intermittent use of instruments and the intermittent pursuit of targets would present no difficulties. This extra flexibility is not available with the alternative optimisation procedures examined earlier although the practical value of this in economic policy-making has not attracted much attention.

50 3.5

3 OPTIMAL-POLICY DESIGN UNCERTAINTY AND POLICY REVISIONS

The policy-section procedures derived in previous sections contain three possible sources of uncertainty: (i) Uncertainty in the variables - the exogenous influences, et, and the random disturbances, st - denoted additive uncertainty, (ii) Uncertainty in the parameters - in the reduced form (2.1.4) this is reflected in A9 B and C, and in the stacked final form it is reflected in R and within the model components of s - denoted multiplicative uncertainty, (iii) The parameters of the probability-density functions of the uncertain components vary over time. In this chapter we restrict ourselves to additive uncertainty. Multiplicative uncertainty is examined in chapter 4. 3.5.1

First-period certainty equivalence

Consider the stacked final-form representation of section 3.4. The optimising value of 5Z depends explicitly on b - and through that on et and et for all of t = 1 , . . . , T, so that its choice must be conditioned on some values for these variables. To that end we replace b by its expectation, conditional on all the information available when (3.4.4) is calculated. This may be justified by the certainty-equivalence theorem: min[£(J)| t - N r x ? m}

(4.2.17)

with

•

+VBQKt+l+K'r+lNt

+ 1Kr+1]

(4.2.18)

ht+i = - ( / H - B X t + 1 ) ' [ ( Q t + 1 + P t — K'

N

rd

4- K'

+ (K'x + 1VBC+VAC)ex

+1

k

V

+ F^ +

fc^F^]

(4.2.19)

These diflference equations can be solved backwards from the boundary conditions provided by (4.2.9) and (4.2.12).

4.3

THE EFFECTS OF MODEL UNCERTAINTY ON POLICY DESIGN

The effects that multiplicative uncertainty can have on the conduct of economic policy can be seen most easily with a scalar example. Although the matrix case was examined in the last section, it is not easy to establish clear-cut conclusions in the general case.

4.3 Model multiplicative uncertainty 4.3.1

69

Parameter variances

Consider the scalar example: yt = atyt-i

+ btxt + ctet + et

(4.3.1)

where in contrast to equation (3.2.1) the coefficients are random variables with means, E(at) = a9 E(bt) = b, E(ct) — c, and variances a2, o\, a2. Possible covariances between the coefficients themselves and the error term are considered later. The objective function is as before, so that for the static, single-period problem the optimal closed-loop is: x, = K , y , - i + * r 2

Kt=-qtab/(qtb

(4.3.2) 2

+ nt + qt>,_!+/c, Kt=-

(4.3.19)

(qtab + qtpabGaab)l{qtb2 +ntqt<j2)

(4.3.20) 2

2

K = - l(qtbc + qtpbc(Jb(ic)et - ntx? - qtbyf + qtphE~\/{qtb + nt + qta )

(4.3.21)

where ptj is the correlation coefficient between i and ;. The covariance between a and b enters into the feedback gain and the covariance between b and c and between b and e appear in the feedforward or tracking gain. In this situation the direction of effect of changes in uncertainty about the impact of policy will no longer be unambiguously negative. Relative to the certainty-equivalent policy where there is no multiplicative uncertainty, whether policy is more or less vigorous in response to unanticipated disturbances will depend upon the strength of the covariation between a and b and upon the sign. If pab is negative and the covariation strong enough, it may well be that the policy is more vigorous than if there were no uncertainty, in so far as changes in a2 are associated with changes in the covariation between a and b. The covariation between b and c and between b and the error term e appear in the tracking gain. If the correlation coefficients are positive, increases in uncertainty about the impact effect of exogenous variables and the additive

72

4 UNCERTAINTY AND RISK

disturbance will produce a more vigorous response than in the certaintyequivalent case. Overall the effect of covariances is to make the effect of multiplicative uncertainty ambiguous in the single-period problem. In the two-period problem, the first-period optimal closed-loop rule is with coefficient covariances: xt = Ktyt-x

+kt

K , = -(Pt + iab + pt + 1pab(Taab)/(pt K=~

2 + 1b

+ nt

-ntxdt

[(A + 1ab + pt + 1 pabaaah)et

-b(qtydt+h

1nt

ht + l = -(a + bKt + l)[qt

+ 1(bKt + l

+cet

+1

-ydt

t + 1)

+ (Kt + Pae°a°e+ K + x GabGaGb~\

(4.3.22)

For the intertemporal problem the first-period optimal decision will be affected by the correlation between a and the additive error, £, as well as the correlation between a and c. But again these only show up in the tracking gain. Once we allow for correlation among the coefficients and the additive error, the question of how multiplicative uncertainty will affect the conduct of economic policy becomes an empirical one. Even when the multiplicative uncertainty is confined to the impact effect of policy and b is correlated with s, whether policy is more or less vigorous than under no multiplicative uncertainty will crucially depend on pbe.

4.3.3

Increasing parameter

uncertainty

It would be useful to be able to give some general results about the effect of uncertainty about policy responses on policy design. Using the 'stacked' approach of section 4.2.3, we can write (from (4.2.6)): x* = xd - (M + F ) " 1 ^ ! + / ) where M = N* + M*Et(R) + Et(R)'M*' + Et(R)'Q*Et(R)

(4.3.23)

4.4 Policy specification under uncertainty

73

F = Et(dR'Q*dR) mx = Et(R)'qi + q2 + M*Et(b) + Et(R)'QEt(b) f=Et{dR'Q*db) An increase in any element V{Ri}) of the covariance matrix V(R), for a fixed value of i= 1,.. . , m T a n d a l l ; = 1 , . . . , nT, implies dFjj>0 since Q* > 0 . But: dx* = - ( M + F)~^F(x* - xd)

(4.3.24)

where dF is a null matrix except where dFn>Q. Consequently dxf < 0 if xf > x? but dxt > 0 if xf < xf. Either way the effect of an increase in the uncertainty about the time Rtj values (for fixed i and any ;) implies \xf — x?| is reduced. Therefore increasing uncertainty about the policy responses usually forces the optimal decisions to follow their ideal values more closely. This is only an approximate result in that the covariances between R and b have been assumed to stay fixed, whereas in practice increases in V(R) may well increase those covariances too. This means that policies would become less vigorous relative to the xd values. But whether that would imply less vigorous intervention is quite another question, and depends on whether the ideal values themselves would imply vigorous policy changes or not. If they do, then increasing uncertainty of this kind should lead to greater intervention. But, in the more likely case that they do not, increasing parameter uncertainty would (as in the exmaple of section 4.3.1) lead to less vigorous interventions.

4.4

POLICY SPECIFICATION UNDER UNCERTAINTY

The broad implication of the argument in section 4.2.1 with respect to certainty equivalence is that if policy-makers do care about the variability of outcomes then it will not be sufficient to consider the deterministic solution as an equivalent solution to the stochastic problem. The policy-maker will need to know the likely degree of variability implied by his policies. Thus, even in the unlikely case in which the policy-maker could write down his relative priorities, the problem would not be fully stated until he also knew what the closed-loop variances were. This might entail a further iteration in the policyformulation process as relative weights are refined in the light of what is revealed by an analysis of the closed-loop variances. For the linear model (3.2.1) we can write analytical expressions for variances of both the targets and the policy instruments. The one-step-ahead open-loop covariance of the vector of endogenous variables is: VE

(4.4.1)

74

4 UNCERTAINTY AND RISK

where R^AR^^A'

+VE

(4.4.2)

with Ro = Vt. The covariances with feedback are: Vyt4i = (A + BKt)Rt(A + BKt)f + Kf

(4.4.3)

If we also allow for disturbances to be transmitted through the exogenous variables which we assume have covariance VCf9 so that we are permitting the covariance of the exogenous variables to vary over time, then the open-loop co variances are: Vyt + l=ARtA'

+ CVeC + VE

(4.4.4)

The closed-loop expressions are more complex because we must also allow for the feedforward term kt which we assume is able to respond to current realisations of the exogenous variables. This means: Vyi+x = (A + BKt)Rt(A + BKt)f + CVeC + VE

(4.4.5)

l

where C = C-lNT + B'{Qt + Qt + l)BY B{Qt + Pt + 1)C. By inspection of these covariances the policy-maker can gain an idea of the likely variability of target variables at any point over the policy horizon and can establish what the choice of particular penalties in the objective function implies for the variability of outcomes. Moreover, the distribution of disturbances from et and et also imply a corresponding covariance in the behaviour of policy variables, i.e.: Vx< = KtVyi_K't + (C - C)Ve(C - CY

(4.4.6)

Thus far we have assumed that the policy-maker is only concerned about how target variables vary, but there is no reason why there should not also be disutility attached to large variations in policy instruments, For example, if interest rates are used as policy instruments there may be large political costs associated with large variances in interest rates because of the effects on the residential loan repayments of households. Moreover, there may be economic costs attached to increases in the variance of policy instruments which would have to be weighed against the benefits arising from the smaller variability of target variables. The linear model we have been examining does not explicitly recognise the effects of variances on behavioural relationships. There may well be some implicit effects through structural parameters, even though these, ordinarily, would have constant variances. The multiplicative-uncertainty case is unlikely to capture these kinds of effects since we do not have the parameter variances dependent upon the variances of yf, xt and et. If active policy stabilisation is going to alter the variability of both targets and instruments, this is likely to

4.5 Risk-averse (variance-minimising) policies

75

undermine the assumption that the underlying structural parameters are relatively invariant. This raises issues and problems which we shall not attempt to address here, though it should be noted that these problems can be interpreted as a lack of structural in variance due to second-moment effects in contrast to the first-moment effects identified by Lucas (1976) and considered in chapter 7 onwards.

4.5

RISK-AVERSE (VARIANCE-MINIMISING) POLICIES

In section 4.2 we showed that certainty-equivalent decisions are invariant to risk. The same policies will always be recommended whatever the potential variability of the target variables. This situation might appear very unsatisfactory to a policy-maker. It has, in fact, been heavily criticised in economic theory, albeit in a particular context. Poole (1970), for example, showed that the proper choice of monetary instrument (the growth of the money stock, or the interest rate) depends on knowing the relative variability of money demand and national income. If the variance of national income is sufficiently large (small) then the money stock (interest rate) is the appropriate instrument, although a certain combination of the two instruments (depending on those variances) is superior to either individually. In making this argument, Poole clearly distinguishes between certainty equivalence 'in the decisions' and 'in the utility sense'. But we have just seen that certainty equivalence in (3.4.4) follows from minimising the first two terms on the right of (4.2.1). So it is the target variances (or risk term) in (4.2.1) which define whether the policy instruments have been selected correctly. If the decision rule is independent of policy target variances, then it will not be possible to pick out the best instruments with respect to the inherent risks, let alone to select optimal values for them. The rest of this chapter will be devoted to designing risk-sensitive decision procedures which react to the variability (and asymmetries) of the target variables. The same distinction between certainty equivalence 'in the decisions' and 'in the utilities' emerges. Our risk-sensitive decisions optimise the expected utilities associated with the performance level, J, actually achieved. These utilities approximate the von Neumann-Morgenstern theory of decision-making under risk. The problem arises because the decision criterion, J, is stochastic and riskneutral. It has been conventional to replace J by its expectation as the decision criterion. If this is unsatisfactory, the obvious remedy is to include the secondand higher-order moments of J. That is, to pick policies which produce not only the best performance in expectation, Et(J), but also one with the smallest variance and the power to ensure that any disturbances are more likely to appear on the more desirable side of the target's distribution than on the less

76

4 UNCERTAINTY AND RISK

favourable side. An appropriate combination of these characteristics of J would define a new decision criterion such that optimising its expectation produces policies consistent with a risk-sensitive decision-maker's tastes. The more moments included, the more are the stochastic characteristics accounted for. Indeed these moments are themselves expectations of increasing powers of the stochastic-optimisation criterion /^ = E f (J-E t (J)Y for ; = 2,3,.... Hence the linear combination Et(J) + Yj=i a./Af/ constitutes an approximation to the expectation of a standard concave utility function if appropriate weights are used for OL}. TO introduce such a combination as an objective is to specify that risk-sensitive decisions must follow from a von Neumann—Morgenstern utility function. We first examine decision rules which would reduce the variability of economic activity. Consider the case where the objective function (3.4.1) is separable between targets and instruments and contains linear penalties, i.e.: J = 1/2(SY'Q*8Y +SX'N*8X)

+ q'15Y +q'25X

(4.5.1)

If optimising the first moment of the distribution of J provides the best expected performance, then minimising its second moment reduces the probability of given variations about Et(J) and hence provides more security and less risk about the actual performance realised. Since risks are reflected in variability, a purely risk-averse decision-maker would aim to minimise the conditional variance Vt(J) = E{[J - E(J)]2\Qt}. Let G = b - E^b). Inserting (3.4.2) into (4.4.1) implies, given Ql9 that: J - E^J) = 8X'R'Q*G

+ E1(b)fQ*G +

1/2[ + 6*)^i26*0)^^0 anc * Et[b(t) + ^ i ^ o 3 replace q and Ex(b) in the linear term. The conditional variance of a(t) in these replacements is x/ = l / l t l (t)~{l/i2{l/ii[l/i2' Inserting these partitions into the final term of (4.6.7) implies the third moments 5* are replaced by: sit) = R{t)Q*t)Et{G(t)G{t)Q*t)G(t)-\

(4.6.8)

The optimally revised decisions are therefore (4.5.3) or (4.6.2) with Qft) for g*, if/t for ij/l9 the replacements for q1 and Ex(b) as above, and (4.6.8) for s*.

4.6.5

The stochastic inseparability of dynamic risk-sensitive decisions

In chapter 3, we pointed out that the constrained-objective function must have an additively recursive structure (at least under a monotonic transformation), so that the decision variables are nested backwards with respect to time, if dynamic programming is to generate optimal decisions. That is a consequence of the necessary conditions for the validity of the principle of optimality (Bellman, 1961, pp. 54-7). The underlying problem may have such a structure since inserting (4.2.8) or (3.2.1) into: J = Z, Jt = lITLih'tQf^

+ Sx'tNtSxt)

which implies that: (4.6.9) which has the required time-nested decision structure. The linearity of the expectations operator, moreover, preserves this decision structure within Et(J) for the certainty-equivalent decisions. But the quadratic nature of variances destroys this nested structure when Vt(J) enters the objective. Indeed:

Vt(J) = I [ Vt(Jt) + 2 X cov(Jr, J,)] t L

j*t

J J

(4.6.10)

where the covariances, cov(Jt,Jj), are non-zero since Jt and Jj are both quadratic functions of xti...,x1 and et,...9e1 (for j>t). These covariance terms cannot be ignored in the sense of optimising 2Kf(Jf) + 2 1 cov(J r , Jt), as would happen in deriving a decision rule for xt (t < T) by dynamic programming. But optimising (4.6.10) with respect to each xt in turn will yield

4.6 Risk-sensitive decisions

81

decision rules directly dependent on earlier decisions as well as on the rules for xj9 j> t, already determined. Thus: xt = f ( x t - l 9 . . .9xx\ Kt

+ l9kt

+ l9..

.9Kt,kt)

(4.6.11)

where the previously determined rules are of the form xt+j = Kt+jYt+j_l + kt+j for j= 1,.. .,T — t. Thus to optimise x 1? say, it will be necessary to solve xt and vice versa, that is, it will be necessary to solve for xl9.. . , x r simultaneously, which is exactly what (4.4.3) does. The joint dependence between x x , . . . , xT here is a result of the stochastic inseparability of the risk-averse objectives over time. This inseparability manifests itself in the fact that i/^, and hence the transformed priorities in Q o , are not block diagonal with respect to time. In fact the t,;th submatrix (IAIX/* where t^=j, contains E(,_!, from which the current feedback value for x, can be determined. The problem becomes the more serious the shorter the intervals over which policy responds to innovations in economic activity. But not only is there a delay in collating data, a number of important macroeconomic time series will be subject to subsequent revision as more accurate estimates become available. Moreover, estimates often become available for aggregated series, such as consumption, before data for the components, such as durable and nondurable consumption. Data may also be collected at a more frequent interval than that of the macroeconometric model. For quarterly models monthly data may provide a partial guide to the outcome for the quarter as a whole, while for yearly models both monthly and quarterly data will provide a good guide to the outcome for the whole year, well before complete published data for the year are available. The forecaster or policy-maker is also aided in obtaining an estimate for the initial state by auxiliary information provided by sources external to the model. Any macroeconometric model used for policy analysis is going to involve a level of abstraction. The practicalities and costs of handling large models place a constraint on the degree of detail they can contain. Rather than try to model every piece of possible systematic information, the model-user

4.7 The Kalman Filter and observation errors

83

will attempt to combine auxiliary information with the results of the model. For example, estimates of the current level of consumption and forecasts of future consumption can be improved by data which are available on retail sales, by recent surveys of consumer intentions or even by anecdotal evidence provided by informal contacts with the retail trade. In this section we examine formal ways of 'filtering' these various kinds of information in order to obtain the best estimate of the current state of the economy and the way in which the quality of the estimate affects the type of policy which is implemented. In chapter 3 the relative merits of a minimal versus a non-minimal statespace realisation were discussed only in terms of the direct economic interpretation of states and the numerical advantages of a minimal realisation. For the non-minimal realisation the observation equation was redundant. But when there are uncertainties about the current state the observation equation can be used as a way of modelling data inadequacies and generating the optimal estimate of the state. Consider the linear model written as: yt = Ayt _! + Bxt + Cet + et = Hyt + rjt

(4.7.2a) (4.7.2b)

where E{r\t) = 0 and E(r]ftrjt) = F t .

This formulation can be used as a flexible device for incorporating various kinds of uncertainty. Note that if the lagged state of the economy is observed without error, rjt = O, H is the identity matrix and equations (4.7.2a) and (4.7.2b) are identical to the original system. The representation described by (4.7.2a) and (4.7.2b) has been used to pool information provided by different models and to combine monthly and quarterly forecasts, to combine auxiliary non-model-based information with model forecasts, to take account of measurement error in initial data estimates, and even to obtain estimates of unobservable expectational variables. Consider, as an example, a situation in which for a quarterly model there are monthly data available for only the first two months of the previous quarter. If we define three elements of the vector z to be the monthly observations, the relevant part of the observation equation relating the jth element of the vector of quarter observations, yt-l9 to the jth, yth + 1 and ;th + 2 elements of the vector of latest measurements provided by z is: 1 1 1 0 0 0

(4.7.3)

0 0 0 with the corresponding diagonal elements of the observation-error co variance matrix being zero for the monthly observations which are available and very large for the monthly observations not yet available.

84

4 UNCERTAINTY AND RISK

If there are no data for a higher frequency than the time interval of the model, it is possible to fill out any missing observations by using the model itself to solve for an estimate for yt_x. The appropriate elements in T would then be the forecast variance-covariance error generated by the model. Feedback occurs in response to innovations in et (ignoring et); this is not observed immediately but is reflected in unanticipated changes in yt _ x, in the form of actual changes in, for example, output, employment and inflation. But if we do not have precise measurements we must rely on more or less accurate estimates in the form of zt _ x. These estimates will change from period to period as new information becomes available. This information is filtered to provide the optimal estimate to the extent that, if certain sources of information, such as preliminary data estimates, provide poor guides to the actual outcome, a relatively low weighting will be attached to them. Filtered estimates can be generated recursively so that previous estimates are updated in the light of the most recent observations. This has the advantage that recursive estimates economise on both the number of calculations which have to be done and the amount of information which needs to be carried from period to period. The method of recursive estimation of the state is referred to as the Kalman Filter. Because of its convenience it has been of particular use in aerospace engineering. But it has also seen many applications in economics. It has been most widely used in the recursive estimation of parameters, used in tests for structural change (Rustem and Velupillai, 1979), for misspecification and for the recursive generation of residuals (Harvey, 1981), as well as for a variety of other estimation and hypothesis-testing problems. It has also been applied to problems arising from the day-to-day use of econometric models in forecasting and policy analysis, the pooling of short-term, medium-term and long-term model forecasts and the filtering of information (Kalchbrenner and Tinsley, 1980). A decision to alter policy at time period t is assumed to be based on information up to that moment. This information is assumed to be incomplete. We thus want the best estimate of yt-l9 conditional on information available up to period r, denoted by yt_ u , such that the estimate is optimal, i.e. we want to minimise The unconventional dating of the conditioning information set relative to the period for which an estimate is required is intended to reflect the disparate nature and uneven character of the information set available. While there may be accurate measurements of current interest and exchange rates, reliable measure of inventory behaviour may only be available for six to nine months before. However, there is no problem in changing the data of the conditioning information set to f — 1.

4.7 The Kalman Filter and observation errors

85

Given our prior knowledge of the conditional probability density of yt_x based on the information set Qt, p(yt-i\Q, we have: l|"r) = £-!,,

(4.7.5)

so that E(yt\Qt) = Ayt_u

+ Bxt + CeUi

(4.7.6)

In this case we assume that the generation of the expectation for et (and for exogenous variables in future periods) is a completely separate issue, though in principle we should also treat the problem of updating the exogenous assumptions as a filtering problem. The conditional covariance of yt-u is defined as: 1 |Q f )

= Af_1

(4.7.7)

so that cov(yt|Qt) = A\t-XA'

+ r, = A,

(4.7.8)

The likelihood function, determined by (4.7.4), together with the normal probability-density function (PDF) of the observation error nn is: p(Q t |^_ 1 ) = constxexp(l/2||z f _ 1 -/fj; f _ 1 J| 2 r r - 1 )

(4.7.9)

The observation-error variance-covariance matrix will be in general timevarying because of possible fluctuations in the type and quality of information available to policy-makers. The prior PDF for the random variable yt_x before observing zt_x is: /7(>;f_1|Q) = constxexp(l/2||j; r _ 1 , t -^_ 1 , f _ 1 || 2 rr 1 )

(4.7.10)

so the conditional PDF, for yt_\ given z,_1? is, by Bayes' law: i,,-A-u-i||2Ar1)

(4.7.11)

Differentiating (4.7.11) with respect to yt-u we have: (HTr^+Ar^-i^ArVr-u-i+HTr1^^

(4.7.12)

Using the matrix lemmas: (/fTt-1H + A r - 1 )" 1 =Ar 1 -A t // / (/fAr 1 //+r t )- 1 //A t ~ 1

(4.7.13)

and (HT-1H + At-1)HT-1=A-1Hf(HAt-1Hf

+ Tt)-1

(4.7.14)

equation (4.7.12) can be rewritten: yt-i,t = Pt-ut-i + Ft(zt-i ~Hyt-^-i)

(4.7.15)

86

4 UNCERTAINTY AND RISK

where The best current estimate of yt _x is therefore a linear combination of the previous best estimate and a correction for the difference between the previous estimate and the latest observations. The filter gain, Fn is independent of the structure of the optimal closedloop rule, which now is: xf = K,JUi., + *i

(4-7.16)

where the precisely observable yt_1 used in the formulation of chapter 3 is replaced by the optimal estimate yt-u provided by (4.7.15). The separation between the structure of the closed-loop rule, represented by the feedback and feedforward terms Kt and kt, and the measurements made of innovations to st does not mean that actual policy innovations will be independent of the quality of information available. Consider the scalar case: y^ayt-i+bXt

+ Bt

(4.7.17)

where observations of yt _ x are provided by: zf_2 =y f _ x + ^ r

(4.7.18)

where E(rjt) = 0 and E(rj'trjt) = o*v The optimal estimate of yt _ x at time period t is: yt-u = yt-u-i

+ [^/K2r + ^)]fe-i -9t-u-i)

(4.7.19)

where o*t is a time-varying measure of the quality of observations of yt _ x. When o*t is zero there are precise observations of yt_x available. But, if for some reason there are doubts about the reliability of recent observations, policy revisions at time period t, relative to what they otherwise would have been on the basis of information available at t — 1, may be more cautious. In some circumstances it is possible to imagine the policy-maker continuing with the previously optimal 'open-loop' policy because of the complete unreliability of the latest measurements. The practical pursuit of a closed-loop policy will, therefore, be affected by the uncertainty attached to current estimates of the state of the economy. The more uncertain these estimates, the less the policy-maker will depart from the policy he originally settled upon on the basis of information available in previous periods.

Risk aversion, priorities and achievements 5.1

INTRODUCTION

The selection of one preferred policy package from a set of candidates involves the use of at least an implicit criterion. In policy studies, specifically those using optimal-control techniques, this criterion is made explicit. Yet a precise and realistic specification of the parameters defining the derivatives of an explicit-criterion function is extremely difficult. This difficulty is compounded by the fact that policy-makers have been reluctant, and usually unable, to specify their relative priorities in advance of a planning exercise. Typically they need to get a feel for how much they can or cannot expect to achieve; they need to establish the trade-offs available between policy goals, and the likelihood of not being able to reach them, in order to set the limits to policymaking and acceptable compromise. If it is hard for policy-makers to set relative priorities in explicit numerical form, it is certainly no easier for their advisory staff. As we have pointed out, the specified objectives will be expressed in the form of an approximation to some generally unknown, or incompletely known, collective-preference function. This preference function is unknown partly because policy-makers are unable to give a full description of their relative priorities for all possible events, but also because they cannot know in advance the preferences, bargaining strengths or solutions acceptable to the other decision-makers who influence economic decisions. If other decision-makers are involved, then to ignore their influences on the final decision might introduce serious errors. An approximate-criterion function has to be evaluated, at least implicitly, at some point of the policy space. In this chapter we shall work with the multiperiod certainty-equivalent decision rules developed in sections 3.4 and 3.5. This is for the convenience of being able to examine the preference structure over all planning periods simultaneously, conditioned on arbitrary information available at any one (and hence all) of the decision points.

88 5.2

5 RISK AVERSION, PRIORITIES AND ACHIEVEMENTS APPROXIMATE PRIORITIES AND APPROXIMATE-DECISION RULES

A quadratic objective function generally can be regarded as no more than a local approximation to the true preferences, evaluated about some particular policy values, just as any model approximates the behaviour of the economy about the observed levels of its variables. If policy values SZ are rationally chosen, then the value of some index of economic performance can be associated with each SZ value. The ability to do no more than rank SZ values in terms of this index (which is the direct consequence of presuming rational choices as a behavioural theory for policymakers) implies that the index is, at most, an ordinal preference function of SZ, say: J = J(SZ)

(5.2.1)

which is convex and written as a loss function so that the optimal policy is the minimiser of (5.2.1). Preferences are expressed as relative penalties on failing to reach the desired or ideal values Zd. This immediately introduces a restriction on the preference function since it implies that the unconstrained minimum of (5.2.1) lies at Zd and so satiation is implied. One way of ensuring that policy choices are always made on the falling portion of the loss function is to set the desired values sufficiently large/small to ensure that the marginal loss is always falling. A second-order expansion of (5.2.1) about some feasible point Z* ignoring the constant, to which minimisation is invariant, and terms other than second order - gives: J^SZ'QSZ + q'SZ

(5.2.2)

where

G= Thus Q is symmetric and positive definite if (5.2.1) is strictly convex and twice differentiable, at least over the feasible region of the model. Now (5.2.2) makes clear that the relative penalties on the 'failures' SZ, appropriate for the specification of J(Z), must always be dependent on the actual dZ values chosen. These penalties may well be sensitive to small changes in dZ values. Indeed there are strong reasons for arguing that this is the typical case in economic policy-making.1 Thus an objective function of the form (5.2.2) cannot normally be expected to be valid for more than a rather small area of the feasible policy space. Moreover, (5.2.2) specifies ideal values of Zd — Q~1q, so that q = 0 if Zd represents a genuinely ideal path for the economy. However, (5.2.2) could also be specified with q non-zero and with Zd replaced by the artificial values Za = Zd + Q~1q. Finally q could be specified as non-zero to

5.2 Approximate priorities and decision rules

89

give preferences which are asymmetric about the true Zd values (because dJ/dZ = q is evaluated at Zd). Presumably q would be chosen so that the constrained optimum 8Z* has the more desirable sign in each element. The first two options are equivalent, whereas the last biases (5.2.2) to give results for the optimal polices over a wider part of the policy space than is available from a second-order approximation to (5.2.1). Whether the final option is used will depend on the amount of prior information there is available on Zd or on J(SZ). The marginal measure of local relative risk aversion 2 for Z t , holding other variables constant, would be:

so that the quadratic approximation already embodies a measure of risk aversion through the choice of the desired values Z d , even though this might conflict with the need to ensure that choices are always being made on the falling portion of the loss function. Taking the quadratic objective function separately, relative risk aversion is: -Z^dZ,)-1 (5.2.4) so that relative risk aversion increases as we approach the unconstrained optimum from one side and diminishes as we approach it from the other. Thus procedures for incorporating risk aversion into policy-making are closely tied up with the specification of the objective function. If we introduce a risk premium, which for many cases (Newberry and Stiglitz, 1981) is approximately equal to one half the variance times the coefficient of absolute risk aversion, then we can link the variance-minimising policies of chapter 4 with standard measures of risk aversion. Thus the size of the risk premium depends on the shape of the preference function and the variance of outcomes. Note that in this section we have included both targets and policy instruments so the question of the variance of the policy-instrument settings is also relevant to the analysis of risk. The risk premium is a measure of the cost of risk in terms of a fall in expected utility. This brings us to our second point about the analysis ofriskin macroeconomic policy-making. The standard analysis of risk in decisionmaking involves questions about how individuals and firms respond to risk. Macroeconomic policy may involve risk for policy-makers because of the chances of being re-elected but, if the policy-maker's objective function is identified with a social-welfare function, then the policy-maker may be averse to overall non-diversifiable market risk. For example, in the capital-asset pricing model (Sharpe, 1964; Lintner, 1965; Mossim, 1966) the cost of capital used to discount a stream of expected cash flows for a particular firm will reflect both the unique risk associated with this stream and overall market risk. For the case where the expected returns of an individual firm are perfectly

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5 RISK AVERSION, PRIORITIES AND ACHIEVEMENTS

correlated with expected market returns the cost of capital is the same as the average market cost of capital, and in turn this cost of capital will reflect nondiversifiable market risk relative to assets with a safe return. Individual lenders and borrowers can reduce the risk attached to investing in fixed and financial assets by diversifying. But the minimum risk is the average market risk and this in turn is determined by macroeconomic factors inducing unpredictable and stochastic cyclical movements in the economy. If economic policy-makers can moderate cyclical movements then it is possible that market risk will be reduced and the average cost of capital fall. This may have beneficial supply side-effects by increasing aggregate investment. Individuals can reduce overall risk by diversification both between safe and risky assets and between alternative risk assets which have negative covariances. In most cases macroeconomic policy-makers cannot reduce policy risks by diversification; although it is possible to have circumstances in which small countries might adopt strategies - such as overseas fixed and portfolio investment - to spread the risks of an internal stabilisation policy.

5.3

RESPECIFYING TtfE PREFERENCE FUNCTION

We first review the question of how to specify relative priorities for the targets and instruments in a multiperiod problem, when the policy-makers' 'true' collective preferences are not known. This will be done for a fixed information set, but later we shall consider the respecification induced by the policymakers' risk sensitivity. Respecifications, for given information, can be carried out using an algorithm due to Rustem et al. (1979) and later extended by Ancot et al. (1982) and Hughes Hallett and Rees (1983). It is based on a variable metric algorithm which uses a 'downhill' property, i.e. J[c5Z(5+1)] < J[