A univariate time series consists of a set of observations on a single variable, y. If there are T observations, they may be denoted by yt, t = 1,…, T. A univariate time series model for yt is formulated in terms of past values of yt and/or its position with respect to time. Forecasts from such a model are therefore nothing more than extrapolations of the observed series made at time T. These forecasts may be denoted by ŷT+l|T, where l is a positive integer denoting the lead time.

No univariate statistical model can be taken seriously as a mechanism describing the way in which the observations are generated. If we are to start building workable models from first principles, therefore, it is necessary to begin by asking the question of what we expect our models to do. The ad hoc forecasting procedures described in section 2.2 provide the starting point. These procedures make forecasts by fitting functions of time to the observations but do so by placing relatively more weight on the more recent observations. This discounting of past observations is intuitively sensible but lacks any explicit statistical foundation. The first part of section 2.3 introduces the idea of a class of statistical models known as stochastic processes. Structural time series models are then built up by formulating stochastic components which, when combined, give forecasts of the required form. It turns out that these models provide a statistical rationale for the ad hoc procedures introduced earlier.

This book presents a coherent and systematic exposition of the mathematical theory of the problems of optimization and stability. Both of these are topics central to economic analysis since the latter is so much concerned with the optimizing behaviour of economic agents and the stability of the interaction processes to which this gives rise. A basic knowledge of optimization and stability theory is therefore essential for understanding and conducting modern economic analysis. The book is designed for use in advanced undergraduate and graduate courses in economic analysis and should, in addition, prove a useful reference work for practising economists.

Although the text deals with fairly advanced material, the mathematical prerequisites are minimised by the inclusion of an integrated mathematical review designed to make the text self-contained and accessible to the reader with only an elementary knowledge of calculus and linear algebra. We strongly urge the reader to peruse the material contained in the mathematical review before proceeding to the main text. Furthermore, Chapter 1 on convexity is to some extent a reference chapter, and can be regarded as an extension to the mathematical review in that it presents certain fundamental properties of convex sets and functions which are used throughout the text. The reader with a basic knowledge of convexity may begin with Chapter 2, where the theory of static optimization is developed. Chapter 4 on comparative statics and duality can be read immediately after Chapter 2 if so desired.

A structural time series model with explanatory variables collapses to a standard regression model when the stochastic components other than the irregular term are dropped. Thus many of the concepts and modelling procedures associated with regression are relevant to the models considered in this chapter. Some of these ideas, particularly those developed in econometrics, are reviewed in section 7.1 and an indication is given as to how they fit in with the structural approach to time series modelling.

Estimation of structural models with explanatory variables is covered in section 7.3. The preceding section lays some of the groundwork by reviewing the methods by which classical regression models may be estimated in the frequency domain. The tests set out in section 7.4 are essentially generalisations of the tests given in chapter 5 and modifications of tests used in regression. A model selection strategy is developed in section 7.5. The applications illustrate how some of the key ideas concerning model selection used in econometrics can be taken on board in the structural approach. This methodology is extended to modelling the effects of interventions in section 7.6 and a number of new diagnostics specifically designed for interventions are introduced.

The state space form is an enormously powerful tool which opens the way to handling a wide range of time series models. Once a model has been put in state space form, the Kalman filter may be applied and this in turn leads to algorithms for prediction and smoothing. The state space form is described in the first section of this chapter, while the second section develops the Kalman filter. Prediction and smoothing are described in sections 3.5 and 3.6 respectively. The Kalman filter also opens the way to the maximum likelihood estimation of the unknown parameters in a model. This is done via the prediction error decomposition and a full account can be found in section 3.4.

The present chapter can be read independently of the rest of the book, and taken as a guide to the uses of the state space models in areas outside engineering. On the other hand, those interested primarily in the practical aspects of structural time series modelling will be reassured to know that they do not have to master all the technical details of the Kalman filter set out here. The most important parts of the chapter with which to become familiar are sections 3.1 and 3.5, the earlier parts of sections 3.2, 3.4 and 3.6, and, for those interested in non-linear models, sub-section 3.7.1. The reader will also benefit by at least skimming through the remaining sections, since there is some reference back to the various algorithms in later chapters and it is useful to have some idea of what these algorithms do and how they fit into the overall picture.

SETS AND MAPPINGS

A set can be regarded as a collection of objects viewed as a single entity. The objects in the collection are called elements or points of the set. If a is an element of set A we write a∈A; a∉A denotes that a is not an element of set A. A set can be defined by listing its elements, A = {1,2,3}, or by stating a common property of its elements; A = {x:x has property P} denotes the set consisting of all objects x that have property P. When two sets A and B have identical elements they are equal, A = B: A ≠ B then denotes that the elements of A and B are not completely identical with each other. A set B is a subset of the set A, denoted B ⊂ A, if for all b∈B, b∈A. Thus, a set is also defined to be a subset of itself. When B ⊂ A and B ≠ A, B is a proper subset of A. Clearly, A = B if, and only if, A ⊂ B and B ⊂ A. The set which contains no elements is called the null or empty set, denoted ø. The null set is a subset of every set. A set can have other sets as its elements. For example, we can consider a set X = {A, B, C} whose elements A, B, C are themselves sets. Here we call X a family of subsets.

INTRODUCTION

In any static optimization problem, the objective function and the constraint functions will contain certain parameters and the optimal solution will depend on the values taken by these parameters. Thus, if any particular parameter value is altered, then we should expect the optimal choice of control variables and the maximum value of the objective function to change. The determination of the effects of parameter variations on the optimal choice of control variables and the maximum value of the objective function is referred to in the economics literature as comparative statics analysis. Section 4.2 is devoted to comparative statics analysis.

Closely related to comparative statics analysis is the theory of duality. At the heart of duality theory in economics is the notion of ‘equivalent representations’. Following Epstein (1981) we may say that:

Thus, in economics, duality refers to the existence of ‘dual functions’ which, under appropriate regularity conditions, embody the same information on preferences or technology as the more familiar ‘primal functions’ such as the utility or production function. Dual functions describe the results of optimizing responses to input and output prices and constraints rather than global responses to input and output quantities as in the corresponding primal functions.

INTRODUCTION

In the econometric model used in the first part of this book it was assumed that economic agents did not base their behaviour upon anticipations of the future; or if they did the process by which they revised their expectations was largely mechanical. This means that economic systems could be treated in principle in an analogous fashion to physical and engineering systems. Economic agents could be treated more or less as automata responding in a regular and reasonably predictable way to stimuli.

Recently the basic theory relating to the behaviour of the economy, especially at the macroeconomic level, has changed. The emphasis has shifted from an essentially backward-looking to a forward-looking perspective. This shift is associated with the rational-expectations hypothesis.

Initially the introduction of forward-looking expectations into macroeconomic models appeared to have serious implications for optimal-control theory. In particular, the neutrality proposition of Sargent and Wallace (1975) suggested that there is no systematic role for a stabilisation policy when expectations are rational. The generality of this proposition has been challenged and it is now recognised that even when expectations are forward-looking it is still possible for a stabilisation policy to have a systematic effect on economic activity. However, this does not leave optimal-control theory unaffected, because the forward-looking ‘acausal’ nature of the economy means that the standard policy-design techniques brought over from engineering are no longer applicable.

INTRODUCTION

One of the most important features revealed by the analysis of the previous two chapters was the extent to which the policy-maker needs to take account of how the private sector will react to expectations of what the policy-maker will do in the future. This raised a number of problems – of a technical nature – because of the inadequacy of recursive techniques based on dynamic programming, and also problems of a more general kind arising from attempting to ensure the ‘consistency’ of policy over time.

In this chapter and the next we examine policy as a dynamic game. Typically this will take the form of a game between competing groups who contribute to the process of policy-making – and this has been widely analysed in bargaining theory – but it could also be a game between different countries.

The two essential features of a dynamic game are the degree of cooperation between participants and the amount of information that any particular participant has about the tastes, intentions and preferences of others. In this chapter we assume that there is no cooperation between players while each player has full information about the tastes, preferences and intentions of other players. The following chapter will then deal with cases of incomplete information, bargaining models and cooperation and social optima.