Structural time series models are models which are formulated directly in terms of components of interest. They have a considerable intuitive appeal, particularly for economic and social time series. Furthermore, they provide a clear link with regression models, both in their technical formulation and in the model selection methodology which they employ. The potential of such models is only now beginning to be realised, and it seems to be an appropriate time to write a book which provides a unified view of the area and points the direction towards future research.

The Kaiman filter plays a fundamental role in handling structural time series models. This technique was originally developed and exploited in control engineering. It has been increasingly used in areas such as economics, and a good deal of work has been done modifying it for use with small samples. Chapter 3 brings these methods together, and it can be read independently of the material on structural time series models. For those who are primarily interested in carrying out applied work with structural time series models, it should perhaps be stressed that the Kaiman filter is simply a statistical algorithm, and it is only necessary to understand what the filter does, rather than how it does it. The same is true of the frequency-domain methods which can be used to construct the likelihood function.

In discussing univariate models, it was argued that the nature of the problem allows fairly strong restrictions to be imposed. These restrictions are not normally enforced within the traditional ARIMA framework. In a multivariate set-up, the number of parameters to be estimated increases rapidly as more series are included and in a vector ARMA model the issues concerned with identifiability become quite complicated; see Hannan (1969). Hence it is even more important to formulate models which take account of the nature of the problem. Apart from saving on the number of parameters to be estimated, such models are also likely to provide more useful information on the dynamic properties of the series.

In section 1.3 a distinction was drawn between multivariate models for cross-sections of time series and multivariate models for interactive systems. This distinction is important in considering the kind of multivariate structural time series models to be entertained. For cross-sections of time series, the class of univariate structural time series models generalises in a rather natural way, as discussed in sections 8.2 to 8.4. However, the fact that several series are now being modelled together suggests the possibility of common factors. Models of this kind are introduced in section 8.5. Section 8.6 examines the way in which control groups can be handled within the statistical framework of multivariate structural time series models, while section 8.7 looks at the handling of various data irregularities.

INTRODUCTION

In this chapter we consider some important mathematical concepts which are frequently used to establish the existence of various types of equilibrium notion in the economics literature. A wide variety of equilibrium notions occur in the economics literature but the confines of space dictate that we consider only some of the more important notions. In Section 3.2, the essential mathematical concepts are presented. These are employed in Section 3.3 to establish the existence of a competitive equilibrium in a private ownership economy with a finite number of agents and commodities, and in Section 3.4, to establish the existence of a Nash equilibrium in n-person non-cooperative games.

EQUILIBRIUM MATHEMATICS

In most economic models the actions taken by an agent are determined by the values of those variables which constitute his economic ‘environment’. If those values uniquely determine the action to be taken by the agent then we can work with functions (point–point mappings). However, when the action to be taken by the agent is not uniquely determined, there is a set of possible actions and we need to work with correspondences (point-set mappings).

Continuity of correspondences

Intuitively, the concept of continuity for a mapping expresses the idea that points ‘close’ to each other in the domain of the mapping are mapped into points which are ‘close’ to each other in the range of the mapping. In this section we show how the intuitive notion of continuity can be made precise in the case of mappings which are correspondences.

A continuous time model is, in some ways, more fundamental than a discrete time model. For many variables, the process generating the observations can be regarded as a continuous one even though the observations themselves are only made at discrete intervals. Indeed a good deal of the theory in economics and other subjects is based on continuous time models. There is thus a strong argument for regarding the continuous time parameters as being the ones of interest. This point is argued very clearly in Bergstrom (1976, 1984).

There are also strong statistical arguments for working with a continuous time model. Although missing observations can be handled by a discrete time model, irregularly spaced observations cannot. Formulating the model in continuous time provides the solution. Furthermore, even if the observations are at regular intervals, a continuous time model has the attraction of not being tied to the time interval at which the observations happen to be made.

The aim of the present chapter is to set out the main structural time series models in continuous time. As with any model formulated at a timing interval smaller than the observation interval, it is important to make a distinction between stocks and flows; see section 6.3. Univariate structural models for stock variables are examined in section 9.2 and the relationship between these models and their discrete time counterparts is explored. An important result to emerge from this exercise is that the structure of the two sets of models is very similar.

The general properties of state space models were set out in chapter 3. The opening section of this chapter shows how the structural models introduced in section 2.3 can be put in state space form. The state space form provides the key to the statistical treatment of structural models. It enables ML estimators of the unknown parameters in a Gaussian model to be computed via the Kaiman filter and the prediction error decomposition. Once estimates of these parameters have been obtained, it provides algorithms for prediction of future observations and estimation of the unobserved components.

Section 4.2 describes various ways in which the unknown parameters in structural models can be estimated in the time domain. Estimation can also be carried out in the frequency domain. This latter approach has a number of attractions and is described in detail in section 4.3. (Note that even if frequency-domain methods are used for ML estimation, the state space form is still needed for prediction and estimation of the unobserved components.) Frequency-domain methods are also important in determining the asymptotic properties of ML estimators. Both asymptotic and small sample properties of estimators for structural models are considered in section 4.5. The preceding material, in section 4.4, is primarily to assure the reader that the models under consideration are identifiable. Finally, sections 4.6 and 4.7 discuss various aspects of prediction and the estimation of unobserved components.

This chapter examines a number of different topics relating to structural time series models. The first two sections deal with certain fundamental questions concerning trend and seasonality, and provide a justification of the statistical models adopted and the reasons for the shortcomings of certain other approaches. Various extensions of the trend and seasonal components of structural models are also considered.

Section 6.3 looks at the consequences of different observation and model timing intervals and shows that the principal structural models are relatively robust to changes in the observation timing interval. Data irregularities are examined in section 6.4. The Kaiman filter is an invaluable tool for handling such problems as missing observations, outliers and data revisions, and the structural approach appears to be the natural way to tackle model formulation.

The potential of state space methods for handling various types of non-linearity and structural change is explored in section 6.5, while the last section sets up models appropriate for dealing with count data and qualitative observations. Again it is argued that the structural approach is the natural way to proceed.

Trends, detrending and unit roots

This section discusses various aspects of trends. The first two sub-sections focus on the fundamental definition of a trend and the way in which a series may be decomposed into a trend and other components. It is assumed that the series in question do not contain components, such as seasonal and daily effects, which tend to repeat their pattern within a given time period.