This book has grown out of lectures given in M.Sc. and third-year undergraduate courses at Birkbeck College and the University of Cambridge by the author since 1980. It is designed as a text for advanced undergraduate and graduate level courses in econometric theory and as a reference book for research workers in applied econometrics.

Since the early 1970s a growing number of applied econometricians began to realise that the ‘textbook’ methodology for econometric modelling was much too rigid and narrow to be directly applicable in practice. The scope of econometric modelling was viewed by the ‘textbook’ methodology as the quantification of theoretical relationships. As a result, textbooks and courses in econometric theory tended to concentrate almost exclusively on the problem of estimation, with passing references to diagnostic testing (testing the assumptions on whose validity the estimation results depend). Econometrics textbooks encouraged the ‘myth’ that the main ingredients for constructing good empirical econometric models were a ‘good’ theoretical model and a menu of estimators (OLS, GLS, 2SLS, LIML, IV, 3SLS, FIML). Armed with these you turn the theoretical model into a statistical model by attaching a white-noise error term and after you help yourself to some observed data series your only problem is one of choosing the best estimator from the available menu. In practice, however, the reality of econometric modelling turned out to be very different, posing numerous problems for which the ‘textbook’ methodology offered no real solutions.

In the previous chapter we discussed the specification of the linear regression model as well as its statistical analysis based on the underlying eight standard assumptions. In the next three chapters several departures from [1]–[8] and their implications will be discussed. The discussion differs somewhat from the usual textbook discussion (see Judge et al. (1982)) because of the differences in emphasis in the specification of the model.

In Section 20.1 the implications of having E(yt,/σ(Xt)) instead of E(yt,/Xt = xt) as the systematic component are discussed. Such a change gives rise to the stochastic regressors model which as a statistical model shares some features with the linear regression model, but the statistical inference related to the statistical parameters of interest θ is somewhat different. The statistical parameters of interest and their role in the context of the statistical GM is the subject of Section 20.2. In this section the so called omitted variables bias problem is reinterpreted as a parameters of interest issue. In Section 20.3 the assumption of exogeneity is briefly considered. The cases where a priori exact linear and non-linear restrictions on θ exist are discussed in Section 20.4. Estimation as well as testing when such information is available are considered. Section 20.5 considers the concept of the rank deficiency of X known as collinearity and its implications. The potentially more serious problem of ‘near collinearity’ is the subject of Section 20.6.

In the previous chapter the axiomatic approach provided us with a mathematical model based on the triplet (S, ℱ, P(·)) which we called a probability space, comprising a sample space S, an event space ℱ (σ-field) and a probability set function P(·). The mathematical model was not developed much further than stating certain properties of P(·) and introducing the idea of conditional probability. This is because the model based on (S, ℱ, P(·)) does not provide us with a flexible enough framework. The main purpose of this section is to change this probability space by mapping it into a much more flexible one using the concept of a random variable.

The basic idea underlying the construction of (S, ℱ, P(·)) was to set up a framework for studying probabilities of events as a prelude to analysing problems involving uncertainty. The probability space was proposed as a formalisation of the concept of a random experiment ℰ. One facet of ℰ which can help us suggest a more flexible probability space is the fact that when the experiment is performed the outcome is often considered in relation to some quantifiable attribute; i.e. an attribute which can be represented by numbers. Real world outcomes are more often than not expressed in numbers. It turns out that assigning numbers to qualitative outcomes makes possible a much more flexible formulation of probability theory. This suggests that if we could find a consistent way to assign numbers to outcomes we might be able to change (S, ℱ, P(·)) to something more easily handled.

‘Why do we need probability theory in analysing observed data?’ In the descriptive study of data considered in the previous chapter it was emphasised that the results cannot be generalised outside the observed data under consideration. Any question relating to the population from which the observed data were drawn cannot be answered within the descriptive statistics framework. In order to be able to do that we need the theoretical framework offered by probability theory. In effect probability theory develops a mathematical model which provides the logical foundation of statistical inference procedures for analysing observed data.

In developing a mathematical model we must first identify the important features, relations and entities in the real world phenomena and then devise the concepts and choose the assumptions with which to project a generalised description of these phenomena; an idealised picture of these phenomena. The model as a consistent mathematical system has a ‘life of its own’ and can be analysed and studied without direct reference to real world phenomena. Moreover, by definition a model should not be judged as ‘true’ or ‘false’, because we have no means of making such judgments (see Chapter 26). A model can only be judged as a ‘good’ or ‘better’ approximation to the ‘reality’ it purports to explain if it enables us to come to grips with the phenomena in question. That is, whether in studying the model's behaviour the patterns and results revealed can help us identify and understand the real phenomena within the theory's intended scope.

In Chapter 22 it was argued that the respecification of the linear regression model induced by the inappropriateness of the independent sample assumption led to a new statistical model which we called the dynamic linear regression (DLR) model. The purpose of the present chapter is to consider the statistical analysis (specification, estimation, testing and prediction) of the DLR model.

The dependence in the sample raises the issue of introducing the concept of dependent random variables or stochastic processes. For this reason the reader is advised to refer back to Chapter 8 where the idea of a stochastic process and related concepts are discussed in some detail before proceeding further with the discussion which follows.

The linear regression model can be viewed as a statistical model derived by reduction from the joint distribution D(Z1 …, ZT; ψ), where, and {Zt,t∈T} is assumed to be a normal, independent and identically distributed (NIID) vector stochastic process. For the purposes of the present chapter we need to extend this to a more general stochastic process in order to take the dependence, which constitutes additional systematic information, into consideration.

In Chapter 21 the identically distributed component was relaxed leading to time varying parameters. The main aim of the present chapter is to relax the independence component but retain the identically distributed assumption in the form of stationarity.

In Section 23.1 the DLR is specified assuming that {Zt,t∈T} is a stationary, asymptotically independent normal process.