Introduction

In the discussion of descriptive statistics in Part I it was argued that in order to be able to go beyond the mere summarisation and description of the observed data under consideration it was important to develop a mathematical model purporting to provide a generalised description of the data generating process (DGP). Motivated by the various results on frequency curves, a probability model in the form of the parametric family of density functions Φ = {f(x; θ), θ ∈ Θ} and its various ramifications was formulated in Part II, providing such a mathematical model. Along with the formulation of the probability model Φ various concepts and results were discussed in order to enable us to extend and analyse the model, preparing the way for statistical inference to be considered in the sequel. Before we go on to consider that, however, it is important to understand the difference between the descriptive study of data and statistical inference. As suggested above, the concept of a density function in terms of which the probability model is defined was motivated by the concept of a frequency curve. It is obvious that any density function f(x; θ) can be used as a frequency curve by reinterpreting it as a non-stochastic function of the observed data. This precludes any suggestions that the main difference between the descriptive study of data and statistical inference proper lies with the use of density functions in describing the observed data. ‘What is the main difference then?’

Econometrics – a brief historical overview

It is customary to begin a textbook by defining its subject matter. In this case this brings us immediately up against the problem of defining ‘econometrics’. Such a definition, however, raises some very difficult methodological issues which could not be discussed at this stage. The epilogue might be a better place to give a proper definition. For the purposes of the discussion which follows it suffices to use a working definition which provides only broad guide-posts of its intended scope:

This definition is much broader than certain textbook definitions narrowing the subject matter of econometrics to the ‘measurement’ of theoretical relationships as suggested by economic theory. It is argued in the epilogue that the latter definition of econometrics constitutes a relic of an outdated methodology, that of the logical positivism (see Caldwell (1982)). The methodological position underlying the definition given above is largely hidden behind the word ‘systematic’. The term systematic is used to describe the use of observed data in a framework where economic theory as well as statistical inference play an important role, as yet undefined. The use of observed data is what distinguishes econometrics from other forms of studying economic phenomena.

Econometrics, defined as the study of the economy using observed data, can be traced as far back as 1676, predating economics as a separate discipline by a century.

The three remaining chapters of this book are concerned with probability models. Just like regression models – indeed like any econometric model – these models determine the probability density function of the dependent variable; the difference lies in the role of the random variation. In regression models this is cursorily treated by the introduction of the disturbance, which is a nuisance variable characterized by nuisance parameters; to eliminate it, and let y take its expected value, is the ideal, or at any rate the standard, case for interpretation and discussion. The interest of the analysis lies precisely in this expected value, which is the “systematic component” of the model, and much ingenuity is spent on its design. In probability models, in contrast, the random variation is the essential phenomenon, and in the construction of the model it is the probability mechanism rather than the systematic component that is elaborated. Probability models are designed to describe individual variation, and they have no immediate bearing on aggregate data. Since they usually account fully for the observed variation there is no need to allow explicitly for neglected factors or other imperfections.

These general characteristics will be demonstrated in the sequel. Like the regression model, most probability models have their origin in biology and in medical research. Numerous studies of transport choice, of the demand for durables, and of occupational choice reflect a lively interest in their application to microeconomic data.

Over the past decades, the advent of high-speed electronic computing has drastically altered the practice of applied econometrics by permitting the adoption of specific models of much greater intricacy and of a higher theoretical content than before. The estimation and testing of these models draw heavily on Maximum Likelihood methodology, and practitioners in the field freely use the theorems of this statistical theory along with dispersed results from linear algebra and the canons of numerical maximization. The aim of this book is to set out the main elements of this common basis of so many current empirical studies. To this end I have collected the various parts, adapted them to the particular conditions of econometric analysis, and put them all together in the form of a separate paradigm.

There is a growing realization that some of the basic theorems have not yet been proved with sufficient generality to cover all the cases that arise in econometrics, and much fundamental work is being undertaken to remedy this defect. At the same time, progress is being made in the design of robust methods that may well, in due course, replace Maximum Likelihood as a general method of inference. The present book does not deal with these new developments. It contains nothing new, but some old things that may surprise. While there is progress in econometrics, some of the fashionable novelties have quite old roots, which are generally ignored.