Applied Discrete-Choice Modelling

David A. Hensher, Lester W. Johnson

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351140744
• Edition: 1
• Topic: Betriebswirtschaft
• Subtopic: Business allgemein

Originally published in 1981. Discrete-choice modelling is an area of econometrics where significant advances have been made at the research level. This book presents an overview of these advances, explaining the theory underlying the model, and explores its various applications. It shows how operational choice models can be used, and how they are particularly useful for a better understanding of consumer demand theory. It discusses particular problems connected with the model and its use, and reports on the authors' own empirical research. This is a comprehensive survey of research developments in discrete choice modelling and its applications.

CHAPTER 1

Introduction

1.1 The Aims of the Book

Applied discrete-choice modelling can be viewed as the use of a variety of statistical techniques to quantify in a meaningful way a relationship between a discrete choice and a set of explanatory variables. The term ‘meaningful’ is fundamental to this statement and is the link with the requirement that a model have a behaviourally reasonably derivation from some theory. A behaviourally reasonable derivation entails the specification of a set of assumptions underlying a theory, which are used in the solution of an estimatable model and a test of a theory's predictive capability. Since a model implies a simplification or abstraction from reality, modelling should be interpreted as an objective, an approach to explaining and predicting the particular features of the analysis unit's behaviour most relevant to the issue under study. Hence the aims in this book of such a modelling exercise are to estimate the effects of changes in (one or more of) the explanatory variables on the probability of choice and to be able to use the estimated models as predictive tools.

The literature on discrete-choice modelling is spread widely across several disciplines, notably statistics, economics, psychology, engineering and sociology. The predominant emphasis of this literature is on the statistical requirements, with a limited concern for the underlying behavioural derivatives. Exceptions are found in economics and psychology. We believe that a knowledge and understanding of the relevance of alternative statistical specifications of discrete-choice models is dependent on exposure to a theoretical framework within which the behavioural assumptions associated with alternative statistical specifications are made explicit. Thus the emphasis in this book is on getting the theory clear and knowing the behavioural strengths and weaknesses of a particular statistical specification. For this reason, a great deal of effort is devoted to the economic and behavioural theory underlying discrete-choice models.

If a particular model formulation is to survive, it should be derived from a theory whose predictions correspond most nearly with actuality (implying it is also operational in a real-world environment (in contrast to a research environment)). One of the difficulties in the survival stakes is the ability of the analyst, as the observer of individual behaviour, to postulate a modelling framework that provides an appropriate reflection of such behaviour. This is a major limitation in modelling; and as will be outlined in the book, we have to consider ways of accommodating this weakness so as to minimise the error in prediction. The translation of the homo economicus assumption (namely, if an object is chosen then it must maximise utility as the chooser perceives it) into a suitable empirical model is the essence of the modelling task.

The satisfying feature of the discrete-choice modelling approaches presented in the following chapters is that they have a definite operational capability, even if there is concern about some of the translational assumptions, and hence, a practitioner will find the style and scope suited for applications in the real-world. We have to, however, guard against the inappropriate use of the models, limiting their application to contexts where the assumptions underlying a theory are likely to be a reasonable approximation to reality. The emphasis is on knowing and applying; hence there is a sense about this book that makes it an introductory manual for practitioners. The range of applications provided and the test case study adequately illustrate the diversity of relevant applications of discrete-choice modelling procedures. It is thus intended that the book exposes, in the simplest manner possible, given the nature of the topic, the range of significant assumptions underlying the structure of the set of discrete-choice models that are of operational relevance, and in so doing anticipate a more selective use of this modelling capability in the spirit in which it has been developed.

1.2 Outline and Scope

The theme of this book is presented in four stages. Stage One introduces the necessary contextual prerequisites (Chapter 2) that provide a link with the established economic theory of consumer demand and a point of departure for the main thrust of the choice theory central to discrete-choice modelling. A paradigm of choice is proposed in Chapter 2.3 which outlines the relationship between the elements of the models in subsequent chapters and the ‘characteristics’ approach to consumer demand, the latter approach being an appropriate source of linkage with established economic theory. We are interested in predicting population behaviour, but argue that the heterogeneous mix of individuals and likely responses in a choice environment supports a microeconomic disaggregate approach to modelling choice behaviour and a procedure for aggregation (post-estimation) to obtain population predictions. The emphasis on the individual in a choice environment is outlined in Chapter 2.

The second stage, embodied in Chapter 3 entails a detailed development of a basic choice model, which will become the point of reference for discussion in Chapters 4 onwards on the adequacy of a range of important behavioural assumptions. The best way to teach a subject is to develop a simple structure initially and then critically appraise it as part of a process of developing alternative and often more realistic structures. The basic choice model in Chapter 3 is a single equation, single decision model. Initially the established demand model with continuous commodities is outlined and shown to be unsuitable when commodities are discrete. An alternative approach is developed to handle discrete commodities, drawing on a specialised set of theory now referred to as random utility theory, and a procedure for maintaining many of the mathematical niceties of established demand theory is discussed, in particular the notion of a margin. A choice-theoretic modelling approach is developed with a minimum of assumptions in order to provide the most general framework. However, in order to solve for an operationally tractable model structure, some strong behavioural assumptions are introduced, notably the independence from irrelevant alternatives property. The structural solution for the basic choice model in its most basic form can be derived from first principles, making a particularly useful reference model. This model is referred to as the basic multinomial logit model.

In Chapter 3 we select one statistical procedure for estimating the parameters of the basic choice model, so that an empirical link between selection probabilities associated with each discrete alternative and the utility associated with each alternative can be established. The maximum-likelihood estimation technique is outlined for any number of alternatives, it being the most appropriate procedure, provided computer subroutines are available. Chapter 3 is concluded with a discussion of the measures used to test for statistical significance, and a simple illustration of the application of the basic choice model. It is essential that the contents of Chapter 3 are understood before going to Chapter 4 and the next stage.

Chapters 4 to 8 introduce the additional conceptual issues necessary in an introduction to applied discrete-choice modelling. In Chapter 3 we have assumed a choice set of a predefined number of alternatives without asking about the actual number and the basis for generating the choice set. How do we generate a choice set? Theoretically a procedure can be postulated for selecting a set of choice sets in a probabilistic choice environment (as outlined in Chapter 4.2), and for selecting elements in each choice set; but from an empirical perspective we are a long way from systematic identification of choice sets. This issue is very important since once a choice set is defined, the analyst's empirical work becomes conditional on the relevance of the choice set, and in the formulation of the basic choice model on the acceptance of each and every alternative being mutually exclusive and collectively exhaustive.

In theory and practice the number of alternatives may be either excessive and/or subsets so similar that there are good reasons for grouping alternatives. This might also be a mechanism for satisfying strong behavioural assumptions related to the similarity of alternatives (the independence from irrelevant alternatives property – introduced in Chapter 3.3 and developed in detail in Chapter 5). In Chapter 4.2 we discuss, within the framework of the basic choice model, ways of handling grouped alternatives. The entire chapter is devoted to extending the basic choice model within a logit context. In Chapter 3 we assumed a single equation choice model based on a single decision. For a number of reasons given in Chapter 4 it is more realistic to consider a decision process as the interaction of a number of choices, which should be modelled either simultaneously or recursively. Alternative decision structures are outlined in Sections 4.3 and 4.4 Ways are suggested to decompose a simultaneous decision structure into a set of sequential-recursive models by invoking separability of choice. The notion of a choice hierarchy is used to tie all the discussion on decision structure together. Section 4.4 is included with an overview of more complex (yet potentially more behaviourally plausible) logit model forms; which is designed in part to highlight the strong assumptions underlying the basic logit choice model. The brief discussion of the similarity of alternatives in a choice set provides a basis for justifying an extensive assessment in Chapter 5 of the validity of the independence assumption in the basic choice model.

Having decided on a suitable decision structure, the analyst has to decide on the form of the utility expression which contains the set of attributes and weights reflecting the relative importance of each attribute. In Chapter 4.5 we take the simple linear-in-parameters additive utility function and discuss alternative specifications of the attributes. An extended discussion of this important element of model specification is given in Appendix B. The chapter concludes with a simple example of two choices which utilises the main tools introduced in this chapter.

A central behavioural assumption of many share models, of which the basic choice model is a member, is that the ratio of the odds of selecting an alternative to that of another alternative remains unchanged as other alternatives are added or deleted from the choice set. This assumption is both a strength and a weakness of such models; it is a strength in that it enables the development of a simply estimated and applied model (see Chapter 3), and it is a weakness in that it is a somewhat restrictive assumption in many applications. Chapter 5 is devoted to a detailed discussion of the independence from irrelevant alternatives (IIA) property, in particular procedures to determine if it is violated in a particular application and if so, proposals to remedy violation. The advantages of a simple model are such that all efforts should be made to assist in maintaining a simple, yet suitable, structure prior to having to move to more complex structures such as multinomial probit (as discussed in Chapter 6), or nested logit (summarised in Chapter 4).

Chapter 6 introduces a range of variants on the basic choice model, giving attention to the alternative statistical estimation techniques such as weighted least-squares regression and maximum likelihood. Modifications to the basic choice model are introduced which are designed to handle situations where pairs of alternatives violate the IIA property. The particular model is known as dogit,* (‘d’ referring to “dodging IIA”). A nonlogit procedure that does not require the IIA property is briefly discussed, referred to as multinomial probit, although we cannot develop it in this book. It is an advanced procedure which has recently been extensively outlined by Daganzo (1979) and evaluated by Horowitz (1980). Chapter 6 is concluded with an outline of ways of testing for alternative functional forms of the linear-in-parameters utility function.

The next two chapters present a range of issues that require some appreciation in a book on choice modelling, even though they are of more general relevance. Since the prime aim of discrete-choice modelling is to develop a forecasting capability, it is necessary to understand the issue of aggregating individual predictions to yield population predictions. The aggregation issue focusses on sources of bias due to defining nonhomogeneous decision units prior to estimation, and the use of aggregate measures of attributes. Two particular levels of aggregation are: (1) the use of group and intergroup averages in the definition of the magnitudes of explanatory variables for each individual and (2) the use of the average level of a variable for all individuals in a sample in obtaining the average probability of choice, both before and after a policy change, and in the calculation of elasticities of choice. A second important general issue is the transferability potential of a model estimated say in one location or time period and applied in another location or time period. Procedures are outlined that argue for transferability of model parameters subject to some modifications. Note that it is the functional form of the model and the estimated coefficients that are transferred, not the levels of the explanatory variables. Section 7.4 discusses alternative sample designs and their implications in choice modelling, and clarifies the sources of error that the modeller must minimize. Sampling is one of the most important elements of modelling, providing a link between the analytical model and the data environment. The last two sections of Chapter 7 illustrate a mechanism for obtaining from a choice model behavioural shadow prices for variables that are not expressed in monetary units; and raises the possibility that behavioural phenomena such as habit and threshold in choice may be important and should be included in the formulation of more relevant choice models.

Up to this point, all explanatory variables have been assumed to be exogenous; however it is reasonable to consider some variables as endogenous. Simultaneous equation models are introduced in Chapter 8 which incorporate endogenous variables, and also the possibility of incomplete information. The latter issue is closely related to endogenous variables, since in practice many situations arise where a variable is only identifiable on a subsample of the data, namely the subsample associated with a particular alternative. An example is a choice model of women working and nonworking where the market wage is only identifiable with working women. Sample separation is assumed and suitable procedures are adopted to handle bias due to selectivity of information. This chapter completes Stage 3.

The final stage includes discussion of a number of applications from a wide range of areas such as employment, child care, transport and education; and a detailed test case study complete with a computer program, a data set listing, directions to use the program and interpretation of the outp...