Modernization is a process of change. Economic development is only one of many interdependent aspects of the modernizing process. This is hard to dispute and yet there are practically no current theories of economic development that include sociological, cultural, and political change as a part of their dynamic fabric. Nor, for that matter, are there noneconomic theories that explicitly account for impact on and feedback from economic variables. Either, it would seem, one believes that economic development causes sociological, cultural, and political modulation, and hence is only concerned with economics, or one feels that sociological, cultural, and political modifications are preconditions for economic change, and therefore only the noneconomic factors are important. The obvious truth of the matter is that economic, sociological, cultural, and political transformations are all occurring simultaneously and any theory that does not explicitly allow for this is subject to rather severe limitations.

The purpose of this chapter is to illustrate a way of constructing models in which both economic and noneconomic change may be analyzed simultaneously. The examples presented are syntheses of the efforts of several authors, each writing from the point of view of his own discipline. No claims of comprehensiveness or correctness are made for the results. Because each author has his own concept of what is important, and because he may leave out elements that others regard as essential, the works upon which the following is based cannot be regarded as representative of the thinking within their respective disciplines. Furthermore, any synthesis necessarily retains much of the restrictiveness and many of the failures of the sources from which it is drawn.

So far attention has been directed primarily toward the achievement of two major goals. First, a general methodology was developed to handle variables that seem incapable of measurement. Rules and guidelines enabling construction and manipulation of relations among such variables are thus available. These relations are basic components of the modern systems approach to analysis in social and behavioral science. Simultaneous equations systems, systems of periodic equations, and models of choice have received particular emphasis. The inescapable conclusion is that, except for measurement, nonquantifiable experience may be approached and understood in much the same way as is traditionally realized when the standard quantitative yardsticks are available.

The second goal was to demonstrate, at least at theoretical levels, how this methodology could be applied. Four examples were given: Political structure was defined as a system of simultaneous equations that may determine political systems or cycles. A dynamic model for planning purposes was presented along with a more complex account of society's process of social, political, cultural, economic, and psychological evolution. Lastly, the firm was modeled and analyzed in terms of the social interactions of the individuals it employs. In all instances, knowledge of structural relations and parameter values would, as in the quantifiable case, permit prediction.

Scientific inquiry, however, involves not only the creation of structure, theoretical propositions, predictions and the like, but also the determination of whether the structure, propositions, and predictions manifest themselves empirically. Accordingly, here in Part III, focus shifts to ways of obtaining specific knowledge of parameters and relations for the purpose of prediction or “empirical verification.”

The origins of this volume emerged from an interdisciplinary course in which I participated while at the University of Pennsylvania during the 1969–70 academic year. Four instructors representing their different fields of study were present: an anthropologist, a political scientist, a sociologist, and myself, from economics. Our forty students were among the brightest freshmen and sophomores the university had to offer. The first term we split the class into four sections, and each instructor exposed ten students to a quick but sophisticated introductory survey of his area. For the second term the class was reunited and a single problem was chosen to be considered by the group as a whole. It was hoped the students would develop an appreciation for social science in general rather than the feeling that our four subjects were distinct and unrelated spheres of knowledge.

The problem selected for the second term was to gain some insight into what it means for a society to modernize. Our attention focused on four books, one from each field: W. H. Goodenough, Cooperation in Change (anthropology), D. E. Apter, The Politics of Modernization (political science), E. E. Hagen, On the Theory of Social Change (sociology), and A. O. Hirschman, The Strategy of Economic Development (economics). Frequently we met with our first-term sections to discuss these books from the point of view of our own disciplines; at other times we met as a single group to educate each other and to obtain an overall perspective. Each student summarized his thoughts in a term paper at the end.

The purpose of this book is to present methods for the analysis of some econometric models in which the dependent variables are either qualitative or limited in their range. These models are commonly encountered in empirical work that analyzes survey data, although we shall also give examples of some time-series models. In a certain sense every variable we consider in practice, at least in econometric work, is limited in its range. However, it is not necessary to apply the complicated analysis described in this book to all these problems. For instance, if we believe that prices are necessarily positive, we might postulate that they have a log-normal distribution rather than the normal. On the other hand, in the limited-dependent-variable models discussed in this book, the variables are limited to their range because of some underlying stochastic choice mechanism. It is models of this kind that we shall be concerned with in this book. Similarly, there are many qualitative variables that are often used in econometric work. These are all usually known as dummy variables. What we shall be concerned with in this book are models in which the dummy variables are endogenous rather than exogenous. The following simple examples will illustrate the types of models that we shall be talking about. These examples can be conveniently classified into three categories: (a) truncated regression models, (b) censored regression models, and (c) dummy endogenous models.

This chapter begins with an analysis of a single explanatory variable that is observed as a dichotomous variable. We discuss the linear probability model and its relationship to the linear discriminant function. We next discuss the logit and probit models and their estimation by maximum likelihood methods based on individual data and minimum χ2 methods based on grouped data. We next consider the unordered polychotomous variable and the multinomial logit model. We then consider polychotomous variables for ordered and sequential responses. For the analysis of polychotomous variables that are not categorized, we consider the Poisson regression model. Finally, we consider estimation of logit models with randomized data and logit and probit models with panel data.

The discussion of the multinomial logit model is continued in Chapter 3. The discussion of discriminant analysis is continued in Chapter 4. The extension of the analysis presented in this chapter to the case of several qualitative (categorical) variables is contained in Chapter 5.

What are discrete regression models?

By discrete regression models we mean those models in which the dependent variable assumes discrete values. The simplest of these models is that in which the dependent variable y is binary (it can assume only two values, which for convenience and without any loss of generality, we denote by 0 and 1). Numerous examples of this were considered in Chapter 1.

This chapter continues with the multinomial logit model discussed in section 2.12. It derives the multinomial logit model from a theory of probabilistic choice. We then discuss its limitations and examine some extensions of this model (the multinomial probit model, the nested logit model, the generalized extreme-value model, etc.).

McFadden's conditional logit model

In the previous chapter we discussed the multinomial logit model as an extension of the simple logit model for dichotomous variables. There it was pointed out that there is a difference in the way the multinomial logit model was derived and discussed in some of the statistical literature and the way it was discussed by McFadden. The latter discussion is related to the hedonic-price problem in econometrics and the theory of probabilistic choice discussed by several psychologists. In this chapter we shall discuss the multinomial logit model and its extensions as developed by McFadden (1973, 1974, 1976a, 1978, 1979, 1982, in press).

We start with the assumption that consumers are rational in the sense that they make choices that maximize their perceived utility subject to constraints on expenditures. However, there are many errors in this maximization because of imperfect perception and optimization, as well as the inability of the analyst to measure exactly all the relevant variables. Hence, following Thurstone (1927), McFadden assumed that utility is a random function.

This book deals with the usual regression and simultaneous-equations models when the dependent variables are qualitative, truncated, or censored. This has been an area of great interest in econometrics in recent years. The book covers in more or less elementary fashion many of the models commonly used in this field. It does not cover the area of analysis of panel data; including this topic would have made the book very unwieldy.

I would like to thank Forrest D. Nelson for our early collaborative work that stimulated my interest in this area. I would like to thank Angus Deaton, Lung-fei Lee, Daniel McFadden, and Robert P. Trost for going through the book in detail, correcting errors, and suggesting methods of improvement. R. P. H. Fishe, David Goldenberg, David Grether, and Forrest Nelson also provided useful comments. They are not, however, responsible for any errors that may remain, nor for any omissions. I would like to thank Betty Sarra for carefully typing the different versions, the National Science Foundation for providing me research funds to work in this area, and the Center for Econometrics and Decision Sciences at the University of Florida for support. I would also like to thank Colin Day and his associates at Cambridge University Press for their careful production of this book.