Tobit Models

10.1 Introduction

Tobit models refer to censored or truncated regression models in which the range of the dependent variable is constrained in some way. In economics, such a model was first suggested in a pioneering work by Tobin (1958). He analyzed household expenditure on durable goods using a regression model that specifically took account of the fact that the expediture (the dependent variable of his regression model) cannot be negative. Tobin called his model the model of limited dependent variables. It and its various generalizations are known popularly among economists as Tobit models, a phrase coined by Goldberger (1964), because of similarities to probit models. These models are also known as censored or truncated regression models. The model is called truncated if the observations outside a specified range are totally lost and censored if we can at least observe the exogenous variables. A more precise definition will be given later.

Censored and truncated regression models have been developed in other disciplines (notably, biometrics and engineering) more or less independently of their development in econometrics. Biometricians use the model to analyze the survival time of a patient. Censoring or truncation occurs when either a patient is still alive at the last observation date or he or she cannot be located. Similarly, engineers use the model to analyze the time to failure of material or of a machine or of a system. These models are called survival or duration models. Sociologists and economists have also used survival models to ana­lyze the duration of such phenomena as unemployment, welfare receipt, employment in a particular job, residence in a particular region, marriage, and the period of time between births. Mathematically, survival models belong to the same general class of models as Tobit models; survival models and Tobit models share certain characteristics. However, because survival models pos­sess special features, they will be discussed separately in Chapter 11.

Between 1958—when Tobin’s article appeared—and 1970, the Tobit model was used infrequently in econometric applications, but since the early 1970s numerous applications ranging over a wide area of economics have appeared and continue to appear. This phenomenon is due to a recent in­

crease in the availability of micro sample survey data, which the Tobit model analyzes well, and to a recent advance in computer technology that has made estimation of large-scale Tobit models feasible. At the same time, many gener­alizations of the Tobit model and various estimation methods for these models have been proposed. In fact, models and estimation methods are now so numerous and diverse that it is difficult for econometricians to keep track of all the existing models and estimation methods and maintain a clear notion of their relative merits. Thus it is now particularly useful to examine the current situation and prepare a unified summary and critical assessment of existing results.

We shall try to accomplish this objective by means of classifying the diverse Tobit models into five basic types. (Our review of the empirical literature suggests that roughly 95% of the econometric applications of Tobit models fall into one of these five types.) Although there are many ways to classify Tobit models, we have chosen to classify them according to the form of the likeli­hood function. This way seems to be the statistically most useful classification because a similarity in the likelihood function implies a similarity in the appropriate estimation and computation methods. It is interesting to note that two models that superficially seem to be very different from each other can be shown to belong to the same type when they are classified according to this scheme.

Sections 10.2 through 10.5 will deal with the standard Tobit model (or Type 1 Tobit), and Sections 10.6 through 10.10 will deal with the remaining four types of models. Basic estimation methods, which with a slight modification can be applied to any of the five types, will be discussed at great length in Section 10.4. More specialized estimation methods will be discussed in rele­vant passages throughout the chapter. Each model is illustrated with a few empirical examples.

We shall not discuss disequilibrium models except for a few basic models, which will be examined in Section 10.10.4. Some general references on dis­equilibrium models will be cited there. Nor shall we discuss the related topic of switching regression models. For a discussion of these topics, the reader should consult articles by Maddala (1980, 1983). We shall not discuss Tobit models for panel data (individuals observed through time), except to mention a few articles in relevant passages.

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