Some Statistical Models for Limited Dependent Variables with Application to the Demand for Durable Goods

Abstract

THIS PAPER DEVELOPS some models for limited dependent variables.2 The distinguishing feature of these variables is that the range of values which they may assume has a lower bound and that this lowest value occurs in a fair number of observations. This feature should be taken into account in the statistical analysis of observations on such variables. In particular, it renders invalid use of the usual regression model. The second section of this paper develops several models for such variables. Like Tobin's [10] model, they are extensions of the multiple probit analysis model.3 They differ from that model by allowing the determination of the size of the variable when it is not zero to depend on different parameters or variables from those determining the probability of its being zero. Estimation and discrimination in the models are considered in Section 3. The models, like their prototypes, seem particularly intractable to exact analysis and large sample approximations have to be used. The adequacy of inferences based on these procedures is explored in Section 4 through a small sampling experiment. Limited dependent variables arise naturally in the study of consumer purchases, particularly purchases of durable goods. When a durable good is to be purchased, the amount spent may vary in fine gradations, but for many durables it is probably the case that most consumers in a particular period make no purchase at all. In Section 5 we apply the models to the demand for durable goods to provide an application of the techniques.