Knowledge that Transforms

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.

where y(t) and x(t) are vectors of G components, z(t) has K components, B is a G x G matrix, and F is a G x K matrix.' The components of z(t) are predetermined at time t and the vector x(t) is not serially correlated, so that (among other requirements) we have

[This note develops a slightly different formulation of one of the basic results presented in a recent paper by Wallace and Hussain [5] on error components models for disturbances in relationships designed to explain cross-sectional observations over time. In their discussion, Wallace and Hussain derive the inverse of the variance-covariance matrix of the disturbances by trial and error. Unfortunately, their formulation does lead to a "natural" interpretation of the generalized least squares estimates, or of the relationships of these estimates to other estimates in the same way diagonalization of the variance-covariance matrix by means of an appropriate orthogonal transformation does. The characteristic roots of the variance-covariance matrix for the disturbances in a three component model which has been studied by Wallace and Hussain are derived here. It is shown how knowledge of these roots and the characteristic vectors associated with them leads to a form of the inverse matrix which may be more readily interpreted, as well as a number of other useful results, including an interpretation of the poor small sample properties of estimates which incorporate dummy variables for each individual.]