Dichotomous Choice Models and Dummy Variables

Abstract

Summary. In an article recently published in this journal, Oksanen demonstrated that the coefficients of observation-specific dummy variables are not estimable in dichotomous logit models. This paper extends Oksanen's result in two ways. First, it is shown that observation-specific dummy variables cannot be estimated in a large class of dichotomous choice models, which include probit as well as logit models. Second, it is shown that the dummy variable need not be observation-specific. The dummy variable can describe a group with more than one member as long as each member of the group makes the same choice. Oksanen's result is thus a special case in which the dichotomous choice model is the logit model, and the observation-specific dummy variable represents a group with one member. In an article recently published in this journal, Oksanen (1986) explores the use of observation-specific dummy variables in linear probability and logit models. An observation-specific dummy variable takes on the value 1 for one observation and 0 for all others. Oksanen demonstrates that, while the coefficients of observation-specific dummy variables can be estimated in linear probability models, the coefficients are undefined and cannot be estimated in logit models. The purpose of this paper is to extend Oksanen's result in two ways. The first is to demonstrate that the coefficients of observation-specific dummy variables cannot be estimated for an entire class of dichotomous choice models which includes logit models as a special case. Second, it is shown that the coefficients of more generally defined dummy variables cannot be estimated in dichotomous choice models. Oksanen's result that coefficients of observation-specific dummy variables cannot be estimated is extended to a large class of dichotomous choice models. The class includes all dichot6mous choice models in which the probability of choice can be expressed as the integral of a continuous density function, f(t), that is positive for all t. This stipulation can also be stated in terms of the cumulative distribution function F. No finite numbers a and b can exist such that F(a) = 0 and F(b)= 1. This class includes probit models as well as logit models.