7 - Semiparametric Estimation of Discrete Choice Models

Summary

Introduction

In many data sets occurring in the social sciences the object of interest is whether a decision has been made or an action carried out. Examples would be whether a durable good is purchased, a bond is issued, or, perhaps, whether a corporation removes its CEO. Generally, an investigator is interested in what it is that precipitates the decision or action, and a range of possible influences (x_i) is entertained. What makes this different from a regular regression problem is that the variable to be explained (y_i) is qualitative rather than quantitative; or, treating the decision outcome as a random variable taking the value zero or unity (reflecting negative and positive decisions or actions), y_i must be a discrete rather than a continuous random variable.

An enormous literature has built up on the types of models that can be employed to explain such data, a masterly exposition of which is Amemiya (1981). Economists imported many of these models from medical and biological statistics, and the traditional ways of estimating unknown parameters have revolved around formulating a suitable density function for the observed data. Section 7.2 reviews this literature, pointing to the fact that estimation has proceeded in either one of two ways. In the first, the discrete nature of the random variable enables one to formulate relationships as a heteroskedastic nonlinear regression. In the second, maximum likelihood methods are employed.