Binary Regression: The Logit Model

Abstract

Categorical regression has the same objectives as metric regression. It aims at an economic representation of the link between covariables considered as the independent variables and the response as the dependent variable. Moreover, one wants to evaluate the influence of the independent variables regarding their strength and the way they exert their influence. Predicting new observations can be based on adequate modeling of the response pattern. Categorical regression modeling differs from classical normal regression in several ways. The most crucial difference is that the dependent variable y follows a quite different distribution. A categorical response variable can take only a limited number of values, in contrast to normally distributed variables, in which any value might be observed. In the simplest case of binary regression the response takes only two values, usually coded as y = 0 and y = 1. One consequence is that the scatterplots look different. Figure 2.1 shows data from the household panel described in Example 1.2. The outcomes “car in household” ( y = 1) and “no car in household” ( y = 0) are plotted against net income (in Euros). It is seen that for low income the responses y = 0 occur more often, whereas for higher income y = 1 is observed more often. However, the structural connection between the response and the covariate is hardly seen from this representation. Therefore, in Figure 2.1 the relative frequencies for owning a car are shown for households within intervals of length 50. The picture shows that a linear connection is certainly not the best choice.