The probit and logit models are regression models for situations in which the dependent variable is a discrete outcome, such as a “yes” or “no” decision. For example, an analyst might be interested in examining the effect of 8th grade math achievement on graduation from high school. The probit model examines the effects of a set of independent variables (Xs) on the probability of success or failure on the independent variables, P(Y). The observed occurrence of a given choice (i.e., success or failure) is taken as an indicator of an underlying, unobservable continuous variable, which may be called “propensity to choose a given alternative.” Such a variable is characterized by the existence of a threshold defining the position at which one switches from one alternative to another. For example, a student’s propensity to graduate from high school may be directly related to his or her 8th grade math achievement, which in turn may depend on family background and motivation factors. Whether a student graduates is likely to depend on whether his or her 8th grade math achievement does or does not exceed his or her threshold. This threshold, which differs across students with the same family background and motivation factors plays the role of a random disturbance.

The probit model is a probability model where:

Prob(event j occurs) = Prob(Y = j) = F[relevant effects: parameters].

Using a model of high school graduation, the respondent either graduates (Y=1) or doesn’t (Y=0) as a function of a set of factors such as parents’ education, family income, academic motivation, and so on, so that:

Prob (Y=1)=F(b'x) Prob(Y=0)=1-F(bx)

The set of parameters b reflect the impact of changes in x on the probability. We could easily estimate this probability model as a linear regression where

y=bx+e

However, this linear probability model presents a number of problems. First, since bx+e must equal zero or one, the variance of the errors depends on b, leading to heteroscedasticity. Second, and more important, we cannot assure that the range of predictions from this model will look like probabilities since we have not constrained bx to be within the zero-one interval.

The inadequacies of the linear probability model suggest that a nonlinear specification is more appropriate. A natural candidate is an S-shaped curved bounded in the interval zero-one. One such curve is the cumulative normal distribution function corresponding to the probit model.1 This model is derived as follows. Let Y* represent an unobservable variable given by

Y*=b'x+e

where e~N(0,1) and ei and ej(i¹j) are independent. The observable binary variable Y is related to Y* in the following way:

Then

E(Y) = p = P(Y = 1) = P(Y*>0) = P(-e