As we have noted, to explain the behavior of a dichotomous dependent variable we will have to use a suitably chosen CDF. The logit model uses the cumulative logistic function, as shown in (15.5.2). But this is not the only CDF that one can use. In some applications, the normal CDF has been found useful. The estimating model that emerges from the normal CDF28 is popularly known as the probit model, although sometimes it is also known as the normit model. In principle one could substitute the normal CDF in place of the logistic CDF in (15.5.2) and proceed as in Section 16.5. Instead of following this route, we will present the probit model based on utility theory, or rational choice perspective on behavior, as developed by McFadden.29
To motivate the probit model, assume that in our home ownership example the decision of the ith family to own a house or not depends on an un-observable utility index Ii (also known as a latent variable), that is determined by one or more explanatory variables, say income Xi, in such a way that the larger the value of the index Ii, the greater the probability of a family owning a house. We express the index Ii as
where Xi is the income of the ith family.
How is the (unobservable) index related to the actual decision to own a house? As before, let Y = 1 if the family owns a house and Y = 0 if it does not. Now it is reasonable to assume that there is a critical or threshold level of the index, call it I*, such that if Ii exceeds I*, the family will own a house, otherwise it will not. The threshold I*, like Ii, is not observable, but if we assume that it is normally distributed with the same mean and variance, it is possible not only to estimate the parameters of the index given in (15.9.1) but also to get some information about the unobservable index itself. This calculation is as follows.
Given the assumption of normality, the probability that I* is less than or equal to Ii can be computed from the standardized normal CDF as30:
Pi = P (Y = 1 | X) = P (I* < Ii) = P (Zi < ft + P2 Xi) = F (p1 + P2 Xi)
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