Latent regression approach:

Qualitative choice situations may also be  analyzed using latent regression models. The outcome of a discrete choice can be  thought of as the result of an  underlying regression. Let us look  at our earlier example of  labor force participation. Let  the propensity to enter the labor force be an unobserved variable Y_i*  = x_iβ  + ε_i.  Here, x_iβ is called the index function. Y_i*  is not observed, but we do observe the individual's decision to be a part of the labor force or not. That is, we observe (Y_i,x_i),  with

Where, c is some positive constant and provides a floor for the propensity variable. By making suitable assumptions about ε_i we can calculate the  probability of  labor force participation given x_i,  and the expectation of Y_i.

We can predict how the probability that Y_i = 1 will change as xi changes. As we shall discuss later this is no longer equal to β. Notice that the binary nature of thc left hand side, dependent variable in  this model  leads us to the result  in this in  turn implies that the impact of changes in  xi on  the mean of Y,  can also be deduced from the above calculations.