Using probit regression coefficients to derive probabilities?

Question

I'm reading a paper here that uses a Probit Regression. I'm not entirely familiar with how this works. But, I'm wondering if there's a way to use the coefficients from Table 3 to derive the probabilities indicated in Tables 4, 5 and 6. And, if so, how I would do that. Thanks.

Answer 1

Take text states on page 6 that $pr( < Highschool) = pr(X*\beta + \epsilon < \delta_1)$ where epsilon follows the standard normal distribution. Therefore, $$pr( < Highschool) = pr( \epsilon > \delta_1 - X*\beta ) = \\1 - pr( \epsilon < delta_1 - X\beta ) = 1 - \phi^{-1}(\delta_1 - X\beta)$$ where $\phi$ is the CDF of the standard normal distribution. Following these steps and estimation your model you can calculate the probability as follows: $$\hat{pr}( < Highschool) =1 - \phi^{-1}(\hat{\delta}_1 - X\hat{\beta})$$