Probit Model

In the probit model, an observed binary variable ${y}_{i}\in \left\{0,1\right\}$ is modeled as the response to an underlying (continuous) latent response ${y}_{i}^{*}$ where

(1)${y}_{i}^{*}={x}_{i}^{\top }\beta +{\epsilon }_{i}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\epsilon }_{i}\sim N\left(0,{\sigma }^{2}\right)$

and

(2)${y}_{i}=\left\{\begin{array}{ll}0,& \text{if}\phantom{\rule{1em}{0ex}}{y}_{i}^{*}\le 0\\ 1,& \text{if}\phantom{\rule{1em}{0ex}}{y}_{i}^{*}>0.\end{array}$

The latent response ${y}_{i}^{*}$ is unobservable. From (1) and (2), it follows that

(3)$\begin{array}{rl}\mathrm{Pr}\left({y}_{i}=1|{x}_{i}\right)& =\mathrm{Pr}\left({y}_{i}^{*}>0|{x}_{i}\right)\\ & =\mathrm{Pr}\left({x}_{i}^{\top }\beta +{\epsilon }_{i}>0|{x}_{i}\right)\\ & =\mathrm{Pr}\left({\epsilon }_{i}>-{x}_{i}^{\top }\beta |{x}_{i}\right)\\ & =1-\mathrm{Pr}\left({\epsilon }_{i}\le -{x}_{i}^{\top }\beta |{x}_{i}\right)\\ & =1-\Phi \left(-\frac{-{x}_{i}^{\top }\beta }{\sigma }\right).\end{array}$

Similarly,

(4)$\mathrm{Pr}\left({y}_{i}=0|{x}_{i}\right)=1-\mathrm{Pr}\left({y}_{i}=1|{x}_{i}\right)=\Phi \left(-\frac{{x}_{i}^{\top }\beta }{\sigma }\right).$