Probit model

The probit model is a popular specification of a binary regression model. Let $Y$ be a binary outcome variable, and let $X$ be a vector of regressors. The probit model assumes that

$Pr(Y=1|X=x)=\Phi (x'\beta )$,

where $\Phi$ the cumulative distribution function of the standard normal distribution. The parameters $\beta$ are typically estimated by maximum likelihood.

The probit model can be obtained from a simple latent variable model. Suppose that

$y^{*}=x'\beta +\epsilon ,$

where $\epsilon |x\sim N(0,1)$, and suppose that $Y$ is an indicator for whether the latent variable $y^{*}$ is positive:

$Y\equiv 1(y^{*}>0).$

Then it is easy to show that

$Pr(Y=1|X=x)=\Phi (x'\beta ).$