Probit Model

In the probit model, an observed binary variable y i{0,1} is modeled as the response to an underlying (continuous) latent response y i where labelProbitLatenty i =x i β+ε iandε iN(0,σ 2) and labelProbitResponsey i={0, ify i 0 1, ify i >0. The latent response y i is unobservable. From (eq:ProbitLatent) and (eq:ProbitResponse), it follows that Pr(y i=1|x i) =Pr(y i >0|x i) =Pr(x i β+ε i>0|x i) =Pr(ε i>x i β|x i) =1Pr(ε ix i β|x i) =1Φ(x i βσ). Similarly, Pr(y i=0|x i)=1Pr(y i=1|x i)=Φ(x i βσ).