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

(1)y i *=x i β+ε iandε iN(0,σ 2)

and

(2)y i={0, ify i *0 1, ify i *>0.

The latent response y i * is unobservable. From (1) and (2), it follows that

(3)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,

(4)Pr(y i=0|x i)=1Pr(y i=1|x i)=Φ(x i βσ).

References