xbeta
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 )
y_i^* = x_i^\top \beta + \varepsilon_i \quad \text{and} \quad \varepsilon_i \sim N(0, \sigma^2)

and

(2)y i={0 , ify i *0 1 , ify i *>0 .
y_i = \begin{cases} 0, & \text{if} \quad y_i^* \leq 0 \\ 1, & \text{if} \quad y_i^* \gt 0. \\ \end{cases}

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 βx i) =1 Pr(ε ix i βx i) =1 Φ(x i βσ).
\begin{split} \Pr(y_i = 1 \vert x_i) &= \Pr(y_i^* \gt 0 \vert x_i) \\ &= \Pr(x_i^\top \beta + \varepsilon_i \gt 0 \vert x_i) \\ &= \Pr(\varepsilon_i \gt -x_i^\top \beta \vert x_i) \\ &= 1 - \Pr(\varepsilon_i \leq -x_i^\top \beta \vert x_i) \\ &= 1 - \Phi\left(-\frac{-x_i^\top \beta}{\sigma}\right). \end{split}

Similarly,

(4)Pr(y i=0 x i)=1 Pr(y i=1 x i)=Φ(x i βσ).
\Pr(y_i = 0 \vert x_i) = 1 - \Pr(y_i = 1 \vert x_i) = \Phi\left(-\frac{x_i^\top \beta}{\sigma}\right).

References

  • Amemiya, T. (1985): Advanced Econometrics. Cambridge, MA: Harvard University Press.
  • Hayashi, F. (2000): Econometrics. Princeton, NJ: Princeton Univerity Press.
  • Maddala, G.S. (1983): Limited-Dependent and Qualitative Variables in Econometrics. New. York: Cambridge University Press.