Probit Model

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

(1)

y_{i}^{*} = x_{i}^{⊤} β + ε_{i} and ε_{i} \sim N (0, σ^{2})

and

(2)

y_{i} = {\begin{array}{ll} 0, & if y_{i}^{*} \leq 0 \\ 1, & if y_{i}^{*} > 0 . \end{array}

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

(3)

\begin{aligned} \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}) \\ = 1 - \Pr (ε_{i} \leq - x_{i}^{⊤} β | x_{i}) \\ = 1 - Φ (- \frac{- x_{i}^{⊤} β}{σ}) . \end{aligned}

Similarly,

(4)

\Pr (y_{i} = 0 | x_{i}) = 1 - \Pr (y_{i} = 1 | x_{i}) = Φ (- \frac{x_{i}^{⊤} β}{σ}) .

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.