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 $y_{i}^{where label ProbitLatent y_{i}^{} = x_{i}^{⊤} β + ε_{i} and ε_{i} \sim N (0, σ^{2}) and label ProbitResponse y_{i} = {\begin{array}{ll} 0, & if y_{i}^{\leq 0 1, & if} > 0 . \end{array} The latent response y_{i}^{is unobservable. From (eq:ProbitLatent) and
(eq:ProbitResponse), it follows that \begin{aligned} \Pr (y_{i} = 1 | x_{i}) & = \Pr (y_{i}^{} > 0 | x_{i}) \\ = \Pr (x_{i}^{⊤} β + ε_{i} > 0 | x_{i}) \\ = \Pr (ε_{i} > - x_{i}^{⊤} β | x_{i}) \\ = 1 - \Pr (ε_{i} \leq - x_{i}^{⊤} β | x_{i}) \\ = 1 - Φ (- \frac{- x_{i}^{⊤} β}{σ}) . \end{aligned} Similarly, \Pr (y_{i} = 0 | x_{i}) = 1 - \Pr (y_{i} = 1 | x_{i}) = Φ (- \frac{x_{i}^{⊤} β}{σ}) . References Amemiya, T. (1985). Advanced Econometrics .
Cambridge, MA: Harvard University Press. Hayashi, F. (2000). Econometrics .
Princeton, NJ: Princeton Univerity Press. Maddala, G. S. (1983). Limited-Dependent and Qualitative Variables in Econometrics .
New. York: Cambridge University Press. Copyright © 2004-2024 Jason Blevins • Modified June 7, 2011 20:40 EST}}$