xbeta
Probit model
In the probit model, an observed binary variable is modeled as the response to an underlying (continuous) latent response where
(1)y_i^* = x_i^\top \beta + \varepsilon_i
\quad \text{and} \quad
\varepsilon_i \sim N(0, \sigma^2)
and
(2)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 is unobservable. From (1) and (2), it follows that
(3)\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 \vert x_i) = 1 - \Pr(y_i = 1 \vert x_i)
= \Phi\left(-\frac{x_i^\top \beta}{\sigma}\right).
References
Revised on April 24, 2009 12:15:30
by
Jason Blevins
(207.192.72.34)