Probit model

Remarks

Equivalent latent variable form: let $y_i^* = x_i^T \beta + \varepsilon_i$ with independent $\varepsilon_i \sim N(0, 1)$, and $y_i = 1$ if $y_i^* > 0$ and $y_i = 0$ otherwise. Then $P(y_i = 1) = P(\varepsilon_i > -x_i^T \beta) = \Phi(x_i^T \beta)$, by the symmetry of the normal distribution. The error variance is fixed at $1$ because $\beta$ and the scale of $\varepsilon_i$ cannot be identified separately from binary data.

It is the generalized linear model with Bernoulli responses and link $g = \Phi^{-1}$, which is not the canonical link.

Example: with $x_i = (1, t_i)$ and $\beta = (-1, 0.5)$, at $t = 2$ one gets $p = \Phi(0) = 1/2$.

Non-example: the model $p_i = 1/(1 + e^{-x_i^T \beta})$ uses the logistic function instead of $\Phi$; it is logistic regression, not the probit model. Its latent errors are logistic, with variance $\pi^2/3$, so its coefficients are typically larger than probit coefficients by a factor of roughly $1.6$ to $1.8$.