Multivariate probit

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Extension of the probit model to estimate several binary variables jointly.

[edit] Example: bivariate probit

$Y 1 and Y 2$ are two binary dependent variables. $\left\{ \begin{array}{ll} Y_1&=1_{(Y^*_1>0)}\\ Y_2&=1_{(Y^*_2>0)} \end{array} \right.$

with

$\left\{ \begin{array}{ll} Y_1^*&=X\beta_1+\varepsilon_1\\ Y_2^*&=X\beta_2+\varepsilon_2 \end{array} \right.$

And:

$\left( \begin{array}{c} \varepsilon_1\\ \varepsilon_2 \end{array} \right)|X \sim \mathcal{N} \left(\left( \begin{array}{c} 0\\ 0 \end{array} \right) , \left( \begin{array}{cc} 1&\rho\\ \rho&1 \end{array} \right)\right)$