In statistics and econometrics, the multivariate probit model is a generalization of the probit model used to estimate several correlated binary outcomes jointly. For example, if it is believed that the decisions of sending at least one child to public school and that of voting in favor of a school budget are correlated (both decisions are binary), then the multivariate probit model would be appropriate for jointly predicting these two choices on an individual-specific basis. J.R. Ashford and R.R. Sowden initially proposed an approach for multivariate probit analysis. Siddhartha Chib and Edward Greenberg extended this idea and also proposed simulation-based inference methods for the multivariate probit model which simplified and generalized parameter estimation.
Example: bivariate probit In the ordinary probit model, there is only one binary dependent variable Y {\displaystyle Y} and so only one latent variable Y ∗ {\displaystyle Y^{*}} is used. In contrast, in the bivariate probit model there are two binary dependent variables Y 1 {\displaystyle Y_{1}} and Y 2 {\displaystyle Y_{2}} , so there are two latent variables: Y 1 ∗ {\displaystyle Y_{1}^{*}} and Y 2 ∗ {\displaystyle Y_{2}^{*}} . It is assumed that each observed variable takes on the value 1 if and only if its underlying continuous latent variable takes on a positive value:
Y 1 = { 1 if Y 1 ∗ > 0 , 0 otherwise , {\displaystyle Y_{1}={\begin{cases}1&{\text{if }}Y_{1}^{*}>0,\\0&{\text{otherwise}},\end{cases}}}
Y 2 = { 1 if Y 2 ∗ > 0 , 0 otherwise , {\displaystyle Y_{2}={\begin{cases}1&{\text{if }}Y_{2}^{*}>0,\\0&{\text{otherwise}},\end{cases}}}
with
{ Y 1 ∗ = X 1 β 1 + ε 1 Y 2 ∗ = X 2 β 2 + ε 2 {\displaystyle {\begin{cases}Y_{1}^{*}=X_{1}\beta _{1}+\varepsilon _{1}\\Y_{2}^{*}=X_{2}\beta _{2}+\varepsilon _{2}\end{cases}}}
and
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