Multinomial Probit - Example: Bivariate Probit

Example: Bivariate Probit

are two binary dependent variables. In the Ordinary Probit model, one latent variable is used, in the bivariate probit model there are two: and . These latent variables are defined as: 
\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_1\beta_1+\varepsilon_1\\
Y_2^*&=X_2\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)

Fitting the bivariate probit model involves estimating the values of . To do so, the Likelihood of the model has to be maximized. This Likelihood is defined as:


\prod P(Y_1=1,Y_2=1)^{Y_1Y_2}P(Y_1=0,Y_2=1)^{(1-Y_1)Y_2}P(Y_1=1,Y_2=0)^{Y_1(1-Y_2)}P(Y_1=0,Y_2=0)^{(1-Y_1)(1-Y_2)}
Substituting the latent variables and in the Probability functions and taking Log's gives: 
\sum Y_1Y_2\ln P(\varepsilon_1>-X_1\beta_1,\varepsilon_2>-X_2\beta_2)
+(1-Y_1)Y_2\ln P(\varepsilon_1<-X_1\beta_1,\varepsilon_2>-X_2\beta_2)
+Y_1(1-Y_2)\ln P(\varepsilon_1>-X_1\beta_1,\varepsilon_2<-X_2\beta_2)
+(1-Y_1)(1-Y_2)\ln P(\varepsilon_1<-X_1\beta_1,\varepsilon_2<-X_2\beta_2)

After some rewriting, the log-likelihood function becomes: 
\sum Y_1Y_2\ln \Phi(X_1\beta_1,X_2\beta_2,\rho)+(1-Y_1)Y_2\ln \Phi(-X_1\beta_1,X_2\beta_2,-\rho)+Y_1(1-Y_2)\ln \Phi(X_1\beta_1,-X_2\beta_2,-\rho)+(1-Y_1)(1-Y_2)\ln \Phi(-X_1\beta_1,-X_2\beta_2,\rho)
Note that is the cumulative distribution function of the bivariate normal distribution. and in the log-likelihood function are observed variables being equal to one or zero.

To maximize the log-likelihood function it is recommended to define the gradient.

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