Consider the following model of data generation:
$\ y_1 = I(y_1^*>0) \ where\ y_1^* = \beta_1X_1+\epsilon_1 $
$\ y_2 = I(y_2^*>0)*y_1 \ where\ y_2^* = \beta_2X_2+\epsilon_2 $
$\ y_3 = k*y_2 \ if \ \gamma_{k-\ 1} \leq y_3^* \lt \gamma_k \ for \ k = 1,2,..K \ where\ y_3^* = \beta_3X_3+\epsilon_3 $
Moreover,
$(\epsilon_1, \epsilon_2, \epsilon_3) \sim N(0,\Sigma) $
Where,
$\Sigma = $$ \begin{pmatrix} 1 & \sigma_1 & \sigma_2\\ \sigma_1 & 1 & \sigma_3\\ \sigma_2 & \sigma_3 & 1\\\ \end{pmatrix} $$ $
Can I sample the latent variables in the following sequential manner?
$ \ y_1^*|y_2^*,y_3^* $
$ \sim TN(\mu_1, \delta_1^2)|_{0,+\infty} \ when \ y_1 = 1 $
$ \sim TN(\mu_1, \delta_1^2)|_{-\infty, 0} \ when \ y_1 = 0 $
$ Where, \ $
$ \mu_1 = \beta_1X_1'+ $$ \begin{pmatrix} \sigma_1\\ \sigma_2 \\ \end{pmatrix}' $$ $$ \begin{pmatrix} \ 1 & \sigma_3\\ \sigma_3 & 1\\\ \end{pmatrix}^{-1} $$ $$ \begin{pmatrix} \ y_2^* - \beta_2X_2'\\ \ y_3^* - \beta_3X_3'\\ \end{pmatrix} $$ $ $ \delta_1^2 = 1- $$ \begin{pmatrix} \sigma_1\\ \sigma_2 \\ \end{pmatrix}' $$ $$ \begin{pmatrix} \ 1 & \sigma_3\\ \sigma_3 & 1\\\ \end{pmatrix}^{-1} $$ $$ \begin{pmatrix} \sigma_1\\ \sigma_2 \\ \end{pmatrix} $$ $
$ \ y_2^*|y_1^*,y_3^* $
$ \sim TN(\mu_2, \delta_2^2)|_{0,+\infty} \ when \ y_1 = 1 \ and \ y_2 = 1 $
$ \sim TN(\mu_2, \delta_2^2)|_{-\infty, 0}\ when \ y_1 = 1 \ and \ y_2 = 0 $
$ \sim N(\mu_2, \delta_2^2) \ when \ y_1 = 0 $
$ Where, \ $
$ \mu_2 = \beta_2X_2'+ $$ \begin{pmatrix} \sigma_1\\ \sigma_3 \\ \end{pmatrix}' $$ $$ \begin{pmatrix} \ 1 & \sigma_2\\ \sigma_2 & 1\\\ \end{pmatrix}^{-1} $$ $$ \begin{pmatrix} \ y_1^* - \beta_1X_1'\\ \ y_3^* - \beta_3X_3'\\ \end{pmatrix} $$ $ $ \delta_2^2 = 1- $$ \begin{pmatrix} \sigma_1\\ \sigma_3 \\ \end{pmatrix}' $$ $$ \begin{pmatrix} \ 1 & \sigma_2\\ \sigma_2 & 1\\\ \end{pmatrix}^{-1} $$ $$ \begin{pmatrix} \sigma_1\\ \sigma_3 \\ \end{pmatrix} $$ $
$ \ y_3^*|y_1^*,y_2^* $
$ \sim TN(\mu_3, \delta_3^2)|_{\gamma_k,\gamma_{k+\ 1}} \ when \ y_3 = k $
$ \sim N(\mu_3, \delta_3^2) \ when \ y_3 = 0 $
$ Where, \ $
$ \mu_3 = \beta_3X_3'+ $$ \begin{pmatrix} \sigma_2\\ \sigma_3 \\ \end{pmatrix}' $$ $$ \begin{pmatrix} \ 1 & \sigma_1\\ \sigma_1 & 1\\\ \end{pmatrix}^{-1} $$ $$ \begin{pmatrix} \ y_1^* - \beta_1X_1'\\ \ y_2^* - \beta_2X_2'\\ \end{pmatrix} $$ $ $ \delta_3^2 = 1- $$ \begin{pmatrix} \sigma_2\\ \sigma_3 \\ \end{pmatrix}' $$ $$ \begin{pmatrix} \ 1 & \sigma_1\\ \sigma_1 & 1\\\ \end{pmatrix}^{-1} $$ $$ \begin{pmatrix} \sigma_2\\ \sigma_3 \\ \end{pmatrix} $$ $
