The QLIM Procedure

Bivariate Limited Dependent Variable Modeling

The generic form of a bivariate limited dependent variable model is

\begin{eqnarray*} y^{*}_{1i} & =& \mb{x}_{1i}’\bbeta _1+ \epsilon _{1i} \\ y^{*}_{2i} & =& \mb{x}_{2i}’\bbeta _2+ \epsilon _{2i} \end{eqnarray*}

where the disturbances, $\epsilon _{1i}$ and $\epsilon _{2i}$, have joint normal distribution with zero mean, standard deviations $\sigma _1$ and $\sigma _2$, and correlation of $\rho $. $y_1^*$ and $y_2^*$ are latent variables. The dependent variables $y_1$ and $y_2$ are observed if the latent variables $y_1^*$ and $y_2^*$ fall in certain ranges:

\begin{eqnarray*} y_1 = y_{1i} \hbox{ if } y^{*}_{1i}\in D_1(y_{1i})\\ y_2 = y_{2i} \hbox{ if } y^{*}_{2i}\in D_2(y_{2i}) \end{eqnarray*}

$D$ is a transformation from $(y_{1i}^*, y_{2i}^*)$ to $(y_{1i}, y_{2i})$. For example, if $y1$ and $y2$ are censored variables with lower bound 0, then

\begin{eqnarray*} y_1 = y_{1i} \hbox{ if } y^{*}_{1i}>0, \quad y_1 = 0 \hbox{ if } y^{*}_{1i} \le 0 \\ y_2 = y_{2i} \hbox{ if } y^{*}_{2i}>0, \quad y_2 = 0 \hbox{ if } y^{*}_{2i} \le 0 \end{eqnarray*}

There are three cases for the log likelihood of $(y_{1i}, y_{2i})$. The first case is that $y_{1i}=y_{1i}^*$ and $y_{2i}=y_{2i}^*$. That is, this observation is mapped to one point in the space of latent variables. The log likelihood is computed from a bivariate normal density,

\[ \ell _{i} = \ln \left[\phi _2(\frac{y_1-\mb{x_1}'\bbeta _1}{\sigma _1}, \frac{y_2-\mb{x_2}'\bbeta _2}{\sigma _2}, \rho )\right] - \ln \sigma _1 - \ln \sigma _2 \]

where $\phi _2(u,v,\rho )$ is the density function for standardized bivariate normal distribution with correlation $\rho $,

\[ \phi _2(u,v,\rho ) = \frac{e^{-(1/2)(u^2+v^2-2\rho uv)/(1-\rho ^2)}}{2\pi (1-\rho ^2)^{1/2}} \]

The second case is that one observed dependent variable is mapped to a point of its latent variable and the other dependent variable is mapped to a segment in the space of its latent variable. For example, in the bivariate censored model specified, if observed $y1>0$ and $y2=0$, then $y1^*=y1$ and $y2^*\in (-\infty , 0]$. In general, the log likelihood for one observation can be written as follows (the subscript i is dropped for simplicity): If one set is a single point and the other set is a range, without loss of generality, let $D_1(y_1)=\{ y_1\} $ and $D_2(y_2) = [L_2, R_2]$,

\begin{eqnarray*} \ell _{i} & =& \ln \left[\phi (\frac{y_1-\mb{x_1}'\bbeta _1}{\sigma _1})\right] - \ln \sigma _1 \\ & +& \ln \left[\Phi \left(\frac{R_2-\mb{x_2}'\bbeta _2 -\rho \frac{y_1-\mb{x_1}'\bbeta _1}{\sigma _1}}{\sigma _2}\right) - \Phi \left(\frac{L_2-\mb{x_2}'\bbeta _2 -\rho \frac{y_1-\mb{x_1}'\bbeta _1}{\sigma _1}}{\sigma _2}\right)\right] \end{eqnarray*}

where $\phi $ and $\Phi $ are the density function and the cumulative probability function for standardized univariate normal distribution.

The third case is that both dependent variables are mapped to segments in the space of latent variables. For example, in the bivariate censored model specified, if observed $y1=0$ and $y2=0$, then $y1^* \in (-\infty , 0]$ and $y2^*\in (-\infty , 0]$. In general, if $D_1(y_1) = [L_1, R_1]$ and $D_2(y_2) = [L_2, R_2]$, the log likelihood is

\[ \ell _{i} = \ln \int _{\frac{L_1-\mb{x_1}'\bbeta _1}{\sigma _1}} ^{\frac{R_1-\mb{x_1}'\bbeta _1}{\sigma _1}} \int _{\frac{L_2-\mb{x_2}'\bbeta _2}{\sigma _2}} ^{\frac{R_2-\mb{x_2}'\bbeta _2}{\sigma _2}} \phi _2(u,v,\rho ) \, du\, dv \]