The HPQLIM Procedure

Limited Dependent Variable Models

Censored Regression Models
Truncated Regression Models

Censored Regression Models

When the dependent variable is censored, values in a certain range are all transformed to a single value. For example, the standard Tobit model can be defined as

$y^{*}_{i} = \mathbf{x}_{i}’\bbeta + \epsilon _{i}$

$y_{i} = \left\{ \begin{array}{ll} y^{*}_{i} & \mr {if} y^{*}_{i}>0 \\ 0 & \mr {if} y^{*}_{i}\leq 0 \end{array} \right.$

where $\epsilon _{i} \sim iid N(0,\sigma ^{2})$ . The log-likelihood function of the standard censored regression model is

$\ell = \sum _{i\in \{ y_{i}=0\} }\ln [1-\Phi (\mathbf{x}_{i}’\bbeta /\sigma )] +\sum _{i\in \{ y_{i}>0\} } \ln \left[\phi (\frac{y_{i}-\mathbf{x}_{i}\bbeta }{\sigma })/\sigma \right]$

where $\Phi (\cdot )$ is the cumulative density function of the standard normal distribution and $\phi (\cdot )$ is the probability density function of the standard normal distribution.

The Tobit model can be generalized to handle observation-by-observation censoring. The censored model on both the lower and upper limits can be defined as

$y_{i} = \left\{ \begin{array}{ll} R_{i} & \mr {if} \; y_{i}^{*} \geq R_{i} \\ y_{i}^{*} & \mr {if} \; L_{i} < y_{i}^{*} < R_{i} \\ L_{i} & \mr {if} \; y_{i}^{*} \leq L_{i} \end{array} \right.$

The log-likelihood function can be written as

$\displaystyle \ell$	$\displaystyle =$	$\displaystyle \sum _{i\in \{ L_{i}< y_{i} < R_{i}\} } \ln \left[\phi (\frac{y_{i}-\mathbf{x}_{i}\bbeta }{\sigma })/\sigma \right] + \sum _{i\in \{ y_{i}=R_{i}\} } \ln \left[\Phi (-\frac{R_{i}-\mathbf{x}_{i}\bbeta }{\sigma })\right] +$
$\displaystyle$	$\displaystyle$	$\displaystyle \sum _{i\in \{ y_{i}=L_{i}\} } \ln \left[\Phi (\frac{L_{i}-\mathbf{x}_{i}\bbeta }{\sigma })\right]$

Log-likelihood functions of the lower-limit or upper-limit censored model are easily derived from the two-limit censored model. The log-likelihood function of the lower-limit censored model is

$\ell = \sum _{i\in \{ y_{i} > L_{i}\} } \ln \left[\phi (\frac{y_{i}-\mathbf{x}_{i}\bbeta }{\sigma })/\sigma \right] + \sum _{i\in \{ y_{i}=L_{i}\} } \ln \left[\Phi (\frac{L_{i}-\mathbf{x}_{i}\bbeta }{\sigma })\right]$

The log-likelihood function of the upper-limit censored model is

$\ell = \sum _{i\in \{ y_{i} < R_{i}\} } \ln \left[\phi (\frac{y_{i}-\mathbf{x}_{i}\bbeta }{\sigma })/\sigma \right] + \sum _{i\in \{ y_{i}=R_{i}\} } \ln \left[1-\Phi (\frac{R_{i}- \mathbf{x}_{i}\bbeta }{\sigma })\right]$

Truncated Regression Models

In a truncated model, the observed sample is a subset of the population where the dependent variable falls within a certain range. For example, when neither a dependent variable nor exogenous variables are observed for $y^{*}_{i} \leq 0$ , the truncated regression model can be specified as

$\ell = \sum _{i\in \{ y_{i}>0\} } \left\{ -\ln \Phi (\mathbf{x}_{i}’\bbeta /\sigma ) + \ln \left[\frac{\phi ((y_{i} - \mathbf{x}_{i}\bbeta )/\sigma )}{\sigma } \right] \right\}$

The two-limit truncation model is defined as

$y_{i} = y_{i}^{*} \mr {if} \; L_{i} < y_{i}^{*} < R_{i}$

The log-likelihood function of the two-limit truncated regression model is

$\ell = \sum _{i=1}^{N} \left\{ \ln \left[\phi (\frac{y_{i}-\mathbf{x}_{i}\bbeta }{\sigma })/\sigma \right] - \ln \left[\Phi (\frac{R_{i}-\mb {x}_{i}\bbeta }{\sigma }) - \Phi (\frac{L_{i}-\mb {x}_{i}\bbeta }{\sigma })\right] \right\}$

The log-likelihood function of the lower-limit truncation model is

$\ell = \sum _{i=1}^{N}\left\{ \ln \left[\phi (\frac{y_{i}-\mb {x}_{i}\bbeta }{\sigma }) / \sigma \right] - \ln \left[1 - \Phi (\frac{L_{i}-\mb {x}_{i}\bbeta }{\sigma })\right] \right\}$

The log-likelihood function of the upper-limit truncation model is

$\ell = \sum _{i=1}^{N}\left\{ \ln \left[\phi (\frac{y_{i}-\mb {x}_{i}\bbeta }{\sigma }) / \sigma \right] - \ln \left[\Phi (\frac{R_{i}-\mb {x}_{i}\bbeta }{\sigma })\right] \right\}$