Limited Dependent Variable Models :: SAS/ETS(R) 12.3 User's Guide

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 of 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- 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

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

Types of Tobit Models

Amemiya (1984) classified Tobit models into five types based on characteristics of the likelihood function. For notational convenience, let denote a distribution or density function, $y^{*}_{ji}$ is assumed to be normally distributed with mean $\mathbf{x}_{ji}’\bbeta _{j}$ and variance $\sigma ^{2}_{j}$ .

Type 1 Tobit

The Type 1 Tobit model was already discussed in the preceding section.

$\displaystyle y_{1i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{1i}’\bbeta _{1} + u_{1i}$
$\displaystyle y_{1i}$	$\displaystyle =$	$\displaystyle y_{1i}^{}\; \; \; \textrm{if $y^{}_{1i} > 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} \leq 0$}$

The likelihood function is characterized as $P(y_{1} < 0)P(y_{1})$ .

Type 2 Tobit

The Type 2 Tobit model is defined as

$\displaystyle y_{1i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{1i}’\bbeta _{1} + u_{1i}$
$\displaystyle y_{2i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{2i}’\bbeta _{2} + u_{2i}$
$\displaystyle y_{1i}$	$\displaystyle =$	$\displaystyle 1 \; \; \; \textrm{if $y_{1i}^{*}>0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y_{1i}^{*}\leq 0$}$
$\displaystyle y_{2i}$	$\displaystyle =$	$\displaystyle y_{2i}^{} \; \; \; \textrm{if $y_{1i}^{}>0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y_{1i}^{*}\leq 0$}$

where $(u_{1i},u_{2i}) \sim N(0,\Sigma )$ . The likelihood function is described as $P(y_{1} < 0)P(y_{1}>0,y_{2})$ .

Type 3 Tobit

The Type 3 Tobit model is different from the Type 2 Tobit in that $y_{1i}^{*}$ of the Type 3 Tobit is observed when $y_{1i}^{*}>0$ .

$\displaystyle y_{1i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{1i}’\bbeta _{1} + u_{1i}$
$\displaystyle y_{2i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{2i}’\bbeta _{2} + u_{2i}$
$\displaystyle y_{1i}$	$\displaystyle =$	$\displaystyle y_{1i}^{}\; \; \; \textrm{if $y^{}_{1i} > 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} \leq 0$}$
$\displaystyle y_{2i}$	$\displaystyle =$	$\displaystyle y_{2i}^{}\; \; \; \textrm{if $y^{}_{1i} > 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} \leq 0$}$

where $(u_{1i},u_{2i})’\sim iid N(0,\Sigma )$ .

The likelihood function is characterized as $P(y_{1} < 0)P(y_{1},y_{2})$ .

Type 4 Tobit

The Type 4 Tobit model consists of three equations:

$\displaystyle y_{1i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{1i}’\bbeta _{1} + u_{1i}$
$\displaystyle y_{2i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{2i}’\bbeta _{2} + u_{2i}$
$\displaystyle y_{3i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{3i}’\bbeta _{3} + u_{3i}$
$\displaystyle y_{1i}$	$\displaystyle =$	$\displaystyle y_{1i}^{}\; \; \; \textrm{if $y^{}_{1i} > 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} \leq 0$}$
$\displaystyle y_{2i}$	$\displaystyle =$	$\displaystyle y_{2i}^{}\; \; \; \textrm{if $y^{}_{1i} > 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} \leq 0$}$
$\displaystyle y_{3i}$	$\displaystyle =$	$\displaystyle y_{3i}^{}\; \; \; \textrm{if $y^{}_{1i} \leq 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} > 0$}$

where $(u_{1i},u_{2i},u_{3i})’\sim iid N(0,\Sigma )$ . The likelihood function of the Type 4 Tobit model is characterized as $P(y_{1} < 0, y_{3})P(y_{1},y_{2})$ .

Type 5 Tobit

The Type 5 Tobit model is defined as follows:

$\displaystyle y_{1i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{1i}’\bbeta _{1} + u_{1i}$
$\displaystyle y_{2i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{2i}’\bbeta _{2} + u_{2i}$
$\displaystyle y_{3i}^{*}$	$\displaystyle =$	$\displaystyle \mathbf{x}_{3i}’\bbeta _{3} + u_{3i}$
$\displaystyle y_{1i}$	$\displaystyle =$	$\displaystyle 1\; \; \; \textrm{if $y^{*}_{1i} > 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0\; \; \; \textrm{if $y^{*}_{1i} \leq 0$}$
$\displaystyle y_{2i}$	$\displaystyle =$	$\displaystyle y_{2i}^{}\; \; \; \textrm{if $y^{}_{1i} > 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} \leq 0$}$
$\displaystyle y_{3i}$	$\displaystyle =$	$\displaystyle y_{3i}^{}\; \; \; \textrm{if $y^{}_{1i} \leq 0$}$
$\displaystyle$	$\displaystyle =$	$\displaystyle 0 \; \; \; \textrm{if $y^{*}_{1i} > 0$}$

where $(u_{1i},u_{2i},u_{3i})’$ are from iid trivariate normal distribution. The likelihood function of the Type 5 Tobit model is characterized as $P(y_{1} < 0, y_{3})P(y_{1} > 0, y_{2})$ .

Code examples for these models can be found in Types of Tobit Models.

Truncated Regression Models

In a truncated model, the observed sample is a subset of the population where the dependent variable falls in 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.

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

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 functions of the lower- and upper-limit truncation model are

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

The QLIM Procedure