9.2 Type II Tobit

Mathematical representation

$$\begin{array}{cc}\begin{array}{ccc}\hfill y_i^{\ast }& ={\mathbf{x}}_i^{\prime }\beta +{𝜀}_i\hfill & \hfill \\ \hfill y_i& =\left\{\begin{array}{ccc}{y}_i^{\ast }\hfill & \hfill \mathrm{if}\hfill & s_i=1\hfill \\ -\hfill & \hfill \mathrm{if}\hfill & s_i=0\hfill \end{array}\right.\hfill \end{array}\hfill & \begin{array}{ccc}\hfill s_i^{\ast }& ={\mathbf{z}}_i^{\prime }\delta +v_i\hfill & \hfill \\ \hfill s_i& =\left\{\begin{array}{ccc}1\hfill & \hfill \mathrm{if}\hfill & s_i^{\ast }>0\hfill \\ 0\hfill & \hfill \mathrm{if}\hfill & s_i^{\ast }\le 0\hfill \end{array}\right.\hfill \end{array}\hfill \end{array}$$

(9.2)

with:

$$\left[\begin{array}{c}\hfill {𝜀}_i\hfill \\ \hfill v_i\hfill \end{array}\right]\sim \mathrm{N}\left(\mathbf{0},{\Omega }^{-1}\right),\text{where}{\Omega }^{-1}\equiv \Sigma =\left[\begin{array}{cc}\hfill \xi +{\gamma }^2\hfill & \hfill \gamma \hfill \\ \hfill \gamma \hfill & \hfill 1\hfill \end{array}\right]$$

(9.3)

• the model is estimated using $N$ observations, for $N_1$ of which the outcome variable is observed ($s_i=1$) and for the remaining $N_0$ the outcome variable is missing ($s_i=0$)
• $y_i$ is the value of the dependent variable in the outcome equation for observation $i$ and it is observed only if $s_i=1$
• $s_i^{\ast }$ is the value of the latent dependent variable in the selection equation for observation $i$
• ${\mathbf{x}}_i$ is a $K\times 1$ vector that stores the values of the $K$ independent variables in the outcome equation for observation $i$
• ${\mathbf{z}}_i$ is an $L\times 1$ vector that stores the values of the $L$ independent variables in the selection equation for observation $i$
• the same variable could appear in both ${\mathbf{x}}_i$ and ${\mathbf{z}}_i$
• $\beta$ is a $K\times 1$ vector of parameters
• $\delta$ is an $L\times 1$ vector of parameters
• $\Omega$ is the precision matrix of the error vector, ${\left[\begin{array}{cc}\hfill {𝜀}_i\hfill & \hfill v_i\hfill \end{array}\right]}^{\prime }$ and $\Sigma$ is the covariance matrix of the error vector
• $\xi$ is the variance of ${𝜀}_i$ conditional on $v_i$
• $\gamma$ is the covariance of ${𝜀}_i$ and $v_i$

Priors

Because $\xi$ is a variance parameter, a Gamma prior is placed on the corresponding precision parameter, $\frac{1}{\xi }$. This is equivalent to placing and inverse-Gamma prior on $\xi$ directly.

The prior for $\gamma$ depends on the value of $\xi$: given $\xi$, $\gamma$ follows a Normal distribution with mean $m_{\gamma }$ and precision $\frac{t_{\gamma }}{\xi }$. This is done so that the prior uncertainty around $\gamma$ scales along with the prior uncertainty around $\xi$.

Syntax

where:

• y is the dependent variable name in the outcome equation, as it appears in the dataset used for estimation
• x1 x2 $\dots$xK is a list of the $K$ independent variable names in the outcome equation, as they appear in the dataset used for estimation; when a constant term is to be included in the model, this must be requested explicitly
• z1 z2 $\dots$zL is a list of the $L$ independent variable names in the selection equation, as they appear in the dataset used for estimation; when a constant term is to be included in the model, this must be requested explicitly

The dependent variable, y, in the dataset used for estimation must contain both numerical values and missing values (“nans"). The values of the variables in the x list are not used during estimation and they could be missing. However, observations with missing values in the z list are dropped prior to estimation.

The optional arguments for the type II Tobit model are:²

Reported Parameters

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a type II Tobit model is created, then the following results are saved in the model item and are accessible via the ‘.’ operator:

Additionally, the following functions are available for post-estimation analysis (see section B.14):

• diagnostics()
• test()
• pmp()
• mfx()

Usually the marginal effects of primary importance in a type II Tobit model are the effects of changes in the independent variables in the outcome equation on the expected value of the dependent variable in the same equation, for the entire population (whether selected or not). These effects, at least for variables included linearly in the model, are the corresponding $\beta$s. Nevertheless, there are two additional types of marginal effects that could be of interest and which are not linear functions of the model’s parameters. The type II Tobit model uses the mfx() function to calculate and report the marginal effects of:

• the variables in the z list on the probability of selection: $\frac{\partial Prob\left(s_i=1|{\mathbf{z}}_i\right)}{\partial z_{i\ell }}$
• the variables in the x list on the expected value of the response variable for the part of the population that is selected³: $\frac{\partial E\left(y_i|{\mathbf{x}}_i,{\mathbf{z}}_i,s_i=1\right)}{\partial x_{ik}}$

The two types of marginal effects can be requested by setting the "type" argument of the mfx() function equal to 1 or 3. The generic syntax for a statement involving the mfx() function after estimation of a type II Tobit model is:

and:

for calculation of the marginal effects on $Prob\left(s=1\right)$ and on $E\left(y|s=1\right)$. The default value of the "type" option is 1. See the general documentation of the mfx() function (section B.14) for details on the other optional arguments.

Examples

Example 1

Example 2

Share this content:

© 2016–20 Grigorios Emvalomatis