  ### 9.1 Type I Tobit

Mathematical representation

 $\begin{array}{cc}\begin{array}{rl}{y}_{i}^{\ast }& ={\mathbf{x}}_{i}^{\prime }\beta +{𝜀}_{i},\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{𝜀}_{i}\sim \mathrm{N}\left(0,\frac{1}{\tau }\right)\\ {y}_{i}& =\left\{\begin{array}{ccc}\ell \hfill & \hfill \mathrm{if}\hfill & {y}_{i}^{\ast }\le \ell \hfill \\ {y}_{i}^{\ast }\hfill & \hfill \mathrm{if}\hfill & \ell <{y}_{i}^{\ast } (9.1)
• the model is estimated using $N$ observations
• ${y}_{i}$ is the value of the dependent variable for observation $i$ and it can assume values in the interval $\left[\ell ,u\right]$; $\ell$ could be $-\infty$ or $u$ $+\infty$
• ${y}_{i}^{\ast }$ is unobserved if ${y}_{i}^{\ast }\notin \left[\ell ,u\right]$
• ${\mathbf{x}}_{i}$ is a $K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector that stores the values of the $K$ independent variables for observation $i$
• $\beta$ is a $K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector of parameters
• $\tau$ is the precision of the error term: ${\sigma }_{𝜀}^{2}=\frac{1}{\tau }$

Priors

 Parameter Probability density function Default hyperparameters $\beta$ $p\left(\beta \right)=\frac{|\mathbf{P}{|}^{1∕2}}{{\left(2\pi \right)}^{K∕2}}exp\left\{-\frac{1}{2}{\left(\beta -\mathbf{m}\right)}^{\prime }\mathbf{P}\left(\beta -\mathbf{m}\right)\right\}$ $\mathbf{m}={\mathbf{0}}_{K}$, $\mathbf{P}=0.001\cdot {\mathbf{I}}_{K}$ $\tau$ $p\left(\tau \right)=\frac{{b}_{\tau }^{{a}_{\tau }}}{\Gamma \left({a}_{\tau }\right)}{\tau }^{{a}_{\tau }-1}{e}^{-\tau {b}_{\tau }}$ ${a}_{\tau }=0.001$, ${b}_{\tau }=0.001$

Syntax

$\left[$<model name> = $\right]$ tobitI( y ~ x1 x2  xK $\left[$, <options> $\right]$ );

where:

• y is the dependent variable name, as it appears in the dataset used for estimation
• x1 x2 $\dots$xK is a list of the $K$ independent variable names, 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 values between the lower and upper censoring points. Observations with missing values in y are dropped during estimation, but if a numerical value beyond the provided bounds is encountered, then an error is produced.

The optional arguments for the type I Tobit model are:1

 Gibbs parameters "chains" number of chains to run in parallel (positive integer); the default value is 1 "burnin" number of burn-in draws per chain (positive integer); the default value is 10000 "draws" number of retained draws per chain (positive integer); the default value is 20000 "thin" value of the thinning parameter (positive integer); the default value is 1 "seed" value of the seed for the random-number generator (positive integer); the default value is 42 Model specification "lb" lower censoring point (it could be set equal to -inf if the dependent variable is not censored from below); the default value is 0 "ub" upper censoring point (it could be set equal to inf if the dependent variable is not censored from above); the default value is $+\infty$ Hyperparameters "m" mean vector of the prior for $\beta$ ($K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector); the default value is ${\mathbf{0}}_{K}$ "P" precision matrix of the prior for $\beta$ ($K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}K$ symmetric and positive-deﬁnite matrix); the default value is $0.001\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}{\mathbf{I}}_{K}$ "a_tau" shape parameter of the prior for $\tau$ (positive number); the default value is $0.001$ "b_tau" rate parameter of the prior for $\tau$ (positive number); the default value is $0.001$ Dataset and log-marginal likelihood "dataset" the id value of the dataset that will be used for estimation; the default value is the ﬁrst dataset in memory (in alphabetical order) "logML_CJ" boolean indicating whether the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood should be calculated (true$|$false); the default value is false

Reported Parameters

 $\beta$ variable_name vector of parameters associated with the independent variables $\tau$ tau precision parameter of the error term, ${𝜀}_{i}$ ${\sigma }_{𝜀}$ sigma_e standard deviation of the error term: ${\sigma }_{𝜀}=1∕{\tau }^{1∕2}$

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

 Samples a matrix containing the draws from the posterior of $\beta$ and $\tau$ x1,$\dots$,xK vectors containing the draws from the posterior of the parameters associated with variables x1,$\dots$,xK (the names of these vectors are the names of the variables that were included in the right-hand side of the model) tau vector containing the draws from the posterior of $\tau$ lb lower censoring point ub upper censoring point logML the Lewis & Raftery (1997) approximation of the log-marginal likelihood logML_CJ the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood; this is available only if the model was estimated with the "logML_CJ"=true option nchains the number of chains that were used to estimate the model nburnin the number of burn-in draws per chain that were used when estimating the model ndraws the total number of retained draws from the posterior ($=$chains $\cdot$ draws) nthin value of the thinning parameter that was used when estimating the model nseed value of the seed for the random-number generator that was used when estimating the model

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

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

The type I Tobit model uses the mfx() function to calculate and report the marginal eﬀects of the variables in the x list on:

• the expected value of the observed dependent variable: $\frac{\partial E\left({y}_{i}|{\mathbf{x}}_{i}\right)}{\partial {x}_{ik}}$
• the expected value of the observed dependent variable, conditional on this being uncensored: $\frac{\partial E\left({y}_{i}|{\mathbf{x}}_{i},\ell <{y}_{i}
• the probability of no censoring: $\frac{\partial Prob\left(\ell <{y}_{i}

The three types of marginal eﬀects can be requested by setting the "type" argument of the mfx() function equal to 1, 2 or 3. The generic syntax for a statement involving the mfx() function after estimation of a type I Tobit model is:

mfx( $\left[$"type"=1$\right]$ $\left[$, "point"=<point of calculation>$\right]$ $\left[$, "model"=<model name>$\right]$ );
mfx( "type"=2 $\left[$, "point"=<point of calculation>$\right]$ $\left[$, "model"=<model name>$\right]$ );

and:

mfx( "type"=3 $\left[$, "point"=<point of calculation>$\right]$ $\left[$, "model"=<model name>$\right]$ );

for calculation of the marginal eﬀects on $E\left(y\right)$, on $E\left(y|\ell , and on $Prob\left(\ell , respectively. 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

myData = import("$BayESHOME/Datasets/dataset8.csv"); myData.constant = ones(rows(myData), 1); tobitI( y1 ~ constant x1 x2 x3 x4 ); Example 2 myData = import("$BayESHOME/Datasets/dataset8.csv");
myData.constant = ones(rows(myData), 1);
myData.y1 = -myData.y1;

myModel = tobitI( y1 ~ constant x1 x2 x3 x4,
"lb"=-inf, "ub"=0,
"m"=ones(5,1), "P" = 0.1*eye(5,5),
"a_tau"=0.01, "b_tau"=0.01,
"burnin"=10000, "draws"=40000, "thin"=4, "chains"=2,
"logML_CJ" = true, "dataset"=myData);

diagnostics("model"=myModel);

mfx("type"=1,"point"="mean","model"=myModel);
mfx("type"=2,"point"="mean","model"=myModel);
mfx("type"=3,"point"="mean","model"=myModel);

1Optional arguments are always given in option-value pairs (eg. "chains"=3).

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