Built using Zelig version r packageVersion("Zelig")
knitr::opts_knit$set( stop_on_error = 2L ) knitr::opts_chunk$set( fig.height = 11, fig.width = 7 ) options(cite = FALSE)
Bayesian Tobit Regression with tobit.bayes
.
Bayesian tobit regression estimates a linear regression model with a censored dependent variable using a Gibbs sampler. The dependent variable may be censored from below and/or from above. For other linear regression models with fully observed dependent variables, see Bayesian regression, maximum likelihood normal regression, or least squares.
z.out <- zelig(Y ~ X1 + X2, below = 0, above = Inf, model = "tobit.bayes", weights = w, data = mydata) x.out <- setx(z.out) s.out <- sim(z.out, x = x.out)
zelig() accepts the following arguments to specify how the dependent variable is censored.
below
: point at which the dependent variable is censored from
below. If the dependent variable is only censored from above, set
below = -Inf
. The default value is 0.
above
: point at which the dependent variable is censored from
above. If the dependent variable is only censored from below, set
above = Inf
. The default value is Inf
.
Use the following arguments to monitor the convergence of the Markov chain:
burnin
: number of the initial MCMC iterations to be discarded
(defaults to 1,000).
mcmc
: number of the MCMC iterations after burnin (defaults to
10,000).
thin
: thinning interval for the Markov chain. Only every
thin
-th draw from the Markov chain is kept. The value of mcmc
must be divisible by this value. The default value is 1.
verbose
: defaults to FALSE. If TRUE
, the progress of the
sampler (every $10\%$) is printed to the screen.
seed
: seed for the random number generator. The default is NA
which corresponds to a random seed of 12345.
beta.start
: starting values for the Markov chain, either a scalar
or vector with length equal to the number of estimated coefficients.
The default is NA
, such that the least squares estimates are used
as the starting values.
Use the following parameters to specify the model’s priors:
b0
: prior mean for the coefficients, either a numeric vector or a
scalar. If a scalar, that value will be the prior mean for all
coefficients. The default is 0.
B0
: prior precision parameter for the coefficients, either a
square matrix (with the dimensions equal to the number of the
coefficients) or a scalar. If a scalar, that value times an identity
matrix will be the prior precision parameter. The default is 0, which
leads to an improper prior.
c0
: c0/2
is the shape parameter for the Inverse Gamma prior
on the variance of the disturbance terms.
d0
: d0/2
is the scale parameter for the Inverse Gamma prior
on the variance of the disturbance terms.
Zelig users may wish to refer to help(MCMCtobit)
for more
information.
rm(list=ls(pattern="\\.out")) suppressWarnings(suppressMessages(library(Zelig))) set.seed(1234)
Attaching the sample dataset:
data(tobin)
Estimating linear regression using tobit.bayes
:
z.out <- zelig(durable ~ age + quant, model = "tobit.bayes", data = tobin, verbose = FALSE)
You can check for convergence before summarizing the estimates with three diagnostic tests. See the section Diagnostics for Zelig Models for examples of the output with interpretation:
z.out$geweke.diag() z.out$heidel.diag() z.out$raftery.diag()
summary(z.out)
Setting values for the explanatory variables to their sample averages:
x.out <- setx(z.out)
Simulating quantities of interest from the posterior distribution given x.out
.
s.out1 <- sim(z.out, x = x.out)
summary(s.out1)
Set explanatory variables to their default(mean/mode) values, with
high (80th percentile) and low (20th percentile) liquidity ratio (quant
):
x.high <- setx(z.out, quant = quantile(tobin$quant, prob = 0.8)) x.low <- setx(z.out, quant = quantile(tobin$quant, prob = 0.2))
Estimating the first difference for the effect of high versus low liquidity ratio on duration(durable
):
s.out2 <- sim(z.out, x = x.high, x1 = x.low)
summary(s.out2)
Let $Y_i^*$ be the dependent variable which is not directly observed. Instead, we observe $Y_i$ which is defined as following:
$$ Y_i = \left{ \begin{array}{lcl} Y_i^ &\textrm{if} & c_1<Y_i^<c_2 \ c_1 &\textrm{if} & c_1 \ge Y_i^ \ c_2 &\textrm{if} & c_2 \le Y_i^ \end{array}\right. $$
where $c_1$ is the lower bound below which $Y_i^$ is censored, and $c_2$ is the upper bound above which $Y_i^$ is censored.
$$ \begin{aligned} \epsilon_{i} & \sim & \textrm{Normal}(0, \sigma^2)\end{aligned} $$
where $\epsilon_{i}=Y^*_i-\mu_i$.
$$ \begin{aligned} \mu_{i}= x_{i} \beta,\end{aligned} $$
where $x_{i}$ is the vector of $k$ explanatory variables for observation $i$ and $\beta$ is the vector of coefficients.
$$ \begin{aligned} \beta & \sim & \textrm{Normal}k \left( b{0},B_{0}^{-1}\right) \ \sigma^{2} & \sim & \textrm{InverseGamma} \left( \frac{c_0}{2}, \frac{d_0}{2} \right) \end{aligned} $$
where $b_{0}$ is the vector of means for the $k$ explanatory variables, $B_{0}$ is the $k\times k$ precision matrix (the inverse of a variance-covariance matrix), and $c_0/2$ and $d_0/2$ are the shape and scale parameters for $\sigma^{2}$. Note that $\beta$ and $\sigma^2$ are assumed a priori independent.
qi$ev
) for the tobit regression model is
calculated as following. Let$$ \begin{aligned} \Phi_1 &=& \Phi\left(\frac{(c_1 - x \beta)}{\sigma}\right) \ \Phi_2 &=& \Phi\left(\frac{(c_2 - x \beta)}{\sigma}\right) \ \phi_1 &=& \phi\left(\frac{(c_1 - x \beta)}{\sigma}\right) \ \phi_2 &=& \phi\left(\frac{(c_2 - x \beta)}{\sigma}\right) \end{aligned} $$
where $\Phi(\cdot)$ is the (cumulative) Normal density function and $\phi(\cdot)$ is the Normal probability density function of the standard normal distribution. Then the expected values are
$$ \begin{aligned} E(Y|x) &=& P(Y^ \le c_1|x) c_1+P(c_1<Y^<c_2|x) E(Y^ \mid c_1<Y^<c_2, x)+P(Y^* \ge c_2) c_2 \ &=& \Phi_{1}c_1 + x \beta(\Phi_{2}-\Phi_{1}) + \sigma (\phi_1 -\phi_2) + (1-\Phi_2) c_2,\end{aligned} $$
qi$fd
) for the tobit regression model is
defined as$$ \begin{aligned} \text{FD}=E(Y\mid x_{1})-E(Y\mid x).\end{aligned} $$
qi$att.ev
) for the treatment group is$$ \begin{aligned} \frac{1}{\sum t_{i}}\sum_{i:t_{i}=1}[Y_{i}(t_{i}=1)-E[Y_{i}(t_{i}=0)]],\end{aligned} $$
where $t_{i}$ is a binary explanatory variable defining the treatment ($t_{i}=1$) and control ($t_{i}=0$) groups.
The Zelig object stores fields containing everything needed to rerun the Zelig output, and all the results and simulations as they are generated. In addition to the summary commands demonstrated above, some simply utility functions (known as getters) provide easy access to the raw fields most commonly of use for further investigation.
In the example above z.out$get_coef()
returns the estimated coefficients, z.out$get_vcov()
returns the estimated covariance matrix, and z.out$get_predict()
provides predicted values for all observations in the dataset from the analysis.
Bayesian tobit regression is part of the MCMCpack package by Andrew D. Martin and Kevin M. Quinn. The convergence diagnostics are part of the CODA package by Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines.
z5 <- ztobitbayes$new() z5$references()
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