For observations beyond ‘k’, a cumulative density is
specified [F

*(y)*] , and the likelihood is a mixture of products of density [*f(y)*] and cumulative density functions [F*(y)*].
Roughly: L = [

*f(y)*]^{d-1}[F*(y)*]^{d}where d = 1 if censoredBelow is the output of a regression modeling academic aptitude (using data from UCLA statistical computing examples- see references in the R-code documentation that follows) as a function of reading and math scores, as well as program participation:

Call:

lm(formula = mydata$apt ~ mydata$read + mydata$math + as.factor(mydata$prog))

Residuals:

Min 1Q Median 3Q Max

-161.463 -42.474 -0.707 43.180 181.554

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 242.735 30.140 8.054 7.80e-14 ***

mydata$read 2.553 0.583 4.379 1.95e-05 ***

mydata$math 5.383 0.659 8.169 3.84e-14 ***

as.factor(mydata$prog)general -13.741 11.744 -1.170 0.243423

as.factor(mydata$prog)vocational -48.835 12.982 -3.762 0.000223 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 62.38 on 195 degrees of freedom

Multiple R-squared: 0.6127, Adjusted R-squared: 0.6048

F-statistic: 77.13 on 4 and 195 DF, p-value: < 2.2e-16

Below is output from a tobit model specified in R:

vglm(formula = mydata$apt ~ mydata$read + mydata$math + as.factor(mydata$prog),

family = tobit(Upper = 800))

Pearson Residuals:

Min 1Q Median 3Q Max

mu -3.2606 -0.69522 0.049445 0.82743 2.7935

log(sd) -1.1309 -0.61020 -0.310926 0.21836 4.8277

Coefficients:

Value Std. Error t value

(Intercept):1 209.5488 32.642056 6.4196

(Intercept):2 4.1848 0.049756 84.1071

mydata$read 2.6980 0.619577 4.3546

mydata$math 5.9148 0.706472 8.3723

as.factor(mydata$prog)general -12.7145 12.409533 -1.0246

as.factor(mydata$prog)vocational -46.1431 13.707785 -3.3662

Number of linear predictors: 2

Names of linear predictors: mu, log(sd)

Dispersion Parameter for tobit family: 1

Log-likelihood: -872.8971 on 394 degrees of freedom

Number of Iterations: 8

Notice the coefficients from the tobit model are larger than those from OLS, indicating the downward bias of the coefficients resulting from OLS regression on a censored dependent variable. Below is the output from a bayesian model, based on the specifications outlined for the MCMCtobit function in the MCMCpack documentation. These results are very similar to those obtained by the previous tobit model output.

Thinning interval = 1

Number of chains = 1

Sample size per chain = 30000

1. Empirical mean and standard deviation for each variable,

plus standard error of the mean:

Mean SD Naive SE Time-series SE

(Intercept) 208.635 33.3365 0.192468 0.213581

mydata$read 2.702 0.6304 0.003640 0.003506

mydata$math 5.930 0.7242 0.004181 0.004368

as.factor(mydata$prog)general -12.733 12.6756 0.073183 0.082116

as.factor(mydata$prog)vocational -46.145 13.9556 0.080573 0.085942

sigma2 4486.277 488.5388 2.820580 2.962118

2. Quantiles for each variable:

2.5% 25% 50% 75% 97.5%

(Intercept) 142.260 186.381 208.659 231.153 273.779

mydata$read 1.482 2.273 2.701 3.120 3.949

mydata$math 4.522 5.438 5.924 6.417 7.357

as.factor(mydata$prog)general -37.558 -21.261 -12.733 -4.175 12.300

as.factor(mydata$prog)vocational -73.634 -55.441 -46.044 -36.719 -18.890

sigma2 3635.560 4140.883 4450.761 4795.448 5536.943

R-Code: (use scroll bar at bottom, or click code and scroll with arrow keys)

# ------------------------------------------------------------------ # | PROGRAM NAME: ex_bayesian_tobit # | DATE: 9/17/11 # | CREATED BY: Matt Bogard # | PROJECT FILE: www.econometricsense.blogspot.com # |---------------------------------------------------------------- # | PURPOSE: comparison of models for censored dependent variables # | 1 - least squares # | 2 - tobit model # | 3 - bayesian model # |------------------------------------------------------------------ # | REFERENCES: # | UCLA Statistical Computing: http://www.ats.ucla.edu/stat/R/dae/tobit.htm # | R Package 'MCMCpack' documentation : # http://mcmcpack.wustl.edu/documentation.html # | # | Literature: # | Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. “MCMCpack: Markov Chain Monte Carlo in R.”, # | Journal of Statistical Software. 42(9): 1-21. http://www.jstatsoft. org/v42/i09/. # | # | Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. # | http:// scythe.wustl.edu. # | # | Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnos- tics for # | MCMC(CODA). http://www-fis.iarc.fr/coda/. # | # | Siddhartha Chib. 1992. “Bayes inference in the Tobit censored regression model." Journal of Econometrics. # | 51:79-99. # | # | # | # ------------------------------------------------------------------ # example tobit model # get data mydata <- read.csv(url("http://www.ats.ucla.edu/stat/r/dae/tobit.csv")) #explore dataset names(mydata) # list var names dim(mydata) # data dimensions hist(mydata$apt) # histogram of dependent variable for academic aptitude # indcates right or upper bound censoring at 'y' = 800 # run model using standard ols regression ols <- lm(mydata$apt~mydata$read + mydata$math + as.factor(mydata$prog)) summary(ols) # tobit model library(VGAM) # load package tobit <- vglm(mydata$apt ~ mydata$read + mydata$math + as.factor(mydata$prog), tobit(Upper=800)) summary(tobit) # note the coefficients for the tobit model are larger, indicating the downward bias # of the OLS estimates # bayesian model library(MCMCpack) # load package bayes.tobit <- MCMCtobit(mydata$apt ~ mydata$read + mydata$math + as.factor(mydata$prog), above = 800, mcmc = 30000, verbose = 1000) summary(bayes.tobit) plot(bayes.tobit) # the empirical (posterior mean) looks very similar to the tobit estimates.