Bayesian Models with Censored Data: A comparison of OLS, tobit and bayesian models

The following R code models a censored dependent variable (in this case academic aptitude) using a traditional least squares, tobit, and Bayesian approaches.  As depicted below, the OLS estimates (blue) for censored data are inconsistent and will not approach the true population parameters (green).

A tobit model will provide better results. A tobit model is estimated via maximum likelihood.

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 censored

Bayesian methods for censored dependent variables are also available, and may provide better estimates than traditional tobit specifications in the face of heteroskedasticity, (which in addition to leading to inaccurate estimates of standard errors in OLS, also leads to inaccurate coefficient estimates in the tobit model). 

Below 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:

Call:
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.

Iterations = 1001:31000
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. 

Created by Pretty R at inside-R.org

Elements of Bayesian Econometrics

 posterior = (likelihood x prior) / integrated likelihood

The combination of a prior distribution and a likelihood function is utilized to produce a posterior distribution.  Incorporating information from both the prior distribution and the likelihood function leads to a reduction in variance and an improved estimator.

As n→ ∞ the likelihood centers over the true β. As a result, with more data the role of the likelihood function becomes more important in deriving the posterior distribution.

P(θ|y) = (P(y|θ)*P(θ) )/(P(y) = ( P(y|θ)*P(θ) ) / ∫  (P(y|θ)*P(θ) )/dθ

P(θ|y) = posterior distribution

P(y|θ) = likelihood function L(θ)

P(θ) = prior distribution

P(y) = ∫  (P(y|θ)*P(θ) )/dθ = the integrated likelihood, a normalizing or proportionality factor

P(θ|y) α P(y|θ)*P(θ) -->  the posterior distribution is a weighted average of the prior distribution and the likelihood function

Deriving the posterior distribution may be an algebraic exercise, but summarization is more difficult, requiring numerical methods. We may often be interested in deriving the posterior expected value of θ.

E(θ|y) = ∫  θ f(θ|y) dθ

This may be approximated numerically using a sequence of random draws   Θ⁽¹⁾ , Θ⁽¹⁾ , …Θ^(G) 

E(θ|y) = 1/G ∑ Θ^(g) 

This can actually be implemented via Markov Chain Monte Carlo methods. For the sequence of draws Θ⁽¹⁾ , Θ⁽¹⁾ , …Θ^(G) , Θ^(g+1)  depends solely on the previous draw Θ^(g)  forming a markov chain which converges to the target density. 

LOSS FUNCTIONS

Define l(Θ^(hat) ,Θ) as the loss from the decision to choose Θ^(hat) to estimate Θ

Ex:  squared error loss l(Θ^(hat) ,Θ) = (Θ^(hat) - Θ)^2

choose Θ^(hat) to minimize E[l(Θ^(hat) ,Θ)] =  ∫  l(Θ^(hat) ,Θ) p(θ|y) dθ

REGRESSION ANALOGY

Recall, that if we specify the following likelihood:

Β^OLS = B^MLE = (X’X)^-1X’y

If we specify a prior distribution for β as P(β) α  exp[(β - m) V^-1 (β - m)/2]

i.e β ~N(m,V)

 and

σ² ~ I gamma(v/2,ς/2) 

Given the specified priors and likelihood, we can express the posterior distribution as:

p( β , σ² |y)  α  p(y|β , σ² ) p(β , σ² )

Bayesian models can be implemented using the package MCMCpack in R.

An example given by Martin (one of the developers of the package) involves modeling murder as a function of unemployment. 

Under the linear regression model we get the following results:

Call:
lm(formula = murder ~ unemp, data = murder)

Residuals:
    Min      1Q  Median      3Q     Max
-5.5386 -2.6677 -0.6324  2.3935  8.9443

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  -0.5509     2.4810  -0.222  0.82521  
unemp         1.4204     0.4535   3.132  0.00296 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.599 on 48 degrees of freedom
Multiple R-squared: 0.1697,    Adjusted R-squared: 0.1524
F-statistic: 9.809 on 1 and 48 DF,  p-value: 0.002957

Using bayesian methods, with standard priors we get very similar results:

Iterations = 1001:11000
Thinning interval = 1
Number of chains = 1
Sample size per chain = 10000

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

               Mean     SD Naive SE Time-series SE
(Intercept) -0.5278 2.5374 0.025374       0.023700
unemp        1.4158 0.4648 0.004648       0.004426
sigma2      13.5547 2.9186 0.029186       0.035616

2. Quantiles for each variable:

               2.5%    25%    50%    75%  97.5%
(Intercept) -5.5103 -2.189 -0.493  1.147  4.402
unemp        0.4957  1.112  1.412  1.719  2.329
sigma2       9.0343 11.496 13.151 15.199 20.355

 Bayesian methods with informative priors (as specified in the earlier discussion above)

Iterations = 1001:11000
Thinning interval = 1
Number of chains = 1
Sample size per chain = 10000

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

              Mean     SD Naive SE Time-series SE
(Intercept) -0.317 0.9147 0.009147       0.008721
unemp        1.392 0.1885 0.001885       0.001908
sigma2      13.306 2.8145 0.028145       0.033750

2. Quantiles for each variable:

              2.5%    25%     50%     75%  97.5%
(Intercept) -2.123 -0.925 -0.3036  0.2991  1.485
unemp        1.026  1.266  1.3907  1.5147  1.769
sigma2       8.916 11.296 12.9415 14.9085 19.802

References:

Andrew D. Martin. "Bayesian Analysis." Prepared for The Oxford Handbook of Political Methodology. Click here for the chapter.

Andrew D. Martin. "Bayesian Inference and Computation in Political Science." Slides from a talk given to the Department of Politics, Nuffield College, Oxford University, March 9, 2004. Click here for the slides, and here for the example R code.

An Introduction to Modern Bayesian Econometrics. Tony Lancaster. 2002. http://www.brown.edu/Courses/EC0264/book.pdf

 

  R-Code :

#   ------------------------------------------------------------------
#  | PROGRAM NAME: EX_BAYESIAN_ECONOMETRCS
#  | DATE: 9-15-11
#  | CREATED BY: MATT BOGARD 
#  | PROJECT FILE: WWW.ECONOMETRICSENSE.BLOGSPOT.COM             
#  |----------------------------------------------------------------
#  | ADAPTED FROM: Andrew D. Martin. "Bayesian Inference and Computation in Political Science." Slides from a talk given to the Department of Politics, Nuffield College, Oxford University, March 9, #      |   2004.   SLIDES:http://adm.wustl.edu/media/talks/bayesslides.pdf  R-CODE :  http://adm.wustl.edu/media/talks/examples.zip           
#  |
#  |
#  | 
#  |------------------------------------------------------------------
 
 
setwd('/Users/wkuuser/Desktop/Briefcase/R Data Sets') 
 
library(MCMCpack)
murder <- read.table("murder.txt", header=TRUE)
names(murder)
dim(murder)
 
# estimation using OLS
lm(murder ~ unemp, data=murder)
summary(lm(murder ~ unemp, data=murder))
 
# posterior with standard priors
post1 <- MCMCregress(murder ~ unemp, data=murder)
print(summary(post1))
 
# posterior with informative priors
m <- matrix(c(0,3),2,1)
V <- matrix(c(1,0,0,1),2,2)
post2 <- MCMCregress(murder ~ unemp, b0=m, B0=V, data=murder)
print(summary(post2))

Created by Pretty R at inside-R.org