Tobit — R cross-check (AER::tobit, survival::survreg)¶

Companion to tobit_python.ipynb / tobit_pymc.ipynb. The Tobit is a censored-Gaussian regression: AER::tobit is the standard interface; it is a wrapper around survival::survreg with a Gaussian distribution and left-censoring.

In [1]:
ok <- suppressWarnings(suppressMessages(require(AER)))
if (!ok) { install.packages('AER', repos='https://cloud.r-project.org'); library(AER) }
d <- read.csv('mroz_hours.csv')
fit <- tobit(hours ~ nwifeinc + educ + exper + expersq + age + kidslt6 + kidsge6, left=0, data=d)
print(summary(fit))
cat(sprintf('\nscale (sigma) = %.1f\n', fit$scale))
# OLS for contrast (biased)
cat('\nnaive OLS (biased toward 0):\n'); print(round(coef(lm(hours ~ nwifeinc+educ+exper+expersq+age+kidslt6+kidsge6, data=d)),2))

Call:
tobit(formula = hours ~ nwifeinc + educ + exper + expersq + age + 
    kidslt6 + kidsge6, left = 0, data = d)

Observations:
         Total  Left-censored     Uncensored Right-censored 
           753            325            428              0 

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept)  965.30528  446.43614   2.162 0.030599 *  
nwifeinc      -8.81424    4.45910  -1.977 0.048077 *  
educ          80.64561   21.58324   3.736 0.000187 ***
exper        131.56430   17.27939   7.614 2.66e-14 ***
expersq       -1.86416    0.53766  -3.467 0.000526 ***
age          -54.40501    7.41850  -7.334 2.24e-13 ***
kidslt6     -894.02174  111.87804  -7.991 1.34e-15 ***
kidsge6      -16.21800   38.64139  -0.420 0.674701    
Log(scale)     7.02289    0.03706 189.514  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Scale: 1122 

Gaussian distribution
Number of Newton-Raphson Iterations: 4 
Log-likelihood: -3819 on 9 Df
Wald-statistic: 253.9 on 7 Df, p-value: < 2.22e-16 


scale (sigma) = 1122.0

naive OLS (biased toward 0):
(Intercept)    nwifeinc        educ       exper     expersq         age 
    1330.48       -3.45       28.76       65.67       -0.70      -30.51 
    kidslt6     kidsge6 
    -442.09      -32.78

Results¶

  • AER::tobit gives the canonical Mroz estimates — educ ≈ +80.6, exper ≈ +131.6 (concave), age ≈ −54.4, kidslt6 ≈ −894, σ ≈ 1122 — matching the from-scratch Gibbs and PyMC pm.Censored exactly.
  • The naive OLS coefficients are visibly smaller in magnitude for every significant effect — the attenuation from ignoring the 43% of women at the zero boundary. (Only kidsge6, weak and insignificant in both fits, bucks the pattern.)
  • Three engines agree (from-scratch augmentation Gibbs ≈ PyMC ≈ AER::tobit); survreg(dist="gaussian", left-censored) is the same fit under the hood, and the same survreg machinery extends to interval censoring (Surv(lo, hi, type="interval2")).