Standardized solution for censored regression

Question

The censReg package in R provides identical results to Mplus for censored regression. Mplus also provides a fully standardized solution. How can I scale the censReg coefficients so they are standardized, (beta weights)? Does it make sense to standardize marginal effects in censored outcome models, (see the margEff.censReg function)?

Answer 1

I will write out the very simplest case of the censored regression model with left-censoring at zero, that is, the classical Tobit model. $$ \begin{align} Y_i^* &= \boldsymbol{X}_i'\boldsymbol{\beta} + U_i, \mathrm{U}_i \sim \mathrm{Normal}(0, \sigma^2) \\ Y_i &= Y_i^*\mathbf{1}_{\left[Y^*_i \geq 0\right]} \end{align} $$

Then, as in Cameron and Trivedi (2005, pg. 542) the marginal effects of the model are $$ \nabla_{\boldsymbol{\beta}}\mathbb{E}(Y_i^* \mid Y_i^* > 0, \boldsymbol{X}_i) = \Phi\left(\frac{\boldsymbol{X}_i'\boldsymbol{\beta}}{\sigma}\right)\boldsymbol{\beta} $$ It is clear to see that you will need separate estimates of the vector of coefficients and the scale of the latent errors ($\sigma$), in order to estimate these marginal effects.

So, for example, in R, we can estimate the standardized coefficients and marginal effects using data and the model described here.

# purpose: censored regression models
# date created: 23rd november 2012

library(censReg)
dfCensReg <- as.data.frame(read.table("ex3.2.dat"))  # read in the data
modelCensReg <- censReg(V1 ~ V2 + V3, data = dfCensReg)  # tobit model
coefCensReg <- coef(modelCensReg, logSigma=F)  # report the scale estimate
coefCensReg/coefCensReg["sigma"]  # standardized coefficients
summary(margEff(modelCensReg)) # marginal effects

which gives you the output:

> coefCensReg/coefCensReg["sigma"]  # standardized coefficients
(Intercept)          V2          V3       sigma 
  0.4811634   1.0032366   0.4618115   1.0000000 
> summary(margEff(modelCensReg)) # marginal effects
   Marg. Eff. Std. Error t value  Pr(>|t|)    
V2   0.732464   0.028181  25.992 < 2.2e-16 ***
V3   0.337169   0.025804  13.067 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1