Re: Re: st: Formally comparing Tobit and Probit estimates

[Christopher Baum <kit.baum@bc.edu>]

<>
On Jan 22, 2014, at 2:33 AM, Nick wrote:

> 1. The coefficients are not even in the same dimensions or units of
> measurement.
> (It's a pity that statistics courses and texts almost uniformly
> neglect such matters.)
> 
> 2. The -tobit- model is linear; the -probit- model is nonlinear.

I wholeheartedly agree with Nick's point #1. However point #2 is misleading and incorrect. The -tobit- model is decidedly 
nonlinear, as its likelihood function is the additively separable sum of a Probit LLF for those observations which are censored
and a regression LLF for those observations that are not. Although the regression terms are the same as one would write
down for linear least squares, the probit terms are not linear in the data. That is why an iterative (maximum likelihood) technique is
needed for its estimation.

The coefficients reported by -tobit- are, like those from -probit-, estimates of the derivative of the latent index variable
with respect to the explanatory variable, which are not themselves of interest. The estimates of the marginal effects for
observed values of the dependent variable are the reported coefficients times the probability of being censored; that is,
they are attenuated toward zero.

The reason why it doesn't make much sense to compare the -probit- and -tobit- coefficients is that for probit, the latent
variable is unobserved for all observations, whereas for tobit, the latent variable is only latent for the censored observations.
Thus the information going into the estimation differs, as  in probit, all observations are coded as 0/1. Of course, there is no 
correspondence between doubly-censored tobit and binomial probit.

These topics are discussed in my book, An introduction to modern econometrics using Stata, chapter 10.

Kit

Kit Baum
Professor of Economics and Social Work, Boston College, Chestnut Hill MA, USA
DIW Research Professor, Department of Macroeconomics, DIW Berlin, Berlin, Germany
baum@bc.edu  |  http://ideas.repec.org/e/pba1.html












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