st: Re: Generating predicted values for OLS with transformed dependent variables

[Phil Schumm <pschumm@uchicago.edu>]

On Apr 12, 2006, at 9:31 AM, Nick Cox wrote:
> As posted earlier, -glm- offers the back-to-basics Jacobin solution  
> as an alternative to this use of Jacobians.
On Apr 12, 2006, at 11:00 AM, Rodrigo A. Alfaro wrote:
> I didn't explore glm command before, then I tried the following:
>
> sysuse auto
> g lnp=ln(price)
> glm price, family(normal) link(log) nohead nolog
> reg lnp, nohead
>
> and the coefficient (let's say mu) is different. Is there somethig  
> than I am missing? a normalization issue?

To expand on Nick's suggestion, one of the primary features of the  
GLM approach (as opposed to modeling a transformed variable) is to  
obtain predictions on the raw (i.e., untransformed) scale.  So GLM is  
absolutely an important alternative to consider if this is a  
requirement.

The reason your results are different is that you've fit two  
different models.  They are:

E[log(price)] = XB    (fit by -regress-, generating B_hat)

and

log(E[price]) = XG    (fit by -glm-)

One can show that under certain conditions, you can consistently  
estimate G by B_hat (except for the intercept), but if those  
conditions aren't met, B_hat will be estimating something different.   
Naively assuming that B_hat estimates G is a common mistake people  
make when interpreting the results of a regression on a transformed  
variable.

The documentation on -glm- in [R] is a good start, but if you're  
using this for anything important, I'd strongly suggest picking up a  
copy of Generalized Linear Models (by McCullagh and Nelder), in  
particular the chapters "An outline of generalized linear models" and  
"Models with constant coefficient of variation".


-- Phil

*
*   For searches and help try:
*   http://www.stata.com/support/faqs/res/findit.html
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/