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


From   Phil Schumm <pschumm@uchicago.edu>
To   statalist@hsphsun2.harvard.edu
Subject   st: Re: Generating predicted values for OLS with transformed dependent variables
Date   Wed, 12 Apr 2006 13:56:57 -0500

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

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