Re: st: Interpretation of interaction term in log linear (non linear) model

["Dimitriy V. Masterov" <dvmaster@gmail.com>]

I would do this in a glm/robust poisson setting to avoid the logging.
See http://blog.stata.com/2011/08/22/use-poisson-rather-than-regress-tell-a-friend/
(especially Austin Nichols' talk cited there), and the references at
http://davegiles.blogspot.com/2011/03/dummies-for-dummies.html, for
some good reasons. Economists do love their logs, but there are better
methods that are starting to catch on with us.

Since I don't have your data, let's take a toy model with the cars.
This is recommended practice on this list that makes it more likely
that you get a good response to your questions. Suppose we ran this
regression:

sysuse auto
poisson price c.mpg##i.foreign, vce(robust)

The expected value of price conditional on continuous mpg, binary
foreign, and their interaction is

E[price|mpg,foreign]=exp{a + b*mpg + c*foreign + d*mpg*foreign}

The derivative of expected price with respect to mpg is
exp{...}(b+d*foreign) using the chain rule. To turn that into a
semielasticity, we need to multiply by 1/(expected price), which is
conveniently exp{...}. We can evaluate both exp{}s at the actual
values of the covariates or representative values like the mean,
median, or mode, but in either case they cancel, leaving you with
(b+d*foreign). I can also evaluate this semielasticity for domestic
cars, which leaves me with (b+d*0)=b. The difference between the two
semielasticities is d, which you can get from the regression table.
This gives you the extra percentage increase in price for an
additional mpg for foreign cars. In this case, it's ~17%.

Here are some examples of how to do this in Stata. For some reason I
could not replicate dydx1 and dydx0 using margins, but the rest seems
to works.

This intuition carries through in the logged y OLS case as well. I
used Chris Baum's levpredict below since E[y|x] != exp{x'b}, but
that's not really necessary.

****************************
sysuse auto, clear

/* Easy Way: Just look at the table */
/* note how glm and poisson are similar */
glm price c.mpg##i.foreign, link(log) family(poisson)
poisson price c.mpg##i.foreign, vce(robust)

/* A Bit Harder Using Margins */
margins                             // this is yhat below
margins, eydx(mpg) at(foreign==1)   // This is eps1 below
margins, eydx(mpg) at(foreign==0)   // This is eps0 below
margins r.foreign, eydx(mpg)        // This is diff_eps


/* Pedagocial, Manual Way */
/*coeflegend allows you to see coeff. names */
poisson price c.mpg##i.foreign, vce(robust) coeflegend

gen double yhat  = exp(_b[_cons] + _b[mpg]*mpg + _b[1.foreign]*foreign
+ _b[1.foreign#c.mpg]*mpg*foreign)
gen double dydx1 = yhat * (_b[mpg] + _b[1.foreign#c.mpg]*1)
gen double eps1  = dydx1 * (1/yhat)
gen double dydx0 = yhat * (_b[mpg])
gen double eps0  = dydx0 * (1/yhat)
gen double diff_eps = eps1 - eps0

sum yhat eps* diff_eps

/* Logged Price */
drop yhat dydx* eps* diff_eps

gen lnp=ln(price)
reg lnp c.mpg##i.foreign, vce(robust)
margins r.foreign, dydx(mpg)

levpredict yhat, duan
gen double dydx1 = yhat * (_b[mpg] + _b[1.foreign#c.mpg]*1)
gen double eps1  = dydx1 * (1/yhat)
gen double dydx0 = yhat * (_b[mpg])
gen double eps0  = dydx0 * (1/yhat)
gen double diff_eps = eps1 - eps0

sum diff_eps
****************************
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