st: Dealing with exposure differences using logarithmic offsets versus rescaling

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

I am trying to test my intuition that logarithmic offsets should yield
the same estimates as dividing the outcome by exposure and using
exposure to weight.

This seems to work for count variable BI with the regular poisson or
its glm equivalent. However, when I try it with a continuous variable
GMB (measuring expenditures) using GLM with gamma family and log link,
the equivalence goes away. I am at a loss for why that is the case.
Any ideas or references would be much appreciated:

Here's my code:

gen lnt1 = ln(t1);
lab var lnt1 "Treatment Offset (ln of Days)";

gen bipd1 = bi1/t1;
gen gmbpd1 = gmb1/t1;

/* Simple Poisson Model for BI */
poisson bi1 i.treat, offset(lnt1) robust;
poisson bipm1 i.treat [iw=t1], robust;

/* Poisson Using GLM for BI */
glm bi1 i.treat, offset(lnt1) link(log) family(poisson) robust;
glm bipm1 i.treat [iw=t1], link(log) family(poisson) robust;

/* GMB using GLM */
glm gmb1 i.treat, offset(lnt1) link(log) family(gamma) robust;
glm gmbpm1 i.treat [iw=t1], link(log) family(gamma) robust;
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