Why are rate ratios different in negative binomial and poisson regressions with one categorical covariate when using the exposure option?

[James Pike]

A simple example of the thing I'm confused over:

clear all
sysuse auto
generate exposure = runiform()

nbreg rep78 i.foreign
poisson rep78 i.foreign //Match

nbreg rep78 i.foreign, exposure(exposure)
poisson rep78 i.foreign, exposure(exposure) //No Match

When introducing the exposure term the coefficients change (which I expect), and they no longer match in the two types of regression. I thought that exposure was handled the same way in both settings so I'm a bit confused over it.

So, why are rate ratios different in negative binomial and poisson regressions with one categorical covariate when using the exposure option, when they match in the case where you don't use exposure?

[George Ford]

I think there's a different variance structure between the two with exposure.

[Leonardo Guizzetti]

Why do you assume they should yield the same results? The NB model assumes additional variance than the (standard) Poisson model does, and that can allow the point estimates to vary. They would only be the same if the alpha parameter in the NB reg model were exactly equal to zero, as is functionally observed in your first example.

Edit to add: the NB model can be viewed as a mixture of Poisson random variables. The exposure information adds further variability than can then allow for a non-zero estimate of the dispersion parameter, alpha.