I've been trying to read up on Poisson regression models, and it looks like it is possible to estimate such a model with a binary outcome. This has come up before on this site here (and somewhat here and there). I am still a bit confused about how to interpret the coefficients when the outcome is binary, and how to specify an offset to facilitate this interpretation. Let's assume that $E[y|x]=\exp(a+\beta x + \gamma d)$, where $x$ is continuous, and $d$ and $y$ are binary. Let's say my random sample is $N=10,000$ and $y=1$ happens 10% of the time and $d=1$ 25% of the time.
- I believe (but would like to verify) that for small values of $\beta$, I can interpret it as an elasticity, so if $\hat \beta=.05$, that's a ~5% increase in $\Pr (y)=1$ for an additional unit of $x$. When $\beta$ is larger, it is more accurate to exponentiate it, so if $\hat \beta=.5$, that corresponds to a 65% increase in $\Pr (y=1)$. With binary $d$ going from 0 to 1, the marginal effect is $\exp(\hat \gamma)-1,$ so for $\hat \gamma=0.3$, we would get a 35% increase.
- Can I interpret $\exp(\alpha)$ as the baseline probability for those with $d=0$ and $x=0$?
- If the outcome $y$ was actual counts or even continuous, I can just change the end to read "increase in $y$" rather than "increase in $\Pr (y=1)$".