# Poisson regression for binary data

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.

1. 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.
2. Can I interpret $\exp(\alpha)$ as the baseline probability for those with $d=0$ and $x=0$?
3. 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 $\Pr (y=1)$".
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Why would you want to use Poisson regression for a dichotomous dependent variable? –  Peter Flom Oct 3 '12 at 17:15
I am also estimating some related count data models, and it would be convenient to have the same underlying model for comparison purposes. Binary also seems like a special case of the count. –  Dimitriy V. Masterov Oct 3 '12 at 17:59
Poisson regression is not such a great idea for binary data, and is a very bad idea if the binary data is not numerical (as is almost always the case, e.g. "Did you vote for candidate $x$?"). Not to mention, under the model, non-zero mass would be given to points other than 0 and 1. Echoing @PeterFlom - why would you want to do this? Is something wrong with using standard techniques for binary data (e.g. logistic regression)? –  Macro Jan 17 '13 at 14:08
I was trying this in the context of an instrumental variable estimator as an alternative to using a maximum-likelihood two-equation probit model, which was performing poorly. Using the Poisson GMM-IV gave me very reasonable estimates. My attempt was inspired a bit by the two papers I mentioned in stats.stackexchange.com/a/38588/7071, which use the Poisson with a continuous outcome. I hoped the P/QMLE intuition would carry over to the binary case, but I have not worked that one out. –  Dimitriy V. Masterov Jan 17 '13 at 18:31
You can use poisson regression with a continuous $Y$ variable because poisson regression provides consistent estimators as long as $E(Y_i|X_i)=exp(X_i\beta)$ --- an amazing, useful, and much-overlooked fact. However, you can't do this for binary $Y$ in general. The reason is the same reason you can't use OLS for binary $Y$ in general. There is nothing constraining $exp(X_i\beta)$ to be between 0 and 1. And, since $E(Y)=P(Y=1)$ for binary $Y$, we must constrain $E(Y)$ to be between 0 and 1. For binary $Y$, the crucial assumption legitimizing poisson regression is false, in general. –  Bill May 9 '13 at 16:21