Fractional dependent variable: Why not use Poisson regression?

In many settings, we are interested in estimating a model with a fractional dependent variable. For example, Papke & Wooldridge (1996) http://faculty.smu.edu/millimet/classes/eco6375/papers/papke%20wooldridge%201996.pdf consider 401(k) plan participation rates, where the rate is defined as $PRATE=\frac{accounts}{emplyees}$. The authors then develop a GLM method to estimate such models. Looking at the count data literature, I wonder one should not run a Poisson regression of $accounts$ on the same set of regressors, and as an offset $employees$. Does this potentially depend on the absolute number of $accounts$?

This is different from a suggested duplicate, What regression model is the most appropriate to use with count data? as my question discusses the correct place of the offset / denominator.

• ... as an offset log(employee) ;-) (if used log-link)! imho ... you've got the same results, but what (in what scale...) you wont (prefere) to interprete it? - just a matter of taste... Commented Jun 29, 2016 at 10:22
• Possible duplicate of What regression model is the most appropriate to use with count data? Commented Oct 11, 2017 at 22:48
• I don't think so. I am asking about count data with a very clear offest / exposure variable and when to model something as rate or count. Commented Oct 13, 2017 at 11:29
• You must use log(employees) as offset. Can you give more details of your application? A very detailed discussion of the How/Why of offset is in stats.stackexchange.com/questions/142338/…, you could also look at stats.stackexchange.com/questions/307369/… (Both are better duplicated than the one proposed above) Commented Oct 13, 2017 at 11:36

The question then should be: wouldn't a binomial regression be more appropriate? (Assuming the same participation rate $p$ for each employee, the number of plans $y$ should be distributed as $Binomial(n,p)$ where $n$ is the number of employees.) IIRC, the reason a binomial regression can't be employed in this case is that the number of employees is not known; only the participation rate itself is known. That rules out binomial regression---and would also rule out Poisson regression with an offset, even if it were appropriate.
• Binomial. An offset doesn't do anything to keep the distribution bounded above; the number of observations cannot, in principle, come from a Poisson distribution. On the other hand, if each employee can have zero or one accounts, and the probability $p$ of having an account is the same for each employee in a group of $n$ employees, the total number of accounts is literally distributed as Binomial(n,p). Commented Oct 19, 2017 at 23:59