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.
 A: One reason not to use Poisson regression here is that, since each employee can have at most one account, the number of accounts is bounded by the number of employees.  A Poisson distribution would allow nonzero probability for the number of accounts exceeding the number of employees.  My understanding is that although Poisson regressions are robust to a lot of violations of assumptions, you'd at least get a loss of efficiency from using a Poisson regression compared to something more appropriate.
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.
