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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.

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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.

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  • $\begingroup$ Thank you for your answer! However, what if we knew the number of employees and each employee could only have zero or one accounts? $\endgroup$
    – Felix H
    Commented Oct 13, 2017 at 16:03
  • $\begingroup$ That's the binomial regression case. $\endgroup$ Commented Oct 13, 2017 at 18:56
  • $\begingroup$ Sure, but then what should be preferable? Binomial or count with some offset? $\endgroup$
    – Felix H
    Commented Oct 19, 2017 at 12:37
  • $\begingroup$ 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). $\endgroup$ Commented Oct 19, 2017 at 23:59

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