Poisson regression can get one so far with so little trouble, why do so many still resist? Especially with panel data. It’s too bad we can’t give it another name to reflect the fact that its a fully robust estimator of conditional mean parameters.
And if anyone makes the uninformed statement that “ the Poisson distributional assumption is too strong” ask them if they ever reject the use of OLS because “normality, homoskedasticity, and serial independence are too strong.” After all, OLS is MLE under those assumptions.
While I talk about this in Chapter 18 of my MIT Press book, the detailed analysis is in my 1999 Journal of Econometrics paper, "Distribution-Free Estimation of Some Nonlinear Panel Data Models." Any kind of y is allowed provided the mean function is correct.

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