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Why would we choose Poisson / NB regression (GLM) over OLS for fitting count data?

Is there a way to show that OLS estimator would lost it consistency and asymptotic normality for count data?

I'm asking this because my sample size ($n$) is extremely large, and I can always increase it when needed (so that the asymptotic behavior of OLS estimator with count data is all that I care about).

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Why would we choose Poisson / NB regression (GLM) over OLS for fitting count data?

This can be more to do with modelling the relationship between the variance and the mean. Recall that in Poisson regression, the variance is equal to the mean. The negative binomial relaxes this slightly by allowing the variance to be quadratic in the mean.

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