Why we cannot use linear regression with count data? We know the counting data can use the Poisson regression or NB regression model, we also know the counting data will violate the normal distribution hypothesis when using Linear regression.
I would like to ask how to prove that the linear regression estimation beta will lead to an estimation error when using count data.
If we still use linear regression models with count data, how to prove that the MSE of the linear regression model will be larger than the MSE of Poisson regression?
 A: Estimating the mean of a normal distribution and the mean of a Poisson distribution are conceptually different. When you have a normal response distribution, you are basically sliding a bell curve up and down the regression line. When you have a Poisson response distribution, the shape changes. You no longer slide some curve up and down the regression line. Remember that, while you are predicting what turns out to be the mean of the Poisson distribution, that parameter you estimate has more control over the distribution than $\mu$ has over a normal variable. (Normal's $\mu$ only changes mean, while Poisson's $\lambda$ changes the mean, variance, and (probably) higher moments, too.)
While we like low MSEs, the reason we use a Poisson or negative binomial GLM is a bit deeper than that. For a Poisson-distributed response, normal simply isn't the correct distribution of our response variable when it is conditioned on our predictors. Inference on the regression parameters ceases to be valid.
