While dealing with data, I very often find my data to be "halfway between discrete and continuous".
For example, let's say my variable of interest is age. In principle, age should be continuous, say $67.34$ years old. But one will just write $67$ on a questionnaire, essentially discretizing the value. Furthermore, my questionnaire usually has a target audience, say the senior citizen group. As a result, the collected ages will be something like $60, 61, 78, \ldots, 96$. That's right: now my supposedly continuous data now becomes a series of integer $\in[60, 96]$.
Strictly speaking, under this kind of scenario, I should not use the generalized linear model, since its $y$ models continuous values. Instead, I should use multinomial logistic regression. Then, my headache is, I have so many ($96-60+1 = 37$) categories!
My main purpose of fitting the model is to do some linear hypothesis testing, e.g., testing if $\beta_1=\beta_2$. Under this consideration, doing multinomial logistic regression causes more trouble, since sometimes the $\beta$'s are not comparable across models. On the contrary, linear hypothesis testing is very straightforward under a general linear model setting.
So what is the price I have to pay if I just go for the general linear model?
A Simpler Version
$y$ is trinomial, $0$ or $1$ or $2$. What happens if I fit a general linear model instead of a trinomial logistic regression, given my purpose of fitting is to compare $\beta$'s?