Regression model for integer response Let the response be $Y_i \in \mathbb{Z}$ and the covariate $X_i \in \mathbb{R}^p$. For counting data where $Y_i$ are restricted to be nonnegative, we have Poisson regression or negative binomial regression. Now, I consider the situation where the response is a kind of integral score which can range from negative to positive integers, and that we don't know the bounds for the response. For example, the number of games that a person won in the past ten years, which can be negative if he/she lost more than won.
Is it still reasonable to use linear regression with normal error? If not, any alternatives should I be looking for? Thanks in advance.

As an alternative we would try using a Skellam distribution as our response distribution. (i.e. our outcome variable is the difference between two Poisson-distributed random variables). It is sometimes used to predict football/rugby goal difference. Do note though that maybe more straightforward to just have to Poisson models and compute their difference. SE.SO has a relevant thread on: How can I fit a Skellam regression?.
