OLS assumption normallity of error term really needed? As the title explains I was wondering whether the additional OLS assumption of having a normally distributed error term isn't redundant if the sample is large enough. I understand that we want the conditional normality of the error term so that our estimator is normally distributed and further, so that one is able to conduct standard inference. However, as we replace the expectations by sample averages (in the estimation process), shouldn't a central limit theorem ensure the normality regardless of how the error terms are distributed as long as the sample is large enough?
Thanks in advance
 A: Normality is actually quite important.  Not in the sense that it must be true, because it never is true, but in the sense that with gross non-normality you should not use OLS, despite asymptotically correct inferences. For example, with grossly outlier-prone processes (substitute "rare, extreme value" for "outlier" to disentangle it from "incorrectly entered data value"), OLS is grossly inefficient compared to likelihood-based methods that model the conditional distributions more accurately. For another example, with highly discrete distributions, Poisson, multinomial logit, ordinal regression, logistic regression, etc., are more appropriate.
When viewing regression as a model for the conditional distribution of $Y$ given $X=x$, which is essential for predicting individual $Y$ values, for scientific integrity of the model, as well as for efficiency of estimates, it is clearly important to try to model that distribution reasonably well. Normality provides a reasonable approximation in many cases, but one should always consider alternatives.
A: You speak about linear regression I suppose. Linear regression can be justified under different set of assumptions, more or less general.

... shouldn't a central limit theorem ensure the normality regardless of
how the error terms are distributed as long as the sample is large
enough?

Asymptotically yes, even if some moment conditions are needed and independence of observations is needed as well.
Sometimes people talk about Normal linear regression, underscoring the fact that the data are normally distributed. Consider that small sample properties can be useful sometimes; in those cases normality assumption is needed. Moreover, sometimes the framework used, like ML, demand precise distributional assumptions.
However the idea to not impose normality and refers to the TLC is quite common (asymptotic theory).
