Help me prove CLT $\implies$ error term in OLS regression is distributed normally How can I prove that the central limit theorem implies that error term in an OLS regression is normally distributed? (A statement to this effect has been mentioned in Econometrics p. 33 by Fumio Hayashi (2000)).
 A: The linear regression model may be written as
$$
y_i=x_i'\beta+u_i
$$
The reference says

In many applications, the error term consists of many miscellaneous
  factors not captured by the regressors. The Central Limit Theorem
  suggests that the error term has a normal distribution.

So, Hayashi asks why $u_i$ ought to be assumed to be normally distributed. 
In my understanding, the logic goes as follows:
The error term $u_i$ captures all influences on $y_i$ that are not already accounted for by the regressors $x_i$. Now, you could think of a great many influences that have not been accounted for. Ideally, the important influences have already been included via the regressors, so that $u_i$ captures "unimportant" factors. The CLT roughly says that a suitably scaled average of many small factors behaves like a normal random variables. So, $u_i$, collecting these factors, would be normally distributed.
Personally, I do not find this logic to be too appealing, though. First, we do not need this normality assumption on the errors once we (as does Hayashi later on, see also the second quotation below) resort to asymptotic approximations (as $n\to\infty$), which is pretty much all we can do once we leave this fairly restrictive setup of the linear model anyhow.
Second, I believe the error term $u_i$ is just the error associated with person/observation $i$, and there is exactly one error for that observation. To me, it is quite a stretch of the imagination to interpret that single error as the average of infinitely many "sub-errors" that would justify a CLT. (Letting $n\to\infty$ is far more plausible to me, as we can conceivably collect more data, though.)
Third (also see below), it is unclear why that logic should be applicable in the practically very relevant case of omitted variables (i.e., important factors being left in the error term).
Indeed, in Chapter 2, Hayashi writes

However, not very often in economics are the assumptions of the exact
  distribution satisfied. The finite-sample theory breaks down if one of
  the following three assumptions is violated: (1) the exogeneity of
  regressors, (2) the normality of the error term (my emphasis), and (3) the linearity
  of the regression equation.

A: There is no such thing as proof that the error term in OLS regression is normally distributed as it's an assumption. It just needs to be stated when we want to test for statistical significance of the $\beta$'s. In fact, the Gauss-Markov theorem doesn't even mention normality of the error term.
What we can do, however, is to conduct model diagnostics (likea q-q plot) in order to verify that such assumption holds.
