Normality of residuals vs. normality of unobserved error in linear regression I frequently encounter people saying that normality is assumed in the "errors" in linear regression. It seems what they mean is "residuals" rather than errors. For example, for MLE, we can arrive at the same result as OLS if we assume the residuals are normally distributed zero mean.
I understand that to arrive at the OLS closed form solution, no assumptions about the distribution of the residuals are needed. However, sometimes we make assumptions about the probabilistic distribution of the residuals for statistical inference, e.g., building confidence intervals on our estimators.
Now my question is, when do we typically assume normality in the unobserved errors, and what insight can we gain from this assumption?
 A: Inference about a coefficient value in simple linear regression is based on the ratio of the coefficient estimate $\hat\beta$ to the standard error of the estimate, $s_{\hat\beta}$. See for example this Wikipedia page on simple linear regression with one predictor variable. That ratio is the statistic on which a test against a distribution is made.
The classic normality assumption for inference in linear regression is about the errors, not the residuals. As the Wikipedia page explains, that assumption about the errors* leads to defined distributions of both the numerator and the denominator in that ratio.
Under the assumption of a normal error distribution, $\hat\beta$ has a normal distribution with variance related to the underlying error variance $\sigma^2$. As the residuals result from a normal distribution of errors, the sum of squared residuals used to determine $s_{\hat\beta}$ is "distributed proportionally to $\chi^2$ with $n − 2$ degrees of freedom" for a simple linear regression that estimates an intercept and one slope.
The numerator and the denominator are then independently distributed with their ratio taking a t distribution. So the classic normality assumption is used not directly in the test procedure for linear regression but in the derivation of the t-test.
If the number of observations is large then the law of large numbers and the central limit theorem hold. Then the ratio $\hat\beta/s_{\hat\beta}$, as a good approximation, does behave as a normal distribution. But this is an assumption about the distribution of the test statistic, not about the underlying errors or residuals (except that the errors have finite mean and variance).
This latter case is one example an asymptotic test that holds in the limit of large numbers of observations. I don't know if much can be said about what "insight can we gain from this [normality] assumption" about "unobserved errors," but very often if you see a statistic tested against a normal distribution you will be seeing the application of an asymptotic test.

*People do check the distribution of the residuals to see whether the assumption of normally distributed errors is reasonable, but as this discussion points out such checking can be of limited usefulness.
