Why we prefer normal distribution of data in linear regression As I recalled, we can also assume the data points come from Laplace distribution and hence it will be the linear regression with absolute error.
Why did so many texts assume the data points came from a normal distribution then?
 A: There is a small point of clarification here:  We actually don't need the error term to be Gaussian (see the Gauss-Markov Theorem), but if we assume it is then some nice connections can be made.
Linear regression is commonly motivated by minimizing the squared error loss.  Squared error is perhaps the simplest loss function to motivate such a model with, so let's go with it for now.  It turns out that minimizing the squared error loss is the same as assuming the likelihood for $y\vert x$ is Gaussian, hence the normality of the error term.
A: The magic word that nobody seems to have mentioned in this thread is efficiency: we know that if the residuals in a linear regression model $ y = X \beta + \varepsilon $ are normally distributed then the usual OLS estimate of $ \hat \beta = (X^T X)^{-1} X^T y $ is efficient (because it's a maximum likelihood estimator).
The OLS estimator still "works" in the sense of being consistent if $ \varepsilon $ is not normally distributed as long as strict exogeneity is satisfied, i.e. as long as $ \mathbb E[\varepsilon \vert X] = 0 $, since
$$ \mathbb E[\hat \beta] = \mathbb E[(X^T X)^{-1} X^T X \beta + (X^T X)^{-1} X^T \varepsilon] = \beta + \mathbb E[ \mathbb E[(X^T X)^{-1} X^T \varepsilon \vert X]] $$
$$ = \beta + \mathbb E[ (X^T X)^{-1} X^T \mathbb E[ \varepsilon \vert X]] = \beta $$
So all you need to use OLS safely is strict exogeneity, but OLS only makes the most efficient use of the data available if $ \varepsilon $ is normally distributed.
A: The main difference with or without normally distributed errors is whether the inference procedures (standard errors, p-values, t-tests etc) hold in small or large samples.
The inference results hold:

*

*If errors are (assumed to be) normally distributed, the normal distribution for the estimates is exact, i.e. it holds even in small samples

*Without the normal assumption, using a normal or student distribution for the estimates is an approximation based on an asymptotic result

A: Linear regression, like least square regression, does not assume Gaussian distributed error terms.
The assumption of Gaussian distributed error terms relates to statistics like hypothesis tests or computations of confidence intervals. For these types of statistics, it is necessary to have more precise assumptions about the hypothetical distribution of the error. And even for these types of statistics the assumption of Gaussian distribution is not really necessary. What we need is that it is approximately Gaussian distributed (Which statistical analysis should I perform if the data sets are not normally distributed?).
