# Assumptions behind simple linear regression model [duplicate]

Let's say we have a simple linear regression model, that is, $y = X\beta + r$ where $y$ is a vector of size n x 1, $X$ a matrix of size n x p, $\beta$ the regression coefficient vector of size p x 1 and $r$ is a residual error of size n x 1.

Most of papers I read consider that $y$ follows a multivariate normal distribution. My question is whether $y$ can follow a distribution different from normal, for example elliptical, etc. Is it necessary that in linear regression, for $y$ to be assumed Gaussian?

I discussed with a researcher and he told me that linear regression models are only suitable (or efficient) for Gaussian assumption. Is this true?

• – Tim Jun 30 '15 at 13:53
• If your errors are iid normal, then yes, linear regression will give efficient estimates. But it's not necessary to have normality to use ordinary least squares linear regression. – Glen_b -Reinstate Monica Jul 5 '15 at 9:05
• Thank you so much for your answer, so OLS is efficient and robust? so OLS can be considered distribution free? – Christina Jul 5 '15 at 9:36

When error-terms are non-normal, you still get many nice properties. The biggest of which is the Gauss-Markov theorem that tells you that the Ordinary Least Square (OLS) estimator, $\hat{\beta}_{OLS} = (X^TX)^{-1}X^TY$, is the best linear unbiased estimator (BLUE).
However, it is only due to the Normality of the error terms that $\hat{\beta}_{OLS}$ is also the Maximum Likelihood Estimator (MLE). That means, you cannot use the asymptotic properties of the MLE. In layman's terms, this means that you cannot use p-values and confidence intervals for $\hat{\beta}_{OLS}$ when the error terms are grossly non-Normal.
In summary, if the error terms are some non-Normal distribution you can use the point estimates for $\hat{\beta}_{OLS}$ but cannot interpret the p-values and confidence intervals that come from MLE theory.