About simple linear regression If we assume a vector $y$ which has normal distribution with mean m and covariance matrix M.
A simple linear regression model is written as $y = \beta_0 + X\beta + e$.
If the mean of y is non zero and has to be estimated, so what can be its estimated value?
In other terms, is there a relationship between the mean vector and the estimated regression coefficients?
Furthermore, what is the $\beta_0$ ? why they consider in most cases that it is equal to 0?
Any help will be very appreciated!
 A: Let me address your questions in inverse order.
Firstly, in most sources I have encountered so far the task is more conveniently reformulated as $y = X^*\beta^* + \epsilon$, where $X^*$ is just $X$ with a column vector of ones appended to the right and $\beta^*$ is your $\beta$ with $\beta_0$ appended to it. In such way, you essentially consider the same model but the explicit intercept term $\beta_0$ now has disappeared. Maybe that's what you took as $\beta_0=0$?
Secondly, the most straightforward and classic way to estimate the explanatory variables $\beta$ is to use the Least Squares method. The solution is then given in terms of the right Moore - Penrose pseudoinverse:
$$
\beta^*_{LS}=(X^{*T}X^*)^{-1}X^{*T}y
$$
As you can see, the sample mean is not explicitly present here.
However, I may be missing something obvious in your question so please correct me if I'm wrong.
EDIT: After reading your question again I've noticed that in your original formulation the errors $e$ are not uncorrelated but rather have a covariance matrix $M$. So, to stay formal I would like to notice that this case can actually be reduced to the non-corellated case by a linear change of variables: write your errors as $e=\sqrt{M}e^*$, where $\sqrt{M}$ is, for example, the Cholesky square root of $M$ and $e^* \sim N(0, I)$, and multiply by $(\sqrt{M})^{-1}$.
