I happen to do a lot of linear regression, the typical use case is predicting $Y$ with a few predictors say $X = [X_1;X_2;X_3]$. So we we find $\hat{\beta}$ such that $$\hat{\beta} = \underset{\beta}{argmin} ||Y-X.\beta||^2$$ But actually what I am more interested in in finding the linear estimation of $Y$ such that $$corr(Y,\hat{Y}) = corr(Y,X.\hat{\beta})$$ is maximal.
I am bothered by the fact that for a given $\beta_0$, $corr(Y,X.a.\beta_0)$ is the same for all $a$, questions are then
Is the optimal linear estimator of $\beta$ in the least square sense also optimal in my correlation term, and if yes should I be bothered that the optimal $\beta$ is not unique ?
