Suppose we have the linear relationship
$$ y= \beta _1 X_1+\beta _2 X_2+\beta _3 X_3 + \cdots + \beta _n X_n $$
and we have an estimate of $y$ given by
$$ \hat{y}= \hat{\beta }_1 X_1+\hat{\beta }_2 X_2+\hat{\beta }_3 X_3 + \cdots + \hat{\beta }_n X_n $$
where $X_1 ... X_n$ are all standard normal distributions.
Can we derive exact formulas for the mean squared error of $\hat{y}$ and for $R^2$, and is there an exact relationship between the two?