R squared for a regression plane without observations

Question

Assuming we have three random variables $X$, $Y$, and $Z$, and we want to estimate a least squares regression plane of the form $Z = a + bX + cY$. We do not know the individual observations, but we know all the means $\mu_X$,$\mu_Y$,$\mu_Z$, variances $\sigma_X^2$,$\sigma_Y^2$,$\sigma_Z^2$ covariances $\sigma_{XY}$,$\sigma_{XZ}$,$\sigma_{YZ}$ and correlations $\rho_{XY}$,$\rho_{XZ}$,$\rho_{YZ}$. Suppose we have estimated $a$, $b$, and $c$. How can we calculate the $R^2$ value given only this information?

Answer 1

By definition, $$ R ^ 2 = 1 - \frac{Var(\epsilon | X, Y)}{Var(Z)} = 1 - \frac{Var(Z - a - bX - cY|X,Y)}{\sigma^2_Z} $$ then, $$ Var(Z - a - bX - cY|X,Y) = \sigma^2_Z + b^2\sigma^2_X - c^2\sigma^2_Y - 2b\sigma_{ZX} - 2c\sigma_{ZY} + 2bc\sigma_{XY}, $$ now, just plug-in your results and reduce the expression.