# Is the RMSE of a regression related to the standard error of its regression coefficients?

Is the root mean square error (RMSE) of a regression related to the standard error of its regression coefficients? If so, how?

Under the classical setting of the multiple regression, we know that $$E||y-\hat{y}||_2^2 = {\rm tr}\big[\sigma^2 (I - X(X^{\rm T} X)^{-1} X^{\rm T})\big] = \sigma^2 (n - {\rm rank}(X)),$$ where the last equality holds from a trace of an idempotent matrix. Also, $${\rm Var}(\hat{\beta}) = \sigma^2 (X^{\rm T} X)^{-1}.$$ From the facts above, it can be said that if covariates are more correlated, then (1) the rank of $$X$$ is small so that the RMSE gets larger, and (2) $$\hat{\beta}$$ would have more variation from multicolinearity.