I am fitting a linear model (normal error structure) with several linear predictors. In R: lm(y ~ x1 + x2).
I need to report how much variation in y my model explains. Which of the following quantities is the best for that: Classical $R^2$, adjusted $R^2$, or crossvalidated $R^2$?
So far I struggle with seeing any use in the classical $R^2$ at all -- it is compromised by potential overfitting, and even if it's not (i.e., the fitted model is actually the "true" model), then my crossvalidated $R^2$ will still be lower than the classical $R^2$. As a consequence, any real-world predictions from the model will have higher variance than what is suggested by the classical $R^2$. Thus, what is the point of the classical $R^2$?