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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 $R^2$?

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classical R^2 describes how well your model sketches your sample, while cross validated R^2 says more about the population you assume to probe by your sample.

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    $\begingroup$ I know, but why should I, or any of my readers, or anyone really, care about how my model sketches my sample? Isn't it always the population that we are interested in? And if yes, why calculating the classical R^2 at all? $\endgroup$ – Petr Keil Jun 29 '18 at 12:13
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    $\begingroup$ well, sometimes your "sample" is the "population" itself, meaning that is all you care. Using CV R^2 is to prevent overfitting, but in this case there is no over fitting, your model in this case is not for statistical inference but a reduction, i.e. I described my sample/population of 20 points with 4 parameters. then the difference between classical R^2 and CV R^2 is because your train and test sets are from different parts of your sample, i.e. incomplete and distorted. $\endgroup$ – Xiaoxiong Lin Jun 29 '18 at 12:26

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