R², what it measures and how it misleads

Question

Define R^2, explain its mechanical properties (what happens as you add regressors, and its relationship to correlation), and discuss why a return-forecasting model with R^2 = 0.005 can be extremely valuable while a model with R^2 = 0.95 can be worthless.

Accepted Answer

Definition. For a fitted regression, R^2 = 1 - \frac{	ext{SSR}}{	ext{SST}} = 1 - \frac{\sumi (yi - \hat{y}i)^2}{\sumi (yi - \bar{y})^2}, the fraction of the variance of y explained by the model, in-sample. In simple regression, R^2 = \hat{ho}{xy}^2, the squared sample correlation; with multiple regressors it is the squared correlation between y and \hat{y}. Mechanical properties. - Adding any regressor never decreases in-sample R^2, even pure noise weakly increases it, since OLS can always set its coefficient to exploit chance correlation. Hence adjusted R^2, which charges a penalty: \bar…

R², what it measures and how it misleads

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