Suppose the $R^2$ value of $x_1$ with $y$ is equal to $0.15$. The $R^2$ value of $x_2$ with $y$ is equal to $0.35$. Can there exist constants $a,b$ such that the variable $X=ax_1+bx_2$ has an $R^2&gt;0.5$ with $y$? I'm thinking no, but I can't find anything on the subject online to verify this. I know in some cases this obviously wouldn't be possible (such as if the sum of the r-squared values of the input variables exceed 1). But can it ever be true? Is there a rule related to this subject?
A very basic question concerning the $R^2$ of OLS regressions   run OLS regression y ~ x1, we have an $R^2$, say 0.3 run OLS regression y ~ x2, we have another $R^2$, say 0.4 now we run a regression y ~ x1 + x2, what value can this regression's R squared be?   I think it's clear the $R^2$ for the multiple regression should be no less than 0.4, but is it possible for it to be more than 0.7?
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