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$\begingroup$ @marlon, indeed, orthogonality always holds: For reg <- lm(y~0+x) we still have, by the OLS properties whuber mentions, that crossprod(resid(reg),x) $\approx0$. But, since the mean of the residuals is not zero here (it must be zero when we have a constant, as $\text{cst}'\hat u=0$ is the orthogonality condition for the constant, so that the sum is zero, and hence also the mean). Hence cor(resid(reg),x) is not zero. $\endgroup$
– Christoph Hanck
Commented Jan 12 at 13:43

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