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Under what conditions does correlation imply causation?
Can somebody illustrate how there can be dependence and zero covariance?

Or could there still be a relationship? Is it possible that the math just works out this way given the way OLS works but there is some relationship? Perhaps a nonlinear one? Anything?

Or does $R^2=0$ imply that X and Y are completely unrelated?

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  • $\begingroup$ There's also very closely related information here and here. Several other places on the site also discuss the difference between uncorrelated and independent but those were the first three I recall at the moment. $\endgroup$
    – Macro
    Jan 28, 2013 at 23:42
  • $\begingroup$ "Does this imply that Y and X are unrelated?" The short answer is "No"; first because you're measuring only linear relationship in the mean, and secondly, sample statistics have variation, so it doesn't necessarily imply that there's no linear relationship. It may just imply a small sample! $\endgroup$
    – Glen_b
    Jan 28, 2013 at 23:46
  • $\begingroup$ And here. And here. $\endgroup$
    – StasK
    Jan 29, 2013 at 0:21

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No, it doesn't mean that they are unrelated. $R^2$ only measures linear relationships. For example, if $X$ is symmetric around zero and $Y = X^2 + \epsilon$, where $\epsilon$ is some error independent of $X$, then the $R^2$ for the regression of $Y$ on $X$ will be zero--even if $\epsilon \equiv 0$, so that there is a perfect deterministic relationship between $X$ and $Y$.

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One counter-example is if $X$ and $Y$ are phase-offset. The following works in R:

t <- seq(0,2*pi,0.1)
x <- sin(t)
y <- cos(t)
summary(lm(y ~ x))$r.squared
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