Regression relation to casual relationship

Question

If the correlation coefficient of two variables is 0, can there still be a causal effect between them? And can the causal relationship between these two variables be studied by regression analysis?

Answer 1

Yes, because the correlation coefficient somewhat captures only the linear dependence between two random variables. As a (famous) counter-example, take $X\sim\mathcal{N}(0,1)$ and $Y=X^2$, then $\mathrm{Cov}(X,Y) = \mathbb{E}(X^3) - \mathbb{E}(X)\mathbb{E}(X^2) = 0$, while $X,Y$ are clearly dependent variables.

To summarize, independence $\Longrightarrow$ uncorrelatedness, but the reverse statement is false. And more important to keep in mind in statistics, correlation is not causation (another well-known counter-example : "All water-drinkers die, but water does not cause death"); the correlation coefficient $\mathrm{Corr}(X,Y)$ may be seen as a "hint" of causal link between the variables $X$ and $Y$.