Correlation does not imply causation, as there could be many explanations for the correlation. But does causation imply correlation? Intuitively, I would think that the presence of causation means there is necessarily some correlation. But my intuition has not always served me well in statistics. Does causation imply correlation?
Solved – Does causation imply correlation
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Correlation is not sufficient for causation. One can get around the Wikipedia example by imagining that those twins always cheated in their tests by having a device that gives them the answers. The twin that goes to the amusement park loses the device, hence the low grade.
A good way to get this stuff straight is to think of the structure of Bayesian network that may be generating the measured quantities, as done by Pearl in his book Causality. His basic point is to look for hidden variables. If there is a hidden variable that happens not to vary in the measured sample, then the correlation would not imply causation. Expose all hidden variables and you have causation.
The quick answer is, no. You can easily come up with non-related data that when regressed, will pass all sorts of statistical tests. Below is an old picture from Wikipedia (which, for some reason has recently been removed) that has been used to illustrate data-driven "causality".
We need more pirates to cool the planet?
For time series, there is a term called "Granger Causality" that has a very specific meaning.
http://en.wikipedia.org/wiki/Granger_causality
Other than that, "causality" is in the eye of the beholder.
Best Answer
As many of the answers above have stated, causation does not imply linear correlation. Since a lot of the correlation concepts come from fields that rely heavily on linear statistics, usually correlation is seen as equal to linear correlation. The wikipedia article is an alright source for this, I really like this image:
Look at some of the figures in the bottom row, for instance the parabola-ish shape in the 4th example. This is kind of what happens in @StasK answer (with a little bit of noise added). Y can be fully caused by X but if the numeric relationship is not linear and symmetric, you will still have a correlation of 0.
The word you are looking for is mutual information: this is sort of the general non-linear version of correlation. In that case, your statement would be true: causation implies high mutual information.