"Correlation does not imply causation" is one of the chestnuts of any forum thread. But all correlation is not the same. A recent study on additive noise model says it can distinguish cause from effect from strictly observational data, blog 
, paper
In the absence of controlled trials, statisticians have widely assumed that it is impossible to determine cause and effect from an observed correlation alone.
[but] Just such a test exists and known as the additional noise model. This assumes that each dataset is made up of the relevant data as well as various sources of noise. Statisticians have shown that the nonlinearity of this process can allow them to determine the direction of cause-and-effect.
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The paper addresses simple cases involving one variable however many cases have multiple variables hence the limits of this approach are evident.
In this case the noise is the signal. See Weak Signal