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If cross correlation peaks at a very high value at positive lag, it does not imply process in time series A causes process time series B. They can both be caused by an unobserved process C with different propagation rate, for example.

Then what can we draw from the fact that cross correlation is high at some lag?

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In this case the framework to investigate is Granger Causality, which seeks to evaluate to what extent one variable has predictive power about another if measured at some earlier time point. As you noted, this is a very soft notion of causality, and perhaps not one which matches an intuitive notion of cause and effect. ( http://www.scholarpedia.org/article/Granger_causality).

A stronger definition relies on some notion of intervention, whereby deliberately modulating the predictive variable gives us some notion about its effect on the predicted variable. This is philosophically contentious, but might be of interest nonetheless ( http://plato.stanford.edu/entries/causation-mani/#Inte).

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