How to establish and prove causality between two variables? I want to establish causality between two variables/attributes of some time series data. 
Is there any method to prove and establish the causality mathematically?
 A: I believe that "proving and establishing causality" is probably somewhat context-dependent. I am familiar with the high bar of academic economics, from when I was in grad school. In other settings, the bar may be lower because "proof" is really a matter of what is acceptable in the eye of the audience to whom you're offering that proof.
Here's my taxonomy of relationships:


*

*Two things are not related

*Correlation: simple mathematical relationship (though the thresholds of un-,weakly, and strongly correlated may be context dependent)

*Association: Beyond correlation (and in fact, regardless of the strength of correlation), some sort of statistical model has found that indeed A and B move together after controlling for some other variables. But the relationship cannot yet be called causal, and/or the direction of causality cannot be proven.

*Causality: This takes association further. There is theoretical support for the causal relationship of A causes B (and not B causes A); and there is also some kind of experimental support: a trial (ideally randomized), simulation or natural experiment, eg two states pass a law in different years and you compare the results of the law's effect on behavior during the time when only one state passed the law.

