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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?

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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.
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  • $\begingroup$ Yes I agree with you. But the thing is I just wanted to know if there is some method or measure with which I could at least tell the audience that this-this thing shows there could be a potential causal relationship between two variables $\endgroup$
    – Sushodhan Vaishampayan
    Sep 18, 2018 at 4:47
  • $\begingroup$ You’ve told us nothing about your data so how can I be more specific than that? My suggestion is that you give your audience theory that supports causality, and the strongest empirical/quantitative evidence you have (if not of your own then perhaps some other study that you can reference). $\endgroup$
    – Chris Umphlett
    Sep 18, 2018 at 9:31
  • $\begingroup$ I am not having my data as such. I just wanted to know how this can be done. For example you can take the data as percentage GDP invested in sports vs the number of medal that country wins in Olympics. So for these kind of data sets, I wanted to know is there any generic method with which I can tell my audience or give them some idea that why I think there might be a causal relationship. $\endgroup$
    – Sushodhan Vaishampayan
    Sep 18, 2018 at 10:50
  • $\begingroup$ In that example the theory is clear: a country that spends more on sports will be better at sports. There should be ample empirical data: countries that have varies their expense, and then the outcomes are known. First, is there a correlation? Easy to discover. For causality if I was in the audience I’d want to make sure you had investigated, and with good reason disregarded, the possibility that countries that do well in the Olympics then decide to raise how much they spend which could make the causality go in the opposite direction. $\endgroup$
    – Chris Umphlett
    Sep 18, 2018 at 10:56
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    $\begingroup$ Thanks for the answer. So I will assume from the above comment that there is no standard and generic method or measure in mathematics which can do the job. We can at best have a correlation measure. Thanks for the help once again. $\endgroup$
    – Sushodhan Vaishampayan
    Sep 18, 2018 at 11:12

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