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I am interested in a method to test causality in the following setup: time series are measured from X1,...,Xn and I want to quantify with a single number the degree to wich this gives information about time series Y1,....,Ym. Do there exist methods to test causality with multiple variables in this sense?

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Welcome to Cross Validated!

I think you're asking for two things, but more information on the context of your problem might lead to better answers.

  1. Is there one number that can quantify the degree of association between a multivariate $\mathbf{X} = (X_1, \ldots, X_n)$ and $\mathbf{Y} = (Y_1, \ldots, Y_m)$.

  2. Does $\mathbf{X}$ "cause" $\mathbf{Y}$?

For (1), can think of the following: You could look at the percentage of explained variance in $\mathbf{Y}$ by $\mathbf{X}$ as a way to quantify the strength of dependence. You can look at the $R^2$ value of the regression $\mathbf{Y} \sim \mathbf{X}$ as a way to quantify this dependence. You could do a simple linear regression.

For (2), you would have to formulate your question more precisely and state the hypothesis that you want to test. Do you want to see if changing at least one (or some subset) of the $X_i$ affects $Y_j$? Or, manipulating all $X_i$ simultaneously affects all $Y_j$ simultaneously?

Do note that inferring causality purely from observational data can be tricky! It involves (untestable) assumptions, or domain knowledge to make the jump from correlations to establishing a causal relationship.

A good thorough book on this subject is Judea Pearl's Causality.

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  • $\begingroup$ both times the second $\endgroup$ – Wouter Devos Sep 27 '18 at 22:32
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    $\begingroup$ Sorry, I didn't get you. Both times the second? $\endgroup$ – Vimal Sep 28 '18 at 16:21
  • $\begingroup$ as an answer to the questions you asked ("for (1)..." and "for (2)..") In both questions I have the second of the two possibilities that you present in mind $\endgroup$ – Wouter Devos Oct 3 '18 at 21:17
  • $\begingroup$ For (2): to check whether manipulating all $X_i$ affects $Y_j$ (extrapolating it to causality might be justifiable depending on your domain), you can formulate the problem as a hypothesis test: You could start simple by setting up a linear regression between $\mathbb{Y}$ and $\mathbb{X}$ and test whether every coefficient in the linear regression is non-zero. $\endgroup$ – Vimal Oct 3 '18 at 22:33

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