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I wish to test the cross-correlation of two time series to measure how "instantaneously related" my time series are. However, I am not quite sure if I should difference my time series first, and make them stationary? Because they obviously aren't:

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And the cross-correlation I get is very different if I difference my time series, or not.

Cross-correlation cff() on original non-differenced data:

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Cross-correlation cff() on differenced data with diff():

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Which one is right? Am I supposed to use cff() on stationary data only? Thank you for the help.

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The idea of how instantantaneously two series requires the forming of a useful ROBUST model. This is called a Transfer Function and sometimes a PDL model or ADynamic Regression. So my answer deals with forming a useful model.

Box and Jenkins suggested pre-filtering where the differencing operator identified as part of the ARIMA process was used as part of the filter to identify model form via the resultant cross-correlation . See https://onlinecourses.science.psu.edu/stat510/node/75 AND http://www.math.cts.nthu.edu.tw/download.php?filename=569_fe0ff1a2.pdf&dir=publish&title=Ruey+S.+Tsay-Lec1 (ignoring the discussion of the CORNER METHOD ) AND http://autobox.com/cms/images/dllupdate/TFFLOW.png . However if both X and Y are I(1) .. it is still possible that Y and X can related without any differencing at all which suggests that alternative approaches to incorporating differencing in the TF model or not be on the table.

cross-correlations between two series are DESCRIPTIVE but not INFERENTIAL ..http://finpko.faculty.ku.edu/myssi/FIN938/Yule.Spurious%20Regression.JRSS_1926.pdf . Apparently GOOGLE folks don't read the classics when they suggest cross-correlations as a way of assessing predictability en route to causation.

why don't you post your data and I will try and help you.

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