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I have two time series that are both non-stationary at level. The ADF test says they have a unit root. When taking the first difference of each time series, they are now stationary. I guess this is denoted as "I(1)".

Now I test for cointegration (Engle-Granger two-step method) of the time series and apply the Granger causality test as provided in statsmodels.

Questions:

  1. Do I apply the cointegration test on the first-difference values? Do I test with the AIC-selected maxlag or without?
  2. When I see that the first-difference time series are cointegrated (both have $p<0.05$), then can I simply apply the statsmodels Granger causality test?
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  • $\begingroup$ Check out the excellent Dave Giles blog posts on Granger causality, cointegration and especially on Granger causality in case of cointegration here and explore the links to other related blog posts in the first paragraph, and here. $\endgroup$
    – Richard Hardy
    Commented Aug 1, 2015 at 17:24

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Question 1: no, you apply the cointegration test on the original series (that you believe to be non-stationnary I(1) ). Regarding selection of lag, one would usually base the choice on a criterion like AIC indeed.

Question 2: again, you would run the cointegration test on the original series. Note that Granger causality does not imply/require cointegration: 2 series can have Granger causality among themselves if they are both I(0), both I(1), or cointegrated.

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    $\begingroup$ You say Granger causality is independent of cointegration. Not quite. Dave Giles notes in his popular blog post "Testing for Granger Causality": If two or more time-series are cointegrated, then there must be Granger causality between them - either one-way or in both directions. However, the converse is not true. So, if your data are cointegrated but you don't find any evidence of causality, you have a conflict in your results. See the post for details. $\endgroup$
    – Richard Hardy
    Commented Apr 30, 2016 at 16:37
  • $\begingroup$ Fair point, thanks for pointing this out! I've changed from "is independent of cointegration" to "does not imply cointegration" $\endgroup$
    – Matifou
    Commented May 1, 2016 at 19:28
  • $\begingroup$ When only one series has a unit root, do I still compute the first difference of both series or only for the one with the unit root? $\endgroup$
    – Mic92
    Commented Aug 4, 2016 at 20:30

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