# Testing first differenced time series - trend or not?

I have a question on testing for stationarity with the ADF test, in order to determine the order of integration.

We do have a time series that looks in plot(df) obviously trending at level and has a high fluctuation ("drift+trend"). We therefore assume, that we have to apply following script from vars to get an appropriate ADF test result:

ur.df(df, type = "trend", selectlags = "BIC")


Assume the test finds that the time series is not stationary (,as the plot already suggested). We therefore proceed testing the first difference of the time series:

ur.df(diff(df), type = "trend", selectlags = "BIC")


Is it okay to consider "trend" in the test, when we know that diff(df) looks stationary in plot(diff(df)?

• The ADF test like all tests have built-in assumptions in order for them to be valid. The big print giveth the small print taketh away! . Find out what these assumptions are . Make sure your data meets the assumptions e.g. no outliers ,no seasonal pulses , no step/level shifts (intercept changes) , no need for transformations like logs , no need to be worried about deterministic change points in error variance etc. – IrishStat Sep 6 '17 at 14:22
• Check out earlier questions tagged with the augmented-dickey-fuller tag to see if your question might have been answered before. – Richard Hardy Sep 6 '17 at 14:24
• I discovered this answer here, though it was not confirmed by anyone else: Link – DanielOY Sep 6 '17 at 15:18
• From that argumentation I would conclude (and Update my post, if you can confirm): The time series has to be visually inspected after first differencing. If it shows no trend, the right model should be: ur.df(diff(df), type = "drift", selectlags = "BIC"). Correct? – DanielOY Sep 6 '17 at 15:20
• @RichardHardy I looked up some of your posts and others on the topic but remain unclear. The time series we're talking about looks very trending and is actually found to be "stationary" when tested with: ur.df(df, type = "trend", selectlags = "BIC"). From that view, would we conclude I(0) or I(1)? It's so hard to believe that I conclude this time series to be I(0). I don't get entirely the discussions on "trend stationarity" and what it really means for my models (i.e. is the problem solved by including "trend" into every of my AR/VAR/VECM models, while proceeding with variables at level? – DanielOY Sep 6 '17 at 16:12