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I have a question regarding testing for cointegration.

The situation is a follows: I have two time series that are both stationary at level when I consider "constant and trend". They are otherwise not stationary. The first differences are all stationary.

With regard to Cointegration testing: Do we test cointegration (Johansen) at level or at first differences? If we test with the level data, is it okay to model with a trend?

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Use levels (not first differences) as an input for the Johansen cointegration test.
If you suspect a deterministic trend in the time series, you may account for that using a specific version of the Johansen cointegration test.
For example, in "urca" package in R, the function for the Johansen cointegration test is ca.jo. You may choose to specify the presence/absence of a deterministic time trend via the argument ecdet. There are more options controlled by other arguments, see here.

However, it seems you do not need to use the Johansen test in the first place. You indicate that the series are stationary at levels if a constant and a trend is allowed for (I suppose this relates to the specification of the augmented Dickey Fuller unit root test). Unless you have a good argument against the presence of a deterministic trend (that is, the deterministic trend is not spurious but real), the series are trend-stationary, and there is no need to proceed with cointegration analysis.

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