Dear Lars,

Theoretically, cointegration under Engle-Granger approach and Johansen approach requests the same integration order of variables. With ARDL approach, we can get cointegration with different order of integration; but with I(0) and I(1), and not with I(2).

 Regards,

We may consider Toda Yamamoto approach.

Thanks for the replies. Exactly why can´t I mix variables with different orders of integration?  I suspect that it has something to do with implementation of the error correction term in subsequent error correction models. Am I right? In the Johansen approach, it seems rather obvious, as it uses matrice calculation of an underlying VAR. But isn't it (different orders of integration) possible with the EG approach (or Phillips-Oularis approach), if I'm only interested in long-run coefficients?

If you first difference your I(2) variable it becomes I(1). You can then apply any of the standard cointegration techniques to the combination of your I(1) variable and your differenced I(2) variable.  Juselius (2006), The cointegrated var model, Oxford, has material on I(2) models within a vecm framework.  Hunter Burke and Canepa (2017), Multivariate Modelling of Non-Stationary Economic Time Series - Palgrave, is a more recent reference which also contains material on I(2) variables.  Most of this material requires some mathematics and is not particularly easy but is the way to go if you wish to proceed farther. My original suggestion may be sufficient for your purpose

Both should be I(1), whether in levels or difference.

The series that is I(2) cannot cointegrate with the other two series because at least two series need to have the highest order of integration (here it is 2) for cointegration among all series to be possible.