I think I understood following so far about testing a cointegration relationship in a time series:
Two time series of same order of integration:
x: I(1), y: I(1)
Apply: Johansen and Juselius (1990) cointegration test
Two time series of arbitrary order of integration:
x: I(0), y: I(1)
Apply: Pesaran et al. (2001) ARDL bounds cointegration test.
Cointegration exists if a linear combination of two time series is stationary.
Question: Can I theoretically apply the cointegration test to two stationary variables (trend-stationary, at level)?
x: I(0), y: I(0)
I would assume that a linear combination of two I(0) time series must be stationary, too.
Lütkepohl and Krätzig (2004) state in their book "Applied Time Series Econometrics":
"Occasionally, it is convenient to consider [cointegration] systems with both I(1) and I(0) variables. Thereby the concept of cointegration is extended by calling any linear combination that is I(0) a cointegration relation, although this terminology is not in the spirit of the original definition because it can happen that a linear combination of I(0) variables is called a cointegration relation."