Co-integration for processes integrated of order zero

Question

I read that two (or more) time series that are integrated of order 1 (or higher) can be co-integrated. I have three time series, and all three of them are stationary processes. Can they be co-integrated?

Answer 1

Cointegration is only relevant for series integrated of order equal or higher than 1:

$I(d), \quad d > 0$.

For example, if you have two series, $x$ an $y$, both integrated of order 1, than cointegration means that there is a combination of the series, e.g. $z = \alpha x + \beta y$ where $z$ is integrated of order zero.

Hence, if $x$ and $y$ are $I(0)$ already, than all $\alpha x + \beta y$ conbinations lead to $z$ integrated of order zero.

This may be easily generalized for three or more series.

Also, for $I(0)$ series, the real problem is wheter the series are correlated, (contemporaneous or lagged dependency may be examined, there may also be a non-linear dependency).