Solved – Cointegration if both variables are I(0)

cointegrationtime series

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."

Best Answer

Since

calling any linear combination that is I(0) a cointegration relation <...> is not in the spirit of the original definition

I will stick to defining a cointegrating combination as one where none of the original variables are I(0).

The case of two series:

  • Two series of different orders of integration will never be cointegrated.
  • Two series both being I(0) cannot be cointegrated.
  • Two series both being I($d$) for $d\geq 1$ can be cointegrated, but they don't have to.

The case of more than two series:

  • Only I($d$) series with $d\geq 1$ can enter the cointegration relationship. Thus I(0) series will never belong in a cointegration relationship.
  • At least two of the series must share the highest order of integration to cointegrate. E.g. three {I(2), I(2), I(1)} processes can cointegrate but {I(2), I(1), I(1)} cannot.

All of this is basic material that should be found in most of the time series textbooks dealing with cointegration. But I understand that keeping track of the discussion in a textbook can be hard, so let this serve as a summary.

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