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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."

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  • $\begingroup$ What about the answers I gave to several of your earlier questions? Do you find them clear? You have not responded there yet. Also, please use the tags appropriately. E.g. the R or the fitting tags are unrelated to this question. Also note that there is special formatting for quotes: they are preceded by the > symbol. You can also find a corresponding button in the editor window. $\endgroup$ – Richard Hardy Sep 5 '17 at 14:01
  • $\begingroup$ I upvoted all your working examples, I thought? Let me check. $\endgroup$ – FenleyK Sep 5 '17 at 14:20
  • $\begingroup$ You don't have to upvote them if they do not deserve that. But let me know what can be improved. Also, satisfactory answers can be accepted by clicking on the tick mark to the left of the answer. Here are the ones from before: 1, 2, 3. Also, what about the current one? $\endgroup$ – Richard Hardy Sep 9 '17 at 16:10
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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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  • $\begingroup$ Exactly that's my point. (2) Why "Two series both being I(0) cannot be cointegrated."? And (2): What does the Pesaran model serve for, if not testing for cointegration of two time series that are integrated of different order? I have recently seen a few economics papers that apply cointegration tests to two I(0) and the Pesaran model to mixes I(0)/(1) models. That' exactly why I'm puzzled. $\endgroup$ – FenleyK Sep 5 '17 at 14:23
  • $\begingroup$ This is one of the sources I refer to: Link $\endgroup$ – FenleyK Sep 5 '17 at 14:32
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    $\begingroup$ @FenleyK, (1) This is about the definition of cointegration. The widespread definition is that you need the series to be integrated to begin with, while an I(0) series is not integrated. Think about the etymology: co-integrated = integrated together. (2) I suppose the Pesaran model serves for the case when there are more than two variables. Then there can be mixed orders of integration, and sometimes I(0) variables are included in modelling although formally they cannot belong in a cointegrating relationship since they are not integrated to begin with. $\endgroup$ – Richard Hardy Sep 5 '17 at 14:46
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    $\begingroup$ Note that I(0) can be considered in the same model with I(1) variables, such as under Pesaran's method, but the I(0) variables cannot be in a cointegrating relationship. A model for a bunch of variables and a cointegrating relationship (characterized by a cointegrating vector) is not the same. $\endgroup$ – Richard Hardy Sep 5 '17 at 14:48
  • $\begingroup$ Thanks for your help! With regard to (1) I then conclude that any paper doing a Cointegration test with two I(0) is complete nonsense? And with regard to (2): The examples on the blog as well as other examples usually consider a system with two variables. Therefore, I am not sure whether this assumption is correct? I feel I(0)/I(1) testing is considered when the order of integration of a variable is not clear after ADF, KPSS and PP tests. The Toda & Yamamoto (1995) Granger approach also points towards testing with different orders of integration. $\endgroup$ – FenleyK Sep 5 '17 at 14:59
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There is no point in searching for a cointegration relationship between two stationary variables, I(0).

When the time series are integrated of order one (or higher), standard linear regression analysis may produce spurious results; in particular, we may erroneously find that one variable is statistically significant for explaining the other variable (Granger and Newbold 1974) [1]. This is where cointegration comes into play.

The focus of cointegration analysis is to search for a linear combination of the time series that has a lower order of integration than the original series. For example, in economic data, it is sometimes observed that the variables are I(1) (stochastic trends) but a linear combination of them is I(0) (stationary). The cointegration relationship reveals a stable relationship between the variables in the long-term and this becomes helpful to develop further economic analysis or to test economic theories.

If the variables are stationary, I(0), then we can rely on standard linear regression analysis and test, for example, whether one variable is a significant predictor of the other variable. (Some kind of cointegration relationship between I(0) variables would not be informative, since any combination of stationary variables will, in principle, be stationary.)

[1] Granger, C. W. J. and Newbold, P. (1974). "Spurious Regressions in Econometrics." Journal of Econometrics, 2(2), pp. 111-120, DOI: 10.1016/0304-4076(74)90034-7.

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