I am carrying the Johansen test on 3 time series variables and eventually estimating a VECM. 2 of my variables are stationary while the other one is nonstationary. I have a few doubts:

  1. Do I need to difference the one time series which is nonstationary and estimate a VECM? Or can I use it the way it is?

  2. For the Johansen test, do I have to run it when all the data is stationary? Or can I run it when 2 of it are stationary and the other one is non stationary?

  • 1
    $\begingroup$ I hope somebody can find a good duplicate. These types of question crop up regularly, and they all can be answered by picking up any time series book and reading the first chapter on cointegration. $\endgroup$ – mpiktas Mar 25 '16 at 7:31

VEC model is a special case of VAR model. It exploits the time-series property of cointegration, i.e. that some linear combinations of unit-root time-series can be stationary. Because of this property VAR model has a special structure, which can be estimated via VECM. To apply VEC model at least two time series must be unit-root time series, because to define cointegration you need at least two unit-root time series. So the answers to your questions are

  1. No you cannot difference the non-stationary time series and then estimate VECM. Because estimating VECM on stationary time series does not make sense.

  2. Johansen's test can be applied only when all the series are non-stationary, because Johansen's test is for testing cointegration, and cointegration is defined only for unit root time series.

  • $\begingroup$ linear combinations of a unit-root time-series I think here "a" should be removed as there have to be several unit-root time series to form a linear combination. $\endgroup$ – Richard Hardy Mar 25 '16 at 10:07
  • $\begingroup$ Fixed it. Thanks. Articles are my weak spot, as my native language does not have those :) $\endgroup$ – mpiktas Mar 25 '16 at 10:49
  • $\begingroup$ Your answer is not exactly correct for 2. You do write: "To apply VEC model at least two time series must be unit-root time series, because to define cointegration you need at least two unit-root time series." which is correct. In the same way you do not need all series to be non-stationary when running the Johansen test. If one series is stationary it can be seen as a cointegrating//equilibrium relation on its own. In the case of OP she can run the Johansen test if two series are non-stationary and one is stationary. In the VECM framework of Johansen the series of interest are usually not $\endgroup$ – Plissken Mar 27 '16 at 15:45
  • $\begingroup$ tested for unit roots before the Johansen test is applied. The reason is that we can test whether or not the series can be seen as a cointegrating relation on their own. If they can be considered as a stationary series on their own this means they do not contain a unit root. This is discussed in the book by Juselius and I think it is also discussed in Johansen's book. $\endgroup$ – Plissken Mar 27 '16 at 15:48
  • $\begingroup$ This is discussed to some extent by Dave Giles here: davegiles.blogspot.fi/2012/01/… . Further, see the first comment here: davegiles.blogspot.fi/2011/05/cointegrated-at-hips.html . We can basically examine all we need in one VECM rather than specifying many models for the data. $\endgroup$ – Plissken Mar 27 '16 at 15:51

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