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I have a few questions about Johansen's cointegration test.

  1. I learnd that all variables must have the same order of integration for cointegration test. Suppose there are 5 variables for one is I(0) and four are I(1). Then I can't make VECM using all 5 variables? Or if I can find cointegration relationship between four I(1) variables by Johansen's test then I can use VECM using all 5 variables?

  2. For 5 variables with I(1). From Johansen's test, I got a result that there are at least 3 cointegration relationships. Then I can use VECM with 5 variables? Is it okay that only partial cointegration relationships for VECM? (I mean, we don't need all n-1 cointegration relationships for VECM with n variables?)

  3. Suppose there are 5 variables for one is I(1) and four are I(2).
    Then I can apply Johansen's test using lag order 2?
    And if I found some cointegration relationships from the above test, it is possible to make VECM using all 5 variables with lag order 2?

guest
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1 Answers1

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  1. The standard cointegration tests do not allow I(0) components. A VECM for a number of I(1) and a single I(0) variable will not work. However, you can include levels of the I(0) variable on the right hand side of the equations, and you can add an equation for levels of it on the left hand side and I(0) components made of the other variables on the right hand side.

  2. If you have 5 I(1) variables, then there can be between zero and 4 cointegrating relationships. A VECM will suit the case of 1 to 4 cointegrating relationships; you do not necessarily need 4.

  3. It is possible that the 4 I(2) variables produce an I(1) combination, so they are cointegrated. Then it is furter possible that this combination cointegrates with the I(1) variable. However, I do not think you can test the I(2) and I(1) variables together at once for cointegration using any standard test.
    Regarding lag order, you probably meant order of differencing, as lag order is not directly determined by the order of integration. Some flavor of VECM should work, but principles similar to these under point 1. should be observed.

Richard Hardy
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  • What do you mean that I can include levels of I(0) on the RHS of the equations? Can you make an example or show the form of equation? – guest Dec 22 '21 at 13:26
  • @guest, see e.g. [this](https://stats.stackexchange.com/questions/148994). I have answered several other related questions, you could look them up, too. A couple of starting points: [here](https://stats.stackexchange.com/search?q=I%280%29+is%3Aanswer+%5Bvecm%5D+user%3A53690) and more broadly, [here](https://stats.stackexchange.com/search?q=is%3Aanswer+%5Bvecm%5D+user%3A53690). – Richard Hardy Dec 22 '21 at 15:02
  • @Ricahrd Hardy, I read your old answers and have one more question. If Johansen's test says there are 3 cointegration relationships for 5 I(1) variables, then should I use all 3 VECM models for my research? In Eviews, I can choose the number of cointegration equation and usually I choose one. Is there any problem with it? – guest Dec 23 '21 at 00:52
  • I also posted another [question](https://stats.stackexchange.com/questions/558093/what-is-the-difference-between-ecm-and-vecm) related to this VECM topic. Hope to get helpful answer from you again. – guest Dec 23 '21 at 01:51
  • @guest, you should use all 3, not 1. Otherwise you will have omitted 2 relevant error-correction terms, and your model will suffer from omitted variable bias. This is similar to how using a VAR on first differences instead of a VECM when the latter is actually appropriate suffers from omitted variable bias, as has been discussed in some previous posts. – Richard Hardy Dec 23 '21 at 06:40