Cointegration – Practical Questions About Cointegration Test: Comprehensive Guide

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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?

Best Answer

  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.

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