Jwan,

     The answer may depend on the kind of cointegration you have.  If you have an I(2) cointegrated model, you may not be able to restrict yourself to the long-run impact matrix of reduced rank to determine the number of cointegrating vectors you have. Also, if you have a multicointegrated model, this matter may be more complicated than you would think.

      You might also have seasonal or periodic cointegration, which would provide additional considerations to answer this question.

     If we assume that you are only referring to I(1) cointegration, then the answer is simpler:  The number of cointegrating vectors in the I(1) case  depends on the results from either the  Johansen trace and/or maximum eigenvalue test.  But that test will depend don the number of autoregressive lags that you include in your model.   According to Luktepohl, that can be determined from a vector autoregression model using information criteria -- the HQ or BIC are the preferred ones.  

        Regards,

           Bob Yaffee

Many thanks for you both Robert and Kotosz, for answering my question.

 PLease let me to make a little clarification about my model to you.

I am trying to assess explanatory variables such as Gross Domestic Product, public fixed capital formation, real exchange rate inflation and real interest rate on private fixed capital formation in Iraq. I am also trying to take two dummy variables which first representing Iran-Iraq War from 1980 to 1989, the second dummy represent international sanction from 1990- After checking for the stationarity , i have run Johansen cointegration test, then I have got this result

Series: LRPI LRGDP LRPUI LREXR LRINTR LINFL DUM1 DUM2

Lags interval (in first differences): 1 to 1

Unrestricted Cointegration Rank Test (Trace)

Hypothesized Trace 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.997307 432.3116 159.5297 0.0000

At most 1 * 0.984753 278.4659 125.6154 0.0000

At most 2 * 0.876431 169.6975 95.75366 0.0000

At most 3 * 0.867038 115.3327 69.81889 0.0000

At most 4 * 0.748059 62.87279 47.85613 0.0011

At most 5 0.501762 27.03021 29.79707 0.1009

At most 6 0.230651 8.916579 15.49471 0.3731

At most 7 0.077562 2.099116 3.841466 0.1474

Trace test indicates 5 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level

**MacKinnon-Haug-Michelis (1999) p-values

Please could you let me whether I can rely on this result and interpret them .

what I think regarding to the  Kotosz answer ther result should be ok because the number of cointegration are less than the number of coitegration.

Your helps are highly appreciate it

Jwan,

   Dear Jwan, 

        I really don't know whether you have an I(1),  I(2), or mullticointegration situation. The inclusion of dummy variables in the cointegration tests may be correct when they constitute structural breaks,  even though this may deplete the power of the tests to find the proper cointegrating rank.  They will play havoc with finite sample situations. But graphing your variables and your cointegrating vectors may reveal whether you have an I(1) or potentially more complicated structure, where you will have cumulation of acceleration rates and changes in levels.  If you have merely one cointegrating vector, chances are that this will not be an I(2) situation.Katrina Juselius has written about this situation in her book, "The Cointegrated VAR", on pages 106-109.  On pages 139-142, she has written about how the inclusion of additive outliers int the cointegrating relations might not cancel and therefore might cumulate.  If this is the case, she says, referring to Neilson and Rahbeck,(then the cumulation of the intervetion dummy might have to be included in the cointegrating relations) ,in which case you would have to use an appropriate test like that of a Zivot Andrews or Bai-Perron test.

       But if your sample size is over 250, you may be able to deal with them.  If they are restricted in the cointegrating vectors, they will have no effect on the asymptotic limiting distributions.  But if they are unrestricted, they may cumulate ( according to the Granger Representation Theorem) generating a different structure in cointegrating vector.  If these are additive outliers, they may have to be be removed before the Johansen tests. But if they represent part of the common trend, they may be part of the perrsistence in the cointegrating relations for which reason they might have to be retained.  If they are not removed, you will have to use a type of Zivot Andrews test assess the unit root with the structural break. This is not the only one you could use but it is one that has been applied under such circumstances.  Bai and Perron have also written on this. 

         Your graphs of the series, their differences, and the cointegrating vectors should be helpful in guiding your decision here.   Also,  you might want to check out the Harvard Hugnes paper of 2003 on "Structural Breaks in Cointegrating VAR models."

           Regards,

                        Robert Yaffee

Dear Robert,

Many thanks for your kind response and providing fully clarification which are definitly did help me regarding draw my conclusion about this model, and yes, I do have an I(1), multicointegration situation.

Many thanks,

Jwan

the number of cointegrating  equations is simply the number of variables less one. So if you have six variables then you can have up to or less than (6-1) cointegrating equations.

Jwan,

  You're welcome.

       Robert

You may find more 

http://www.iei.liu.se/nek/ekonometrisk-teori-7-5-hp-730a07/labbar/1.233753/dfdistab7b.pdf

I support the answer of Professor Balázs Kotosz (second answer).. If the number of variables are = n, the max number of cointegrations should be n-1

Good luck

Ahmed

As said by the colleagues, the number of cointgration relationships must be less than the number of variables investigated.

It all depends on the number of variables in the equation. the number of cointegrating equations should be less than the number of variables.

Suppose you have n non-stationary variables. Then, you could have at most n-1 cointegrating relationship. Note that nonstationarity is important here. If you have stationary variables, you  will get trivial cointegrating vectors. They include only the respective variable, while the coefficients of the other variables are 0. So, if you have n stationary variables, you will get exactly n (trivial) cointegration vectors.

In general, the cointegration vectors gives you the basis of the cointegration space. Linear combinations of them produce again long run relations. Therefore, after estimating the cointegration space, you should proceed with tests on linear restrictions, etc, in order to identify your cointegration vectors.

The maximum number of co-integration vectors are one less than the total variables in VAR model. suppose we have 3 variables in our VAR model, in this situation maximum no of relationships are 2. If we get 3 co-integration vectors in the case of 3 variable model then its means that variables are I(0). In this situation no need to perform co-integration analysis. 

please consider cointegration always in a system! 

also, I would recommend you to start testing bi-variate cases (2 timeseries) at first. So you can also try to translate the results into an economic context and make sure it makes sense. 

With increasing the amount of time-series in your system, you create a more complex structure & it will become much harde to analyse! 

So my recommendation is to start analyses with 2 time-series and just be able to answer the simple question if they are cointegrated (or not) - and what this implies... 

After you feel comfortable enough with this, you can extend your analysis to multivariate setting - and complement the previous results with a multivariate system. 

It all depends on the number of variables in the equation. the number of co-integrating equations should be less than the number of variables by at least one. It means that if the number of your series is 6. then the maximum number of co-integration is 4

Good luck

Hi,  for my case is that the Johansen cointegration test shows that i have 2 cointegrating vectors for 2 variables. So this suggests that both of my variables are I(0) right? However, from the augmented dickey fuller test results, one of my variable is stationary at first difference while the other stationary at its level. May I know what causes this to happen? Thank you.

It depends on the number of variables you have. It usually is one less than the number of variables in your model.

Where i can find references of the statement that maximum cointegration vectors are n-1 ?

How about 4 variables? possible combination is 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4. six in total... please enlighten us...

The number of cointegrating vectors that are possible when applying the Johansen test depends on the number of variables in the system and whether they all have unit roots. If there are three variables each with unit roots, there are at most two cointegrating vectors. More generally, if there are n variables which all have unit roots, there are at most n-1 cointegrating vectors.

The number of cointegrating vectors that are possible when applying the Johansen test depends on the number of variables in the system and whether they all have unit roots. If there are three variables each with unit roots, there are at most two cointegrating vectors. More generally, if there are n variables which all have unit roots, there are at most n-1 cointegrating vectors.