The Johanesen determination of cointegration rank uses a sequence of nested hypotheses.  The concept of nested hypotheses is not well explained in econometrics texts.  It is well covered in Johansen's text 'Likelihood-based inference in cointegrated vector autoregressive models, Oxford University Press, 1995.  
 
The trace test is
 
H0(r) : at most r cointegrating vectors against the alternative of 
H1(r): more than r cointegrating vectors
 
The maximum eigen value test
 
H0(r) exactly r cointegrating vectors
H1(r) exactly r+1 cointegrating vectors
 
Thus the trace test is nested.  If using the trace test H0(r) is rejected one can proceed to test H0(r+1).  This can be done because the tests are nested.
 
If you use the maximum eigen-value test and you reject H0(r) then it is not logical to test H0(r+1) as you have already accepted this by rejecting H0(r).
 
In practice the two tests are used to complement one another.
 
One should also pay attention to the economics of the situation.  Often economic arguements provide a strong rational for the determination of the number of cointegrating vectors.  These relationships do determine the long run relationship between the economic variables.  Johansen without theory is dangerous.    If there are two or more than two cointegrating vectors you will have to use economic theory to identify them.  You may of course test overidentifying restrictions within a Johansen framework.
 
These finer points are not, in general well under stood by economists.  I hope that this provides some clarification.
 
               

 
On 22/12/06, javier garcia enriquez <javigarcia83@yahoo.es> wrote:
Peter Summers wrote:
 

"Javi,
 
What you really need to do is consult an econometrics textbook, not the gretl list."
 
Sorry, but I think you're wrong. If you have read the Johansen's article and other autor's articles, you can see the topic that I question isn´t clear in the literature, especially with the lambda max test. Depending what software do you use, the null hypothesis is different and, in statistics, the interpretation depends completely of the null hypothesis. So, to interpretate Gretl´s or other program's output we really need the null hypothesis that they use, don't you agree?
 
Thanks to all, especially to Allin and Merry Christmas
Javi
 
PD: sorry for my bad English :p
 



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John C Frain
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