Dear Mark,

Thanks for your response. What is your opinion (other stat-gurus as well)? is there anything wrong in my understanding here?

Regards,

"Leeds, Mark (IED)" <Mark.Leeds@morganstanley.com> wrote: hi megh : that material is extremely difficult for me but I would
recommend 
reading  one more of the following to get a better idea of cointegration
in the multivariate case.


Johansen or Hamilton text   ( very difficult for me )
Hayashi( less difficult )
Enders  ( not difficult )

I think you're A[0] is the pi matrix in the econometric literature. 

Assuming this is the case ( if not, then
Everything below is incorrect and disregard it )

If there are n equations in the system and the rank of pi is n, then
there are no cointegrating vectors. 
If there are n equations and the rank of pi is r, then there are n-r
cointegrating vectors. If there
are n equations and the and the rank of pi is 1, then there are n-1
cointegrating
relationships which is the maximum that there can be. I don't think the
rank of pi can be less 
Than 1 but the determinant of pi can be zero if all it's rows ( columns
) are not linearly independent.

This probably doesn't help a heck of a lot but if you want to some other
book references,
Let me know. A really nice readable explanation of cointegration using
the matreix results (
Which I always find difficult )
Is by Lehmann and the title has the work "desiderata" in it. I don't
have it in front of me
But if you google "lehmann desiderata", I'm sure it will pop up.




-----Original Message-----
From: r-help-bounces@stat.math.ethz.ch
[mailto:r-help-bounces@stat.math.ethz.ch] On Behalf Of Megh Dal
Sent: Sunday, August 05, 2007 6:33 AM
To: r-help@stat.math.ethz.ch
Subject: [R] Understanding of Johansen test.

Dear all,
   
  I am struggling to understand the johansen test procedure in the
context of co-integration in time series. Yes I understand that this
forum is not directly statistics related but still I am posting here
hoping that I would get som help.
   
  The error correction representation of a VAR[p] model can be written
as: 
  Delta y[t] = A[0]*y[t-1] + A[1]*Delta y[t-1] +..............
   
  where, y[t] is a vector of n variables.
   
  It is said that "if the variables in system are all co-integrated,
then Rank of A[0] will be different from zero"
   
  My understanding is following : suppose, y[t] is of order 3 and p = 1
   
  Then Delta y[t] = A[0]*y[t-1]  + epsilon[t]
   
  Hence : Delta y1[t] = a[11]*y1[t-1] + a[12]*y1[t-1] +a[13]*y1[t-1] +
epsilon1[t]
              Delta y2[t] = a[12]*y1[t-1] + a[22]*y1[t-1] +a[23]*y1[t-1]
+ epsilon2[t]
              Delta y3[t] = a[31]*y1[t-1] + a[32]*y1[t-1] +a[33]*y1[t-1]
+ epsilon3[t]
   
  But is rank of A[0] is 0 then it is possible to find non-zero coef for
all of above three equations such that : a[11]*y1[t-1] + a[12]*y1[t-1]
+a[13]*y1[t-1] = 0
                  a[12]*y1[t-1] + a[22]*y1[t-1] +a[23]*y1[t-1] =  0
                  a[12]*y1[t-1] + a[22]*y1[t-1] +a[23]*y1[t-1] = 0
   
  therefore number of co-integrating relationship is 3 am I correct?
   
  Therefore in my understanding : if variables in a system show some
co-integrating relationship thenrank should be close to zero.
   
  Am I making any mistakes? Can anyone here clarify me?
   
  Regards,
  Megh
   

       
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