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)" 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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