# [R-SIG-Finance] [R-sig-finance] Rank

Eric Zivot ezivot at u.washington.edu
Fri Jul 3 19:20:09 CEST 2009

```I suggest that you look at Johansen's book on cointegration

http://www.amazon.com/Likelihood-Based-Inference-Cointegrated-Autoregressive-Econometrics/dp/0198774508/ref=sr_1_1?ie=UTF8&s=books&qid=1246640896&sr=8-1

His treatment is the most complete and will answer all of your questions.
A nice empirical/practical follow up to this book is the recent one by K. Jusalius (his wife)

http://www.amazon.com/Cointegrated-VAR-Model-Applications-Econometrics/dp/0199285675/ref=pd_sim_b_2

The issue here is because the Pi matrix has rank 2 there are only two cointegrating relationships among the Y's of the form beta1'(Y1, Y2, X1, X2, X3) and beta2'(Y1, Y2, X1, X2, X3) where the coefficients on the X's are not all zero.

The X variables are unmodeled - which in the cointegration literature means that they are weakly exogenous wrt to the cointegration parameters in the VECM. If there is now feedback from the Ys to the Xs then the reduced form relationship for DX(t) does not involve Y and the Xs are then strongly exogenous. In particular, the error correction coefficients on these variables (alphas) are zero so that the system has the form like

DY1(t) = a1*beta1'(Y1, Y2, X1, X2, X3) + lags of DY(t) and DX(t) + e1(t)
DY2(t) = a2*beta2'(Y1, Y2, X1, X2, X3) + lags of DY(t) and DX(t) + e1(t)
DX(t) = lags of DX(t) + e3(t)

Notice in this type of representation there is no cointegration among the Xs because the reduced form for the Xs is not a VECM.
Now because the X's are unmodeled, there is the possibility that there are cointegration relationships among the X's that do not involve the Ys. I think this is causing the confusion. In general, it is assumed that such relationships do not exist when the VECM is specified with unmodeled variables.

None of this discussion has to do with finance

On Fri, 3 Jul 2009, RON70 wrote:

>
Oh John, it is page 399, sorry.

John C. Frain wrote:
>
> Your pi matrix is 2 by 5 and therefore must be of rank <= 2 and you
> can not have more than two cointegrating relationships betwween the
> y's.   Page 408 of my copy of Lutkepohl(2005) deals with Multiplier
> analysis and Optimal Control
>
> Best Regards
>
> John
>
> 2009/7/3 RON70 <ron_michael70 at yahoo.com>:
>>
>> I am not sure why you are saying c.i. relationships can not be more than
>> n.
>> Quote from Lutkepohl, page : 408 : "Because the error correction term now
>> involves all the cointegration relations between the endogenous and
>> unmodelled variables,it is possible that r>K. ", here he defined K as
>> number
>> of endo. variables in the system................any idea?
>>
>> However your 1st point is valid, I should have added diff. operator on
>> the
>> left side, it was a typo.
>>
>> PS. I understand some ppl here previously suggested not to read Lutkepohl
>> 1st, however except few things I am getting comfortable-reading on that,
>> atleast easier than Hamilton, perhaps I have only softcopy of Hamilton
>> ;).
>>
>>
>> matifou wrote:
>>>
>>> 2009/7/3 RON70 <ron_michael70 at yahoo.com>
>>>
>>>>
>>>> This is a finance related question in the sense that I have come
>>>> accross
>>>> this
>>>> kind of problem in Co-Integration matrix construction in a VECM. I am
>>>> explaing how :
>>>>
>>>> Suppose I have 2 endogeneous variables and 3 exogeneous variable all
>>>> are
>>>> I(1) and assumed to have cointegration relationships among them. Let
>>>> say
>>>> the
>>>> DGP is
>>>>
>>> what do you mean by exogenous?
>>>
>>>
>>>>
>>>> y[t] = alpha * t(beta) * (y[t-1] : x[t-1]) + ..................
>>>
>>> left should be differenced
>>>
>>>>
>>>>
>>>> pi = alpha * t(beta)
>>>>
>>>> Obviously dimension of y vector is 2 and x vector is 3. Therefore there
>>>> could be more than 2 cointegrating relationships in that.
>>>
>>> if you have more than two cointegrating relationships: I would say x is
>>> not
>>> exogeneous
>>>
>>> Hence rank of pi
>>>> is in principle more than 2. As number of co-integrating relationships
>>>> is
>>>> estimated on looking at rank of pi matrix. However number of rows there
>>>> is
>>>> :
>>>> 2.
>>>>
>>> I am trying to understand this scenario here. In this case, can usual
>>>> VECM estimation procedure work? More important to me is to understand
>>>> rank
>>>> of pi is more than it's row number.
>>>>
>>>> Thanks
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> Enrico Schumann wrote:
>>>> >
>>>> > that's not a finance question, but the rank can at most be the min of
>>>> n
>>>> > and
>>>> > m.
>>>> >
>>>> > -----Ursprüngliche Nachricht-----
>>>> > Von: r-sig-finance-bounces at stat.math.ethz.ch
>>>> > [mailto:r-sig-finance-bounces at stat.math.ethz.ch] Im Auftrag von RON70
>>>> > Gesendet: Freitag, 3. Juli 2009 03:22
>>>> > An: r-sig-finance at stat.math.ethz.ch
>>>> > Betreff: [R-SIG-Finance] [R-sig-finance] Rank
>>>> >
>>>> >
>>>> > Hi, i have a small matrix related question which most of you find
>>>> trivial
>>>> > however I am not getting through. Suppose I have a matrix of
>>>> dimension
>>>> > (nxm), n < m. Is it in principle possible to have the rank of that
>>>> matrix
>>>> > greater than n? Is it possible to have some example?
>>>> >
>>>> > Thanks,
>>>> > --
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>>>> >
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>>>> >
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>>
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>
>
>
> --
> John C Frain, Ph.D.
> Trinity College Dublin
> Dublin 2
> Ireland
> www.tcd.ie/Economics/staff/frainj/home.htm
> mailto:frainj at tcd.ie
> mailto:frainj at gmail.com
>
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