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

John Frain frainj at tcd.ie
Fri Jul 3 22:29:44 CEST 2009 

I agree with what is said below.  Even though the rank of pi is 2 it
can be factorised into a product of two matrices alpha which is 2 by 3
and a beta which is 3 by 5.  In macroeconometrics, in practice,  such
a scheme has been used to incorporate a disequilibrium in one sector
of a macro model (modelling the X,s) into another sector (modelling
the Y's).  We did necessarily even use the first differences of the
Y's inmodelling the X's.

I agree that this is probably a matter of time series macoeconometrics
rather than finance and I should probably leave it at that,

Best Regards

John

2009/7/3 Eric Zivot <ezivot at u.washington.edu>:
> 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_ot nec2
>
> 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, X 2, 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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>>
>>
>>
>> --
>> 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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-- 
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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