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Multivariate EWMA covariance estimator?

6 messages · Neuman Co, Berend Hasselman

Neuman Co
# 
Hi,
since I want to calculate the VaR of a portfolio consiting of 4 assets
(returns saved into "eonreturn","henkelreturn" and so on) I have to
estimate the covariance matrix. I do not want to take the rectangular
version with equal weights, but the exponentially weighted moving
average in a multivariate version. I want to estimate a covariance
matrix at every time point t. Then I want to comput the VaR at this
time point t. Afterwards, I will look at the exceedances and do a
backtest.

I tried to implement it as follows (data attached):

lambda<-0.9

summe2<-0
dummy2<-0
covestiexpo<-list(NA)
meanvalues<-NA
for(i in 101:length(eonreturn)){
meanvalues<-matrix(c(mean(eonreturn[(i-100):(i-1)]),mean(henkelreturn[(i-100):(i-1)]),mean(siemensreturn[(i-100):(i-1)]),mean(adidasreturn[(i-100):(i-1)])),4)
for(a in 1:100){
dummy2<-lambda^(a-1)*t(datamatrix[(i-a),]-t(meanvalues))%*%(datamatrix[(i-a),]-t(meanvalues))
summe2<-summe2+dummy2
}
covestiexpo[[i]]<-(1-lambda)*summe2
}


So the covestieexpo[[101]] would be the covariance estimate for the
101th day, taking into account the last 100 observations. Now, the
problem is, that there seems to be something wrong, since the
covariance estimates are cleraly wrong, they seem to be too big. At
the beginning, compared to the normal covariance estimate the
difference is as follows:

covestiexpo[[101]]
            [,1]        [,2]        [,3]        [,4]
[1,] 0.004559042 0.002346775 0.004379735 0.003068916
[2,] 0.002346775 0.001978469 0.002536891 0.001909276
[3,] 0.004379735 0.002536891 0.005531590 0.003259803
[4,] 0.003068916 0.001909276 0.003259803 0.003140198



compared to cov(datamatrix[1:100,])
             [,1]         [,2]         [,3]        [,4]
[1,] 0.0018118239 0.0007432779 0.0015301070 0.001119120
[2,] 0.0007432779 0.0008355960 0.0009281029 0.000754449
[3,] 0.0015301070 0.0009281029 0.0021073171 0.001269626
[4,] 0.0011191199 0.0007544490 0.0012696257 0.001325716

So already here, it is obvious, that something is not correct, if I
look at a period far ahead:

covestiexpo[[1200]]

          [,1]      [,2]      [,3]      [,4]
[1,] 0.5312575 0.1939061 0.3419379 0.2475233
[2,] 0.1939061 0.3204951 0.2303478 0.2022423
[3,] 0.3419379 0.2303478 0.5288435 0.2943051
[4,] 0.2475233 0.2022423 0.2943051 0.4599648


you can see, that the values are way too large, so where is my mistake?



Thanks a lot for your help!
--
Neumann, Conrad
Berend Hasselman
#
On 02-06-2013, at 15:17, Neuman Co <neumancohu at gmail.com> wrote:

            

      
      
      
    

        
Without actual data this is an unverifiable statement.
But you probably have to move the statement

summe2 <- 0

to inside the i-forloop just before the a-forloop.

summe2 <- 0
for(a in 1:100){
?

so that summe2 is initialized to 0 every time you use it as an accumulator in the a-forloop.
Furthermore there is no need to initialize dummy2. It gets overwritten continuously.

Berend

  
  


  


      

    

    

      
      

        


  

        

      Neuman Co
    

    

      
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Thanks a lot for your answer, one more question:
I now use 100 values, so not infinity values. That means I cut some
values off, so the weights will not sum up to one. With which factor
do I have to multiply the (1-lambda)*summe2 to rescale it? So that I
do not always underestimate the variance anymore?

2013/6/2 Berend Hasselman <bhh at xs4all.nl>:

            

      
      
      
    

        

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Berend Hasselman
    

    

      
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On 02-06-2013, at 19:03, Neuman Co <neumancohu at gmail.com> wrote:

        

            

      
      
      
    

        
I don't know but maybe something like this

1/sum(lambda^((1:100)-1))/(1-lambda)

which in your case is 1.000027

Berend

            

      
      
      
    

  
  


  


      

    

    

      
      

        


  

        

      Neuman Co
    

    

      
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Again, a big thanks for your answer.

On this webpage:
http://financetrainingcourse.com/education/2010/03/master-class-calculating-value-at-risk-var-final-steps/

I found, that I have to rescale by dividing the weights calculated in
Step B2 by 1-?n

The "?" is the lambda, since the webpage cannot display it, I also
found it on another webpage, therefore, I changed my code to the
following:

dummy2<-lambda^(a-1)/(1-lambda^100)*t(datamatrix[(i-a),]-t(meanvalues))%*%(datamatrix[(i-a),]-t(meanvalues))


Do you think this is correct?

One further question: You also told me, that I do not have to
initialize my dummy2, what does this mean? I wrote dummy2<-0 because I
have to create this variable before using it for the loop?

2013/6/2 Berend Hasselman <bhh at xs4all.nl>:

            

      
      
      
    

        

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Berend Hasselman
    

    

      
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On 02-06-2013, at 19:45, Neuman Co <neumancohu at gmail.com> wrote:

        

            

      
      
      
    

        
I'm not going to do homework.
You have to scale the weights. So you need to scale lambda^(a-1) by their sum for a=1:100 if I understand correctly.

            

      
      
      
    

        
Exactly that: you don't have to initialize dummy2.
You are assigning a value to dummy2 for each value of a.
You are thus either creating a new object dummy2  or overwriting any existing object dummy2 and thus destroying a previous value.

            

      
      
      
    

        
You are not using it on the righthand side of an expression. You are assigning to it.
You only need to initialize dummy2 if you have an expression involving dummy2 on the righthand side e.g.  dummy2 <- dummy2 + ??

Berend