Advances in Risk Management

246 MONITORING COVARIANCES OF ASSET RETURNS

(b) The covariance matrix ofvˆ ̃is equal to:

Var(vˆ ̃)=

n−p

2









(n−p

2

− 1

)



(

n−p

2

)

(

I+H ̃( 22 −)b ̃−

1

(^2) G ̃b−
1
2
′)

• 
  
  
  
  
  (
  n−p− 1
  2
  )
  
  (n−p
  2
  )
  
  
  
  
  2
  H ̃( 22 −) ̃b−^12 Fb ̃−^12 ′
  
  
  
  
  where the matricesGandFare given in the Appendix.
  We make use of these results in the next section, where the multivariate
  and simultaneous control charts are constructed for detecting changes in
  the covariances of returns.

13.3 MULTIVARIATE STATISTICAL SURVEILLANCE

The covariance matrix of asset return for a given horizon of interest is esti-
mated from the returns of higher frequency. Using non-overlapping samples
of data permits us to construct a sequence of covariance matrix estimators
that are independent through time. This point has already been discussed in
the financial literature. For instance, Schwert (1989) proposed estimating the
variance of monthly returns using daily data, while Andersen, Bollerslev,
Diebold and Ebens (2001) made use of this approach for the approximation
of daily variances and covariances from high-frequency return data.
For a given sample of asset returnsX 1 ,...,Xnwe constrainmsub-
samples of sizen ̃, that is{X(1);j}nj ̃= 1 ,{X(2);j} ̃nj= 1 ,...,{X(m);j}nj ̃= 1 , whereX(i);j=

Xn ̃(i−1)+j,i=1,...,mandj=1,...,n ̃.For thei-th subsample the estima-

tors for#,w, andv, namely#ˆ(i),wˆ(i), andvˆ(i), are defined using (13.2),
(13.3), and (13.4) correspondingly. From Theorem 1 it follows that under the
assumption of no change in the covariance matrix of returnsvˆ(i)are inde-
pendently identicallyt-distributed withn ̃−pdegrees of freedom, with the
mean vector 0 , and the covariance matrix (n ̃−p)I/(n ̃−p−2). We consider
theq+q(q+1)/2 dimensional vector:

η(i)=(vˆ(i);1,...,vˆ(i);q,vˆ(i);1vˆ(i);1,vˆ(i);1vˆ(i);2,...,vˆ(i);q− 1 vˆ(i);q,vˆ(i);qvˆ(i);q), (13.6)

which is used to construct control schemes for detection changes in the mean
vector and the covariance matrix ofvˆ(i)simultaneously. In case of no breaks
in the returns covariances the expected value of the vectorsη(i)is:

μη=E(η(i))=

(

0,...,0,

n ̃−p

n ̃−p− 2

,0,...,0,

n ̃−p

n ̃−p− 2

)

(13.7)