OLHA BODNAR 253A large value ofsimPPCUSUMtis a hint that a change in the covariance
matrix of the asset returns has occurred. Furthermore, this change leads to
the reconstruction of the GMVP. The control limit is determined as described
above. It is obtained within an extensive Monte Carlo study.
The simultaneous MEWMA control statistic is given by
simMEWMAt=max
j=1,...,pMEWMA
(j)
twhereQ
(j)
t is defined as in section 13.5. Application of the asymptotic covari-
ance matrix in constructing the quadratic formsQ
(j)
t leads to the asymptotic
analog of the MEWMAcontrol scheme which we denote by simMEWMAas.
As usual for both control designs the control limits are obtained within an
extensive Monte Carlo study.
13.5 A COMPARISON OF THE MULTIVARIATE AND
SIMULTANEOUS CONTROL CHARTS
In this section we compare the multivariate control charts and simultaneous
control schemes.
13.5.1 Structure of the Monte Carlo Study
Without loss of generality, in this section the in-control process is taken to
be a four-dimensional Gaussian process {Xt} with zero mean vector and the
covariance matrix as:
#=
0 .84813 0.3726 0.18718 0. 15418
0 .3726 1.52624 0.31376 0. 35488
0 .18718 0.31376 1.83864 0. 28748
0 .15418 0.35488 0.28748 2. 06115
To calculate#we made use of monthly data from Morgan Stanley Capital
International for equity markets returns of four developed countries (the
USA, the UK, Japan and Germany). This choice is not restrictive because
in the in-control state the proposed statisticsvˆhas the same distribution
independent of the constant matrixL. As a result, the calculated control lim-
its can be used for the non-singular matrix#. Note, that in case where the
number of elements in each subsample{X(i);j}nj ̃= 1 is large enough, the dis-
tribution of the vectorvˆ(i)is very accurately approximated by the standard
normal distribution. Thus, we can use the control limits that are calculated
for detecting changes in the mean vector and the covariance matrix of the
standard normally distributed random vector.
In our simulation study we setn ̃=20. This choice corresponds to estima-
tion of the covariance matrix of the four weeks (roughly monthly) returns