Advances in Risk Management

250 MONITORING COVARIANCES OF ASSET RETURNS

problem is that the direction is unknown and therefore the statistic cannot be
directly applied. Ngai and Zhang (2001) proposed to solve this problem by

estimatinga 0 byaˆ 0 , and approximateCat^0 byCtaˆ^0. Hereˆa 0 is the value at which

Cat^0 attains its maximum on the unit circle, for example,Ctaˆ^0 =max||a|| 2 = 1 Cat.
They proved that max||a|| 2 = 1 Cat=PPCUSUMtwith:

PPCUSUMt=max{0,||St−1,t||#η−k,||St−2,t||#η− 2 k,...,||S0,t||#η−tk}

(13.13)

fort≥1.St−v,tand||St−v,t||#ηare defined in section 3.2.
When a process is appeared to be out-of-control at timet 0 , then it exists

t 1 <t 0 such that

√

S′t 1 ,t 0 #−η^1 St 1 ,t 0 −(t 0 −t 1 )k=max||a|| 2 = 1 Cta 0 >h 3 , whereh 3 is

a preselected value. Then the direction of the shift is estimated by:

aˆ 0 =

#

−^12

η St 1 ,t 0

S′t 1 ,t 0 #−η^1 St 1 ,t 0

13.3.5 Multivariate EWMA control chart

The EWMA control chart, first introduced by Roberts (1959), was adapted
to multidimensional observations by Lowry, Woodall, Champ and Rigdon
(1992). In an ARL comparison the authors showed that the properties of
the multivariate EWMA chart are similar to or even better than those of the
multivariate CUSUM charts of Crosier (1988) and Pignatiello and Runger
(1990). Additionally, the design of the multivariate EWMA chart is much
simpler than that of the multivariate CUSUM charts. Prabhu and Runger
(1997) gave recommendations on the choice of the EWMA parameter.
We define the MEWMA recursion for the vectorη(t)by:

Zt=Rη(t)+(I−R)Zt− 1 t≥ 1

whereR=diag(r 1 ,r 2 , ...,rq+q(q+1)/ 2 ), 0<rj<1,j=1,...,q+q(q+1)/2.
Rewriting gives:

Zt=(I−R)tZ 0 +R

∑t−^1

j= 0

(I−R)tη(t−j)

=(I−R)t(Z 0 −μη)+R

∑t−^1

j= 0

(I−R)t(η(t−j)−μη)+μη

since (I−R)t+R

∑t− 1

j= 0 (I−R)

t=I. Hence, in the in-control scenario the

mean of the vectorZtisE 0 (Zt)=(I−R)t(Z 0 −μη)+μη.In the following it is
always assumed that the processZtstarts in the target valueμη,Z 0 =μη.