Page 32 - Textos de Matemática Vol. 47

22 H. FERREIRA
A natural estimator for R(X, , I ) is
Rˆ ( X , , I ) = 1 Xn 0@ _p _ Fˆ j ( X ( k ) ) ^p _ Fˆ j ( X ( k ) ) 1A ,
niiii k=1 j=1 i2Ij j=1 i2Ij
and, in particular for p = d, 0
1 Rˆ(X, ) = @ Fˆ j (X(k)) Fˆ j (X(k))A .
1WXn _d ^d njjjj
j=1
If we denote Mˆk(Ij) j = Fˆ j (X(k)) then we can write
1 Xn _p
Rˆ(X, ,I) = Mˆk(Ij) j
X
( 1)|T|+1
1 Xn _ n k=1 j2T
Mˆk(Ij) j .
k=1 j=1 i2Ij i i
n k=1 j=1
;6=T ✓{1,...,p}
The strong consistency of the terms of this sum is stated in the proof of the Proposition 3.8 of Ferreira and Ferreira (2012b) and the asymptotic normality can be deduced from the Theorem 6 in Fermanian et al. (2004).
As an application of this estimation procedure we consider for X some financial stock markets grouped in the three big world markets I1 =Europe, I2 =USA and I3 =Far East, as considered in Ferreira and Ferreira (2012b). The data are monthly maximums of the negative log-returns of the closing values of the stock market indexes CAC 40 (France), FTSE100 (UK), SMI (Swiss), XDAX (German), Dow Jones (USA), Nasdaq (USA), SP500 (USA), HSI (China) and Nikkei (Japan), from January 1993 to March 2004.
In the Table 1 we present the estimates corresponding to M¯(A)=1Xn _Fˆi(X(k))
which we need to compute Rˆ(X, 1, I) for I = {I1, I2, I3}. Table 1. I1 =Europe, I2 =USA and I3 =Far East.
We obtain then Rˆ(X,1,I) = 0.321, Rˆ((XI1,XI2),1,{I1,I2}) = 0.172, Rˆ((XI1,XI3),1,{I1,I3}) = 0.222 and Rˆ((XI2,XI3),1,{I2,I3}) = 0.247, sug- gesting a stronger dependence between I1 and I2.
nj k=1 i2A
A
I1
I2
I3
I1 [ I2 0.739
I1 [ I3 0.770
I2 [ I3 0.744
I1 [I2 [I3 0.801
M¯ ( A )
0.692
0.614
0.626