A skewed GARCH-type model for multivariate financial time series 147By assumption the distribution ofztis multivariate skew-normal:
μ 3 (xt|σt)=√
2
π(
4
π− 1
)
Dt(
δ⊗δT⊗δT)
(Dt⊗Dt). (12)Consider now the following mixed moments of order three:
E(
XitXjtXkt∣
∣σt)
=
√
2
π(
4
π− 1
)
(δiσit)(
δjσjt)
(δkσkt) i,j,k= 1 ,...,p. (13)We can use definitions ofandσtto write the above equations in matrix form:
μ 3 (xt|σt)=√
2
π(
4
π− 1
)
(σt)⊗(σt)T⊗(σt)T. (14)Ordinary properties of tensor products imply that
μ 3 (xt|σt)=√
2
π(
4
π− 1
)
(
σt⊗σtT⊗σtT)
(⊗). (15)
By assumptionE
(
σitσjtσht)
<+∞fori,j,h= 1 ,...,p, so that we can take
expectations with respect toσt:
μ 3 (xt)=√
2
π(
4
π− 1
)
E
(
σt⊗σtT⊗σtT)
(⊗). (16)
The expectation in the right-hand side of the above equation equalsμ 3 (σt). Moreover,
sinceP(σit> 0 )=1, the assumptionE
(
σitσjtσht)
<+∞fori,j,h= 1 ,...,p
also implies thatE(σit)<+∞fori= 1 ,...,pand that the expectation ofxtequals
the null vector. As a direct consequence, the third moment equals the third cumulant
ofxtand this completes the proof.
5Dataanalysis
This section deals with daily percent log-returns (i.e., daily log-returns multiplied by
100) corresponding to the indices DAX30 (Germany), IBEX35 (Spain) and S&PMIB
(Italy) from 01/01/2001 to 30/11/2007. The mean vector, the covariance matrix and
the correlation matrix are
⎛
⎝
− 0. 0064
0. 030
0. 011
⎞
⎠,
⎛
⎝
1 .446 1.268 1. 559
1 .268 1.549 1. 531
1 .559 1.531 2. 399
⎞
⎠ and⎛
⎝
1 .000 0.847 0. 837
0 .847 1.000 0. 794
0 .837 0.794 1. 000
⎞
⎠, (17)
respectively. Not surprisingly, means are negligible with respect to standard deviations
and variables are positively correlated. Figure 1 shows histograms and scatterplots.