A skewed GARCH-type model for multivariate financial time series 145HenceE(
ZiZjZk)
is in theith row and in thejth column of thekth matrixMk.
There is a simple relation between the third central moment and the first, second and
third moments of a random vector [9, page 187]:
Proposition 5.Letμ=μ 1 (z),μ 2 (z),μ 3 (z)the first, second and third moments
of the randomvector z. Then the third central moment of z can be represented by
μ 3 (z)−μ 2 (z)⊗μT−μT⊗μ 2 (z)−μ
[
μV 2 (z)]T
- 2 μ⊗μT⊗μT,where AV
denotes the vector obtained by stacking the columns of the matrix A on top of each
other.
3 The multivariate skew-normal distribution
We denote byz∼SNp( ,α)a multivariate skew-normal random vector [2] with
scale parameter and shape parameterα. Its probability density function is
f(z; ,α)= 2 φp(z; ) (
αTz)
, z,α∈ Rp, ∈ Rp×Rp, (1)where (·)is the cumulative distribution function of a standard normal variable and
φp(z; )is the probability density function of ap-dimensional normal distribution
with mean 0pand correlation matrix .Expectation, variance and third central mo-
ment ofz∼SNp( ,α)(i.e., its first three cumulants) have a simple analytical form
[1, 8]:
E(z)=√
2
πδ, V(z)= −2
πδδT, μ 3 (z)=√
2
π(
4
π− 1
)
δ⊗δT⊗δT,(2)whereδ= α/
√
1 +αT α. As a direct consequence, the third cumulant ofz∼
SNp( ,α)needs onlypparameters to be identified, and is a matrix with negative
(null) entries if and only if all components ofδare negative (null) too.
The second and third moments ofz∼SNp( ,α)have a simple analytical form
too:
μ 2 (z)=E
(
zzT)
= , (3)
μ 3 (z)=√
2
π[
δT⊗ +δ(
V
)T
+ ⊗δT−δ⊗δT⊗δT]
. (4)
Expectation ofzzTdepends on the scale matrix only, due to the invariance prop-
erty of skew-normal distributions: ifz∼SNp( ,α)thenzzT∼W( , 1 ), i.e., a
Wishart distribution depending on the matrix only [11]. As a direct consequence,
the distribution of a functiong(·)ofzsatisfyingg(z)=g(−z)does not depend
either onαor onδ.
The probability density function of theith componentziofz∼SNp( ,α)is
f(zi)= 2 φ(zi) ⎛
⎝√δizi
1 −δ^2 i