Mathematical and Statistical Methods for Actuarial Sciences and Finance

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A skewed GARCH-type model for multivariate financial time series 145

HenceE

(

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

π

δ, V(z)= −

2

π

δδT, 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


⎠, (5)
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