Mathematical and Statistical Methods for Actuarial Sciences and Finance

146 C. Franceschini and N. Loperfido

whereδiis theith component ofδandφdenotes the pdf of a standard normal
distribution. The third moment of the corresponding standardised distribution is

√

2 ( 4 −π)

⎛

⎝√ δi

π− 2 δi^2

⎞

⎠

3

. (6)

Hence positive (negative) values ofδilead to positive (negative) skewness. More-
over, positive values ofδilead toFi( 0 )> 1 −Fi( 0 )whenx>0, withFidenoting
the cdf ofzi.

4 A skewed GARCH-type model

In order to describe skewness using a limited number of parameters, we shall introduce
the following model for ap-dimensional vector of financial returnsxt:

xt=Dtεt,εt=zt−E(zt), zt∼SNp(,α), Dt=diag

(

σ 1 t,...,σpt

)

(7)

σkt^2 =ω 0 k+

∑q

i= 1

ωikxk^2 ,t−i+

q∑+p

j=q+ 1

ωjkσk^2 ,t+q−j, (8)

where ordinary stationarity assumptions hold and{zt}is a sequence of mutually
independent random vectors.
The following proposition gives the analytical moment of the third cumulant
μ 3 (xt)of a vectorxtbelonging to the above stochastic process. In particular it shows
thatμ 3 (xt)is negative (null) when all the elements in the vectorδare negative (null)
too.

Proposition 6.Let{xt,t∈Z}be a stochastic process satisfying (10), (11) and
E

(

σitσjtσht

)

<+∞for i,j,h= 1 ,...,p. Then

μ 3 (xt)=μ 3 (xt)=

√

2

π

(

4

π

− 1

)

μ 3 (σt)(⊗), (9)

where=diag

(

δ 1 ,...,δp

)

andσt=

(

σ 1 t,...,σpt

)T

.

Proof.We shall writeμ 3 (y|w)andμ 3 (y|w)to denote the third moment and the
third cumulant of the random vectory, conditionally on the random vectorw.From
the definition of{xt,t∈Z}we have the following identities:

μ 3 (xt|σt)=μ 3 {Dt[zt−E(zt)]|σt}=μ 3 (Dtzt|σt). (10)

Apply now linear properties of the third cumulant:

μ 3 (xt|σt)=Dtμ 3 (zt)(Dt⊗Dt). (11)