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)