Mathematical and Statistical Methods for Actuarial Sciences and Finance

150 C. Franceschini and N. Loperfido

structure and their heavy tails make it difficult to understand its sampling properties.
On the contrary, the sign test for symmetry possesses very appealing sampling prop-
erties, when the location parameter is assumed to be known [10, page 247]. When
dealing with financial returns, it is realistic to assumed to be known and equal to zero,
for theoretical as well as for empirical reasons. From the theoretical point of view, it
prevents systematic gains or losses. From the empirical point of view, as can be seen
in (17), means of observed returns are very close to zero. The following paragraphs in
this section state model’s assumptions, describe the sign test for symmetry and apply
it to the data described in the previous section.
We shall assume the following model for ap-dimensional vector of financial
returnsxt:xt=Dtεt,E(εt)=0,Dt=diag

(

σ 1 t,...,σpt

)

and

σkt^2 =ω 0 k+

∑q

i= 1

ωikxk^2 ,t−i+

q∑+p

j=q+ 1

ωjkσk^2 ,t+q−j, (22)

where ordinary stationarity assumptions hold and{εt}is a sequence of mutually
independent random vectors. We shall test the hypotheses

H 0 ijk:Fijk( 0 )= 1 −Fijk( 0 ) versus H 1 ijk:Fijk( 0 )< 1 −Fijk( 0 ) (23)

fori,j,k= 1 , 2 ,3, whereFijkdenotes the cdf ofεt,iεt,jεt,k. Many hypothesesHaijk
fori,j,k= 1 , 2 ,3anda= 0 ,1 are equivalent to each other and can be expressed in

a simpler way. For example,H
ijj
0 andH

ijj
1 are equivalent toFi(^0 )=^1 −Fi(^0 )and
Fi( 0 )< 1 −Fi( 0 ), respectively, whereFidenotes the cdf ofεt,i. Hence it suffices
to test the following systems of hypotheses

H 0 i:Fi( 0 )= 1 −Fi( 0 ) versus H 1 i:Fi( 0 )< 1 −Fi( 0 ),i= 1 , 2 , 3 (24)

and

H 0123 :F 123 ( 0 )= 1 −F 123 ( 0 ) versus H 1123 :F 123 ( 0 )< 1 −F 123 ( 0 ). (25)

Letx^1 ,x^2 andx^3 denote the column vector of returns in the German, Spanish and
Italian markets. The sign test rejects the null hypothesisH 0 ijkif the numbernijkof
positive elements in the vectorxi◦xj◦xkis larger than an assigned value, where

“◦” denotes the Schur (or Hadamard) product. Equivalently, it rejectsH 0 ijkifzijk=
2

√

n

(

fijk− 0. 5

)

is larger than an assigned value, wherefijkis the relative frequency

of positive elements inxi◦xj◦xkandnis its length. UnderH
ijk
0 ,nijk∼Bi(n,^0.^5 )
andzijk∼N( 0 , 1 ), asymptotically.
Table 2 reports the relative frequencies of positive components inx^1 ,x^2 ,x^3 and
x^1 ◦x^2 ◦x^3 , together with the corresponding test statistics andp-values.
In all four cases, there is little evidence supporting the null hypothesis of symme-
try against the alternative hypothesis of negative asymmetry. Results are consistent
with the exploratory analysis in the previous section and motivate models describing
multivariate negative skewness.