Mathematical and Statistical Methods for Actuarial Sciences and Finance

(Nora) #1
A skewed GARCH-type model for multivariate financial time series 149

We shall measure skewness using the following indices, defined as:

A 1 =

∑n

i= 1

(

xi−x
s

) 3

, A 2 =

q 1 − 2 q 2 −q 3
q 3 −q 1

and A 3 =

3

s

(x−q 2 ), (18)

whereqiis theith quartile (i= 1 , 2 ,3). Their values for the three series are reported
in Table 1.


Ta b le 1 .Skewness coefficients
A 1 A 2 A 3
S&PMib − 0. 227 − 1. 049 − 0. 095
Ibex35 − 0. 046 − 1. 104 − 0. 081
Dax30 − 0. 120 − 1. 076 − 0. 089

All indices suggest negative skewness. In order to assess multivariate skewness,
we shall consider the third cumulant and the third moment. The third sample cumulant
is


m 3 (X)=

1

n

∑n

i= 1

(xi−m)⊗(xi−m)T⊗(xi−m)T, (19)

wherexiis the transpose of theith row of then×pdata matrixXandmis the mean
vector. The third sample cumulants of the above data are





0 .394 0.201 0.396 0.201 0.092 0.242 0.396 0.242 0. 414

0 .201 0.092 0.242 0.092 0.088 0.146 0.242 0.146 0. 216

0 .396 0.242 0.414 0.242 0.146 0.216 0.414 0.216 0. 446


⎠. (20)

The most interesting feature of the above matrix is the negative sign of all its
elements. The third moment has a similar structure, since all entries but one are
negative:





0 .422 0.173 0.340 0.173 0.025 0.199 0.340 0.199 0. 3940

0 .173 0.025 0.190 0.025 -0.053 0.035 0.199 0.035 0. 109

0 .340 0.190 0.390 0.199 0.035 0.109 0.394 0.109 0. 365


⎠. (21)

We found the same pattern in other multivariate financial time series from small
markets.


6 Sign tests for symmetry


This section deals with formal testing procedures for the hypothesis of symmetry.
When testing for symmetry, the default choice for a test statistic is the third stan-
dardised moment, which might be inappropriate for financial data. Their dependence

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