A skewed GARCH-type model for multivariate financial time series 149We shall measure skewness using the following indices, defined as:A 1 =
∑ni= 1(
xi−x
s) 3
, A 2 =
q 1 − 2 q 2 −q 3
q 3 −q 1and 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. 089All 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∑ni= 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