The name is absent

ln∕_t — In ∕⅛-ι, where I_t indicates the price at time t, obtaining a sample of T = 854
returns. Table 2 reports preliminary statistics for the four return series, and gives the
mean, median, minimum, maximum, standard deviation and skewness and kurtosis
indices.

The nonparametric densities with normal kernel of the distributions of the returns is
given in hg. I². From the latter hgure, also if the distributions have fat tails, it is difficult
to identify their shape. Moreover, since multivariate skewness and curtosis indices are
equal, respectively, to 2.246 and 61.523, the data are not normally distributed.

Finally, if we express the depedence structure of the four indices with the correlation
index, we obtain the following matrix;

Note the high dependence of SfePCOMP with NASCOMP and MSACWFL.

Table 2: Summary statistics for weekly returns.

4.2 Parameter estimation and results

We have adapted to the four series of returns the KS distribution with marginals skew
Student t, generalized secant hyperbolic and exponential power seen in the previous
section. According to the IFM (inference for margins) method (Joe and Xu, 1996), the
parameters of the marginals have been estimated distinctly from the parameters of the
KS function. In other words, the process of estimation has been divided in the following
steps:

i) estimating the location, scale, skewness and kurtosis parameters of marginal dis-
tributions using the maximum likelihood method:

ii) estimating the KS parameter ɑ, always with the maximum likelihood method,
given the estimations performed in step (i).

²The bandwidth of the nonparametric densities has been obtained with the Sheather and Jones method
(Sheater and Jones, 1991).

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