Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

3.20 λU(N) for the upper tails of index S&P 500 and the nine assets plotted
in blue and λL(N) for the lower tails plotted in red using non-parametric
approach given by equation ˆ λ +,− �
= 1/ max 1, l
�ˆν with coefficients ˆν,
β
l and β for rolling time horizon windows of S = 2500 considered data
points from N = (1 . . .S), (2 . . .S + 1), . . ., (5736 − S + 1 . . .5736) or a
total time interval from July 1985 to Mars 2008. . . . . . . . . . . . . . 48
3.21 λU(N) for the upper tails of index S&P 500 and the nine assets plotted
in blue and λL(N) for the lower tails plotted in red using non-parametric
approach given by equation ˆ λ +,− �
= 1/ max 1, l
�ˆν with coefficients ˆν,
β
l and β for rolling time horizon windows of S = 1600 considered data
points from N = (1 . . .S), (2 . . .S + 1), . . ., (5736 − S + 1 . . .5736) or a
total time interval from July 1985 to Mars 2008. . . . . . . . . . . . . . 49
3.22 λU(N) for the upper tails of index S&P 500 and the nine assets plotted
in blue and λL(N) for the lower tails plotted in red using non-parametric
approach given by equation ˆ λ +,− �
= 1/ max 1, l
�ˆν with coefficients ˆν,
β
l and β for rolling time horizon windows of S = 800 considered data
points from N = (1 . . .S), (2 . . .S + 1), . . ., (5736 − S + 1 . . .5736) or a
total time interval from July 1985 to Mars 2008. . . . . . . . . . . . . . 50
3.23 λU(N) for the upper tails of index S&P 500 and the nine assets for rolling
time horizon windows of size S = 2500 plotted in black, S = 1600 plotted
in blue, and S = 800 plotted in red using non-parametric approach given
by equation ˆ λ +,− �
= 1/ max 1, l
�ˆν with coefficients ˆν, l and β from
β
N = (2501 − S . . .S), (2502 . . .S + 1), . . .,(5736 − S + 1 . . .5736). . . . 51
3.24 λL(N) for the lower tails of index S&P 500 and the nine assets for rolling
time horizon windows of size S = 2500 plotted in black, S = 1600 plotted
in blue, and S = 800 plotted in red using non-parametric approach given
by equation ˆ λ +,− �
= 1/ max 1, l
�ˆν with coefficients ˆν, l and β from
β
N = (2501 − S . . .S), (2502 . . .S + 1), . . .,(5736 − S + 1 . . .5736). . . . 52
3.25 Return data for index S&P 500 ranging from January 1950 to April 2008 55
3.26 Autocorrelation for a return series on the left and for a squared return
series on the right of index S&P 500 ranging from January 1950 to April
2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.27 Scale factors ĈY (k) = k
N · (Yk,N) ˆν of the index S&P 500 and Ĉε(k) =
k
N · (εk,N) ˆν of the nine assets’ residues plotted for the upper tails of the
bigger data set ranging from July 1985 to April 2008 with k=1, 2,...,
458 (k/N = 0% . . .8%). . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.28 Scale factors ĈY (k) = k bˆ · (Yk,N)
N γ n of the index S&P 500 and Ĉε(k) =
k bˆ · (εk,N)
N γ n of the nine assets’ residues plotted for the upper tails of the
bigger data set ranging from July 1985 to April 2008 with k=1, 2,...,
458 (k/N = 0% . . .8%). . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.29 Fraction of scale factors
� Ĉε(k)
ĈY (k)
� 1/α
= εk,N
Yk,N
of the nine assets’ residues
and the index S&P 500, plotted for the upper tails of the bigger data
set ranging from July 1985 to April 2008 with k=1, 2,..., 458 (k/N =
0% . . .8%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59