Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

3.65 λU,m(m) for upper tails of index S&P 500 and the nine assets plotted in
green and λL,m(m) for lower tails plotted in black using a non-parametric
approach given by equation ˆ λL,m = 1 �m k j=1 1R (j)
m1≤k and R(j)
m2≤k and ˆ λU,m =
� 1 m
k j=1 1R (j)
for rolling time horizon windows of S =
m1 >m−k and R(j)
m2 >m−k
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. . . 121
3.66 λL,m(N) for lower tails of index S&P 500 and the nine assets for rolling
time horizon windows of size S = 2500 plotted in blue, S = 1600
plotted in green, and S = 800 plotted in red using a non-parametric
approach given by equation ˆ λU,m = 1 �m k j=1 1R (j)
from
m1 >m−k and R(j)
m2 >m−k
N = (2501 − S . . .S), (2502 . . .S + 1), . . .,(5736 − S + 1 . . .5736). . . . 122
3.67 λU,m(N) for lower tails of index S&P 500 and the nine assets for rolling
time horizon windows of size S = 2500 plotted in blue, S = 1600 plotted
in green, and S = 800 plotted in red a using non-parametric approach
given by equation ˆ λL,m = 1 �m k j=1 1R (j) from N = (2501 −
m1≤k and R(j)
m2≤k S . . .S), (2502 . . .S + 1), . . ., (5736 − S + 1 . . .5736). . . . . . . . . . . . 123
4.1 Non-parametric probability density distribution functions of index S&P
500, assets CVX (Chevron Corp.), and a synthetic sample estimated
using a Gaussian box kernel for a time interval of N = 2507 data points
ranging from January 1991 to December 2000. . . . . . . . . . . . . . . 149
4.2 Non-parametric probability density distribution functions of index S&P
500, assets CVX (Chevron Corp.), and a synthetic sample plotted on
a semi-log scale and estimated using a Gaussian box kernel for a time
interval of N = 2507 data points ranging from January 1991 to December
2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149