Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

3.39 Empirical complementary cumulative distribution functions F εi,empirical =
k plotted in red and parametric complementary cumulative distribu-
N
tion functions F εi,parametric = Ĉε, c · ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for
c = 0.005 · N plotted in black for k/N = 4% . . .0% (from left to right)
in dependence of returns for the lower tails of the nine assets for a time
interval ranging from January 1991 to December 2000. . . . . . . . . . 72
3.40 Empirical complementary cumulative distribution functions F εi,empirical =
k plotted in red and parametric complementary cumulative distribution
N
functions F ε,parametric = Ĉε · ε−ˆν with Ĉε = 1 �k k j=1 Ĉε plotted in black
for k/N = 4% . . .0% in dependence of returns for the lower tails of the
nine assets for a time interval ranging from January 1991 to December
2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.41 Empirical complementary cumulative distribution functions F εi,empirical =
k plotted in red and parametric complementary cumulative distribu-
N
tion functions F εi,parametric = Ĉε, c · ε−ˆν with Ĉε,c = 1 �k k−c j=c Ĉε for
c = 0.005 · N plotted in black for k/N = 4% . . .0% in dependence of
returns for the lower tails of the nine assets for a time interval ranging
from January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . 74
3.42 ˆ βSI(k) of the upper tails plotted in blue and of the lower tails plotted in
black calculated by least square method applied on linear additive single
factor model: X = β·Y +ε and first SI condition: Y ≥ Y (k)∩X ≥ X(k)
in dependence of threshold k. Reference ˆ β calculated for all data is given
in green. Y denotes the index return vector of S&P 500 and Y denotes
asset return vector of the nine assets for a time interval ranging from
January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . . . . 80
3.43 ˆ βSI(k) of the upper tails plotted in blue and of the lower tails plotted in
black calculated by least square method applied on linear additive single
factor model: X = β · Y + ε and second SI condition: Y ≥ Y (k) in
dependence of threshold k. Reference ˆ β calculated for all data is given
in green. Y denotes the index return vector of S&P 500 and Y denotes
asset return vector of the nine assets for a time interval ranging from
January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . . . . 81
3.44 ˆ βSI(k) of the upper tails plotted in blue and of the lower tails plotted in
black calculated by least square method applied on linear additive single
factor model: X = β · Y + ε and third SI condition: X ≥ X(k) in
dependence of threshold k. Reference ˆ β calculated for all data is given
in green. Y denotes the index return vector of S&P 500 and Y denotes
asset return vector of the nine assets for a time interval ranging from
January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . . . . 82
3.45 ˆ βSI(k) of the upper tails plotted in blue and of the lower tails plotted in
black calculated by least square method applied on linear additive single
factor model: X = β · Y + ε and fourth SI condition: Y ≥ Y (k) ∪ X ≥
X(k) in dependence of threshold k. Reference ˆ β calculated for all data
is given in green. Y denotes the index return vector of S&P 500 and Y
denotes asset return vector of the nine assets for a time interval ranging
from January 1991 to December 2000. . . . . . . . . . . . . . . . . . . . 83