Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

3.51 ˆ λ + (k) for the lower tail of the nine assets and the index S&P 500 is
plotted in black for ˆ β(k) calculated by least squares method applied to
linear additive single factor model: X = β·Y +ε and adapted to the first
condition: Y ≥ Y (k) ∩ X ≥ X(k), ˆν(k = 4%) and ˆ l(k) in dependence
of threshold k for k/N = 3% . . .10%, in blue for ˆ β(k) and ˆ l(k) adapted
to the first SI condition for comparison, in green for Reference ˆ β and
ˆl(k), and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index
return vector of S&P 500 and X denotes asset return vector of the nine
assets for a time interval ranging from January 1991 to December 2000. 91
3.52 ˆ λ + (k) for the upper tail of the nine assets and the index S&P 500 is
plotted in black for ˆ β(k) calculated by least squares method applied to
linear additive single factor model: X = β · Y + ε and adapted to the
second condition: Y ≥ Y (k), ˆν(k = 4%) and ˆ l(k) in dependence of
threshold k for k/N = 3% . . .10%, in blue for ˆ β(k) and ˆ l(k) adapted
to the first SI condition for comparison, in green for Reference ˆ β and
ˆl(k), and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index
return vector of S&P 500 and X denotes asset return vector of the nine
assets for a time interval ranging from January 1991 to December 2000. 92
3.53 ˆ λ + (k) for the lower tail of the nine assets and the index S&P 500 is
plotted in black for ˆ β(k) calculated by least squares method applied to
linear additive single factor model: X = β · Y + ε and adapted to the
second condition: Y ≥ Y (k), ˆν(k = 4%) and ˆ l(k) in dependence of
threshold k for k/N = 3% . . .10%, in blue for ˆ β(k) and ˆ l(k) adapted
to the first SI condition for comparison, in green for Reference ˆ β and
ˆl(k), and for Reference ˆ λ + (k = 4%) is given in red. Y denotes the index
return vector of S&P 500 and X denotes asset return vector of the nine
assets for a time interval ranging from January 1991 to December 2000. 93
3.54 ˆ βSI(N) plotted in black for the upper tails and plotted in green for the
lower tails for 9 assets given by X and index S&P 500 given by Y using
the linear single factor model: X = β · Y + ε and the first SI condition:
Y ≥ Y (k) ∩ X ≥ X(k) 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. . . . . 100
3.55 ˆ βSI(N) plotted in black for the upper tails and plotted in green for the
lower tails for 9 assets given by X and index S&P 500 given by Y using
the linear single factor model: X = β·Y +ε and the second SI condition:
Y ≥ Y (k) 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. . . . . . . . . . . . . . 101
3.56 λ + (N) for upper tails of index S&P 500 and the nine assets plotted
in blue and λ − (N) for lower tails plotted in red using non-parametric
approach given by equation ˆ λ +,− = 1/ max
�
1,
l
ˆβ +,−
SI
� ˆν
with coefficients
ˆν, l, and ˆ β +,−
SI 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. ˆ β +,−
SI were calculated
using the linear single factor model: X = β · Y + ε and the first SI
condition: Y ≥ Y (k) ∩ X ≥ X(k). . . . . . . . . . . . . . . . . . . . . . 102