Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

3.17 Relative deviations between coefficients of upper and lower tail dependence
estimated by the non-parametric approach of Sornette & Malevergne
using ˆ β calculated for the whole sample and ˆ βSI calculated for the
tail according to three different conditions for index S&P 500 and the
nine assets. The tail represents the most extreme 4 % of data during
a smaller time interval ranging from January 1991 to December 2000
(k = 100) and during a bigger time interval ranging from July 1985 to
Mars 2008 (k = 229). * denotes zero values of tail dependence calculated
by ˆ βSI, ’NaN’ denotes 0/0, and ’inf’ denotes ·/0. . . . . . . . . . . . . . 76
3.18 Results for ˆ βSI using the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) and
the second SI condition: Y ≥ Y (k) for upper and lower tails calculated
by least squares method applied on linear additive single factor model:
X = β · Y + ε and for resulting upper and lower tail dependence coefficients
ˆ λ + and ˆ λ− with corrected ˆl (c . . .k) and uncorrected ˆl (1 . . .k).
The tail represents the most extreme 4% of data during a time interval
ranging from January 1991 to December 2000 (k = 100, c = 13). *
denotes ˆ β < 0 → λ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.19 Results for ˆ βSI using the first SI condition: Y ≥ Y (k) ∩ X ≥ X(k) and
the second SI condition: Y ≥ Y (k) for upper and lower tails calculated
by least squares method applied on linear additive single factor model:
X = β · Y + ε and for resulting upper and lower tail dependence coefficients
ˆ λ + and ˆ λ− with corrected ˆl (c . . .k) and uncorrected ˆl (1 . . .k).
The tail represents the most extreme 4% of data during a time interval
ranging from July 1985 to Mars 2008 (k = 229, c = 29). * denotes
ˆβ < 0 → λ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.20 Establishing the uncertainty of non-parametrically estimated upper and
lower tail dependence coefficients ˆ λ by creating 1000 bootstrap samples
of historical return data tables for S&P 500 index and corresponding
asset returns and calculation of quantiles, extreme values, and standard
deviations of the results. The tails represent the most extreme 4% (excluding
highest 0.5% for λc) of the return values during a time interval
from January 1991 to December 2000. ˆ βj have been calculated on the
first SI condition: Y ≥ Y (k) ∩ X ≥ X(k). ’inf’ denotes ·/0. . . . . . . . 95
3.21 Establishing the uncertainty of non-parametrically estimated upper and
lower tail dependence coefficients ˆ λ by creating 1000 bootstrap samples
of historical return data tables for S&P 500 index and corresponding
asset returns and calculation of quantiles, extreme values, and standard
deviations of the results. The tails represent the most extreme 4% (excluding
highest 0.5% for λc) of the return values during a time interval
from July 1985 to Mars 2008. ˆ βj have been calculated on the first SI
condition: Y ≥ Y (k) ∩ X ≥ X(k). ’inf’ denotes ·/0. . . . . . . . . . . . 96
3.22 Establishing the uncertainty of non-parametrically estimated upper and
lower tail dependence coefficients ˆ λ by creating 1000 bootstrap samples
of historical return data tables for S&P 500 index and corresponding
asset returns and calculation of quantiles, extreme values, and standard
deviations of the results. The tails represent the most extreme 4% (excluding
highest 0.5% for λc) of the return values during a time interval
from January 1991 to December 2000. ˆ βj have been calculated on the
second SI condition: Y ≥ Y (k). ’inf’ denotes ·/0. . . . . . . . . . . . . 97