Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

and finally I obtained:
y(r) =
� � 1
α 1
, (3.50)
1 − r
which allowed me to transform our uniformly distributed data into data distributed
according to relation (3.48) with α = 3.
Putting now the transformed values into equation (3.47) to calculate Hill’s estimator,
I could test whether the output was a tail index ˆν close enough to three representing
the true value α of the tail index for our transformed data.
Conducting this procedure several times allowed me to check whether the standard
deviation fulfills the above described relation of:
std(ˆν)
α ∼ = 1
√ k , (3.51)
which, given α = 3 and n = 100, says that std(ˆν) ∼ = 0.3 and therefore all ˆν should be
within an interval of 3 ±2∗0.3 if the error follows a Gaussian distribution as requested.
I performed the above described sampling 10’000 times and received a mean value of
mean(ˆν) = 3.06, which shows a small bias of 2%, and a 95% deviation-quantile of 0.63,
which is close to the above calculated 0.6. Therefore I conclude that our smaller sample
size with about 100 return data points perceived as tail data already agrees well with
the general assumption of Hill’s estimator to be asymptotically normally distributed.
However, performing this little affirmation we stressed the assumption of using independent
and identically distributed (i.i.d.) data samples, which unfortunately is not
the case for real financial data. Applying Engle’s test for the presence of ARCH effects
[17] (1982) to our real financial time series the result shows significant evidence in
support of GARCH effects (heteroscedasticity) 4 . As shown by Kearns and Pagan [22]
(1997) for heteroskedastic time series the variance of the estimated tail index can be
seven times larger than the variance given by the asymptotic normality assumption.
For comparison I also implemented the OLS (ordinary least squares) log-log rank-size
tail index estimate proposed by Gabaix [23] (2006) defined by:
log(Rank − 1/2) = a − ˆb γ n · log(Size), (3.52)
with the tail index given by ˆb γ n . A standard error of the OLS estimate ˆb γ � n of the slope
2
was estimated to n ˆb γ n in the paper. Assuming a tail index equal to three as above,
this yields a standard error of 0.42, which is higher compared to Hill’s estimator.
We can now perform the same little check as above to see how Gabaix’s ˆb γ n estimator
performs for i.i.d. data samples. Sampling 10’000 times, I obtained a mean value of
mean( ˆb γ n) = 3.015, which only constitutes 0.5% bias and a 95% deviation-quantile of
0.83. This perfectly agrees with the estimated standard error given in the paper [23]
(2006) but still is a little higher than the standard deviation of Hill’s estimator.
Now we can apply both, Hill’s estimator ˆν and Gabaix’s ˆb γ n estimator, on real data
for comparison and subsequently perform bootstrap sampling with replacement (Monte
Carlo simulation) to find out, whether the fact, that real financial time series exhibit
internal dependence, in terms of time varying volatility clustering (large price changes
come in clusters), impact deviation quantiles or in other words: whether Hill’s estimator
4 Engle’s ARCH test is described in subsection (3.2.3)
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