Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

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γ b n ν(k) 4.2 4 3.8 3.6 3.4 3.2 3 S&P 500 BMY CVX HPQ KO MMM PG SGP TXN WAG 2.8 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 k/N Figure 3.2: Tail index estimators ˆ b γ n (Gabaix’s estimator) for the upper tails of index S&P 500 and 9 major assets included plotted in dependence of threshold k on a time interval ranging from July 1985 to April 2008. 3.5 3 2.5 S&P 500 BMY CVX HPQ KO MMM PG SGP TXN WAG 2 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 k/N Figure 3.3: Tail index estimators ˆν (Hill’s estimator) for the lower tails of index S&P 500 and 9 major assets included plotted in dependence of threshold k on a time interval ranging from July 1985 to April 2008. 26
γ b n 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 S&P 500 BMY CVX HPQ KO MMM PG SGP TXN WAG 1.4 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 k/N Figure 3.4: Tail index estimators ˆ b γ n (Gabaix’s estimator) for the lower tails of index S&P 500 and 9 major assets included plotted in dependence of threshold k on a time interval ranging from July 1985 to April 2008. lower tail. Furthermore, it is shown in [23] (2006) that although the standard error of the Hill’s estimator is less than that of the tail index estimate ˆb γ n, its (small sample) bias in the case of deviation from the power law is (typically) greater than that of ˆb γ n . Perhaps, we indeed under-estimate the lower tail such that the exponent is probably smaller than given by Hill’s estimator. To check whether our assumption of equal tail indexes of asset returns, residues, and index returns is accurate with our estimation methods and data sets, I performed a simple test of means at the 95% confidence level for any given quantile. Values that reject the equality hypothesis are asterisked. As we can see, the results were inconclusive. Anyway, the performed tests were based on the assumption of asymptotic normality in the distribution of the estimates with the standard deviations of � 2/n · ˆb γ n and ˆν/√ k, which might be too restrictive. The true distribution is probably ’wilder’ than Gaussian as explained above. This would expand the confidence interval. It is nevertheless interesting to notice that in the negative tails of ˆ b γ n , although the equality hypothesis was rejected for all assets and residues because estimates for assets and residues were unexeptionally lower than for the S&P 500 index, tail index estimates of asset returns are nearly identical to tail indexes of their corresponding residues obtained by regression on the index S&P 500. As we can obtain in the upper parts of the tables (3.6) and (3.7) where the smaller samples with 100 data points considered as tail data are shown, 95% quantiles of the upper tails show good accordance to our i.i.d. estimates, whereas 95% quantiles of lower tails tend to be somewhat higher. In terms of deviation quantiles Hill’s estimator still outperforms Gabaix’s ˆ b γ n estimator. What we can observe in the lower parts of the tables is that deviation quantiles become obviously smaller when we augment sample sizes. This counts for both estimators. 27
Page 1 and 2: Eidgenössische Technische Hochschu
Page 3 and 4: Abstract Recent events i.e. severe
Page 5 and 6: Appendix 160 M-Files . . . . . . .
Page 7 and 8: 3.12 β(N) estimated for 9 assets X
Page 9 and 10: 3.30 Fraction of scale factors �
Page 11 and 12: 3.46 ˆ βSI(k) calculated by the f
Page 13 and 14: 3.57 λ + (N) for upper tail of ind
Page 15 and 16: List of Tables 3.1 Estimated values
Page 17 and 18: 3.11 Estimated values of upper and
Page 19 and 20: 3.23 Establishing the uncertainty o
Page 21 and 22: 4.10 Estimated upper and lower tail
Page 23 and 24: 5.1 Descriptive statistics of histo
Page 25 and 26: 1.2 Thesis Outline The study of ext
Page 27 and 28: 2.1 Multivariate Extreme Value Dist
Page 29 and 30: and X is positively upper orthant d
Page 31 and 32: We refer to Appendix B of [3] (2007
Page 33 and 34: Chapter 3 Concepts for the Estimati
Page 35 and 36: or expressed through copula C: 1
Page 37 and 38: Parametric Joint-Tail Models Depend
Page 39 and 40: First of all we notice that only fo
Page 41 and 42: 19 m=2507 bs=1000 upper tail lower
Page 43 and 44: 3.2 Approaches according to Sornett
Page 45 and 46: is equal to the weakest tail depend
Page 47: ν(k) 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2
Page 53 and 54: 3.2.2 Implementation of the Non-Par
Page 55 and 56: l(k) mean,c 4.5 4 3.5 3 2.5 2 1.5 1
Page 57 and 58: Non-Par Par up lo up lo up lo up lo
Page 59 and 60: λ(k) λ(k) mean 0.14 0.12 0.1 0.08
Page 61 and 62: coefficient estimates. For the bigg
Page 65 and 66: β β β 1 0.9 0.8 0.7 0.6 BMY 0.5
Page 67 and 68: l l l 2.6 2.4 2.2 2 1.8 BMY 1.6 100
Page 69 and 70: ν S&P 500 8 7 6 5 4 3 2 1 0 1000 2
Page 71 and 72: 49 λ λ λ 0.2 0.15 0.1 0.05 BMY 0
Page 73 and 74: 51 λ λ λ 0.12 0.1 0.08 0.06 0.04
Page 75 and 76: Error Bars in Dependence of Time To
Page 77 and 78: Return 0.1 0.05 0 −0.05 −0.1
Page 79 and 80: 3.2.4 Implementation of the Paramet
Page 81 and 82: (C ε /C Y ) 1/α (C ε,mean /C Y,m
Page 83 and 84: F par ,F emp F par ,F emp 61 F par
Page 87 and 88: To summarize for upper tails: tail
Page 89 and 90: λ(k) mean λ(k) mean,c 0.1 0.09 0.
Page 97 and 98: 3.3 β-Smile Improvement As an addi
Page 99 and 100:
77 β β + SI β − SI λ + λ + S
Page 101 and 102:
I also implemented the β-smile imp
Page 103 and 104:
81 β, β SI β, β SI β, β SI 1.
Page 105 and 106:
β, β SI 83 β, β SI β, β SI 1
Page 107 and 108:
85 β SI , β β SI , β β SI , β
Page 109 and 110:
β SI , β 87 β SI , β β SI , β
Page 111 and 112:
As an additional information, ˆν
Page 113 and 114:
91 λ λ λ 0.06 0.05 0.04 0.03 0.0
Page 115 and 116:
93 λ λ λ 0.2 0.15 0.1 0.05 BMY 0
Page 117 and 118:
95 constr 1 m=2507 bs=1000 upper ta
Page 119 and 120:
97 constr 2 m=2507 bs=1000 upper ta
Page 121 and 122:
Observation of λ by Rolling Time W
Page 123 and 124:
101 β β β 1.5 1 0.5 0 −0.5 BMY
Page 125 and 126:
103 λ λ λ 0.2 0.15 0.1 0.05 BMY
Page 127 and 128:
and the corresponding estimator for
Page 129 and 130:
ˆλU,m ˆλL,m ˆλ EV T U,m ˆλ
Page 131 and 132:
EVT λ , λ U,m U,m EVT λ , λ U,m
Page 133 and 134:
Estimation of optimal Threshold k T
Page 135 and 136:
λ BMY λ KO 113 λ SGP 0.4 0.3 0.2
Page 137 and 138:
The right hand side of figure (3.62
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117 m=5736 bs=800 upper tail lower
Page 141 and 142:
119 λ λ λ 0.35 0.3 0.25 0.2 0.15
Page 143 and 144:
121 λ λ λ 0.5 0.4 0.3 0.2 0.1 BM
Page 145 and 146:
123 λ λ λ 0.5 0.4 0.3 0.2 0.1 BM
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sometimes were significantly differ
Page 149 and 150:
Chapter 4 Application of Concepts t
Page 151 and 152:
with the index can have much strong
Page 153 and 154:
Index Asset Abbrev. AORD Australian
Page 155 and 156:
Index Asset Abbrev. MERVAL Acindar-
Page 157 and 158:
C. W. β Non-Par Par up lo up lo up
Page 159 and 160:
C. W. β N-Par Par up lo up lo up l
Page 161 and 162:
C. W. β N-Par Par up lo up lo up l
Page 163 and 164:
141 ˆλ + SI1 95% Q 90% Q ˆ λ +
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exchange rate of the US$ and the Sw
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A Eur GBP BrR CHFR CHY indoRu indRu
Page 169 and 170:
4.3 Application to Synthetic Time S
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Density Density 60 50 40 30 20 10 F
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151 βorig ˆβSI 2 bias rel. bias
Page 175 and 176:
153 βorig ˆλ +,− (α,βorig) +
Page 177 and 178:
155 βorig ˆλ +,− EV T (ˆν,β
Page 179 and 180:
synthetic time series we detected a
Page 181 and 182:
[17] Engle, R.F. (1982). Autoregres
Page 183 and 184:
Approach according to Poon, Rocking
Page 185 and 186:
Non-Parametric Approach according t
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%estimate lambda% for j=1:n-1; if(b
Page 189 and 190:
Approaches according to Schmidt & S
Page 191 and 192:
Composition of Synthetic Data Set c
Page 193 and 194:
Descriptive Statistics The kurtosis
Page 195 and 196:
N Mean Std Skew. Kurt. Statistic St
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Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

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