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Measuring Extreme Tail Association

Bikramjit Das, Xiangyu Liu

Abstract

Simultaneous occurrences of extreme events need not imply symmetric or reciprocal tail dependence. However, most existing measures of extremal dependence are inherently symmetric and hence often fail to capture directional influence in tail association. We introduce a rank-based measure of Extreme Tail Association (ETA) for bivariate data quantifying such directional influence of one variable on another in extreme tail regions. The proposed estimator is easily computable, consistent with its population counterpart, and asymptotically normal under mild conditions, allowing for statistical inference. We further develop a formal test for asymmetry in tail association based on a multiplier bootstrap procedure. The practical relevance of the methodology is illustrated using data on extreme price movements in major cryptocurrencies. Beyond providing a flexible tool for extremal association, the proposed framework offers a substantive argument for investigating causal relationships in extreme scenarios.

Measuring Extreme Tail Association

Abstract

Simultaneous occurrences of extreme events need not imply symmetric or reciprocal tail dependence. However, most existing measures of extremal dependence are inherently symmetric and hence often fail to capture directional influence in tail association. We introduce a rank-based measure of Extreme Tail Association (ETA) for bivariate data quantifying such directional influence of one variable on another in extreme tail regions. The proposed estimator is easily computable, consistent with its population counterpart, and asymptotically normal under mild conditions, allowing for statistical inference. We further develop a formal test for asymmetry in tail association based on a multiplier bootstrap procedure. The practical relevance of the methodology is illustrated using data on extreme price movements in major cryptocurrencies. Beyond providing a flexible tool for extremal association, the proposed framework offers a substantive argument for investigating causal relationships in extreme scenarios.
Paper Structure (24 sections, 17 theorems, 96 equations, 13 figures, 2 algorithms)

This paper contains 24 sections, 17 theorems, 96 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Extreme Tail Association Coefficient (sample)
  3. Properties of the sample ETA Coefficient
  4. Measuring association and asymmetry for extreme values
  5. Extreme Tail Association coefficient
  6. Properties of the population ETA coefficient
  7. Tail Copula and the ETA measure
  8. Examples
  9. Asymptotic behavior
  10. Consistency of $\eta_{k,n}$ and $\Delta_{k,n}$
  11. Asymptotic normality of $\eta_{k,n}$ and $\Delta_{k,n}$
  12. Testing for tail asymmetry
  13. Asymptotic behavior of the (multiplier) bootstrap estimator
  14. Algorithm for the testing procedure
  15. Data Analysis
  16. ...and 9 more sections

Key Result

Proposition 2.3

Suppose $(X,Y)\sim F$ with continuous marginal distributions $F_X$ and $F_Y$, respectively. If $\Lambda$ denotes the tail copula as in eq:tailcopup and $\eta(X|Y)$ exists, then we have the representation Furthermore, $0\le \eta(X|Y) \le 1$.

Figures (13)

  • Figure 1: Plot of $\eta(X|Y), \eta(Y|X)$ and $\Delta(X,Y)$ for \ref{['ex:gumbel']}.
  • Figure 2: Bootstrap p-values using $B=100$ multiplier bootstrap samples for simulated data from \ref{['subsec:datasimul']} for testing $\mathcal{H}_0:\Delta(X,Y)=0$ versus $\mathcal{H}_1:\Delta(X,Y)\neq 0$. The red trajectory is p-values across different choice of the intermediate sequence $k\in\mathcal{K}=\{100,110, \ldots, 500\}$. The blue line marks the reference significance level $\alpha=0.05$.
  • Figure 3: Cryptocurrency data from \ref{['subsec:datacrypto']}: Plot of $\eta_{k,n}(X|Y^{C})$ (red curve) and $\eta(Y^{C}|X)$ (blue curve) for the positive log-returns with different choices of $k\in \mathcal{K}$.
  • Figure 4: Cryptocurrency data from \ref{['subsec:datacrypto']}: Plot of $\eta_{k,n}(X^{-}|Y^{-C})$ (red curve) and $\eta_{k,n}(Y^{-C}, X^{-})$ (blue curve) for the negative log-returns with different choices of $k\in \mathcal{K}$.
  • Figure 5: Cryptocurrency data example: the p-values for testing $\mathcal{H}_0:\Delta(X,Y^C)=0$ versus $\mathcal{H}_1:\Delta(X,Y^C)\neq 0$ in the right tail (extreme positive returns). The red curve plots the p-values as a function of the intermediate sequence $k\in \mathcal{K}$, and the blue horizontal line indicates the reference significance level $\alpha=0.05$.
  • ...and 8 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Remark 1
  • Corollary 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • ...and 32 more