[Paper Review] Asymptotic Independence ex machina -- Extreme Value Theory for the Diagonal BEKK-ARCH(1) Model

Sebastian Mentemeier, Olivier Wintenberger|arXiv (Cornell University)|Jul 24, 2019

Financial Risk and Volatility Modeling22 references4 citations

TL;DR

This paper establishes that in the diagonal BEKK-ARCH(1) and related SRE models, components with distinct tail indices become asymptotically independent in extreme values despite sharing the same driving innovation. Using vector scaling regular variation, it proves that extremal dependence vanishes when tail indices differ, even under perfect dependence in the innovation structure.

ABSTRACT

We consider multivariate stationary processes $(\boldsymbol{X}_t)$ satisfying a stochastic recurrence equation of the form $$ \boldsymbol{X}_t= \mathbb{ M}_t \boldsymbol{X}_{t-1} + \boldsymbol{Q}_t,$$ where $(\boldsymbol{Q}_t)$ are iid random vectors and $$ \mathbb{M}_t=\mathrm{Diag}(b_1+c_1 M_t, \dots, b_d+c_d M_t) $$ are iid diagonal matrices and $(M_t)$ are iid random variables. We obtain a full characterization of the multivariate regular variation properties of $(\boldsymbol{X}_t)$, proving that coordinates $X_{t,i}$ and $X_{t,j}$ are asymptotically independent even though all coordinates rely on the same random input $(M_t)$. We describe extremal properties of $(\boldsymbol{X}_t)$ in the framework of vector scaling regular variation. Our results are applied to some multivariate autoregressive conditional heteroskedasticity (BEKK-ARCH and CCC-GARCH) processes.

Motivation & Objective

• To characterize the multivariate regular variation properties of diagonal stochastic recurrence equations (SREs) with common driving innovation.
• To investigate the joint extremal behavior of components in multivariate GARCH-type models where marginal tail indices differ.
• To resolve the paradox that components relying on the same innovation can exhibit asymptotic independence in extremes.
• To extend vector scaling regular variation (VSRV) to stationary Markov chains for modeling non-standard extremal dependence.
• To provide a theoretical foundation for modeling heterogeneous tail behavior in financial time series with distinct risk profiles during crises.

Proposed method

• Analyzes the diagonal SRE model $\boldsymbol{X}_t = \mathbb{M}_t \boldsymbol{X}_{t-1} + \boldsymbol{Q}_t$, where $\mathbb{M}_t = \mathrm{Diag}(b_1 + c_1 M_t, \dots, b_d + c_d M_t)$ and $M_t$ i.i.d.
• Applies the Kesten-Goldie theorem to derive marginal tail indices $\alpha_i$ satisfying $\mathbb{E}[|b_i + c_i M_0|^{\alpha_i}] = 1$.
• Uses vector scaling regular variation (VSRV) to describe joint tail behavior, replacing non-standard regular variation for Pareto-like marginals.
• Derives the spectral tail process $\widetilde{\boldsymbol{\Theta}}_t$ via recursive dynamics $\widetilde{\boldsymbol{\Theta}}_t = \mathbb{M}_t \widetilde{\boldsymbol{\Theta}}_{t-1}$ under the $\alpha_1$-tilted measure.
• Employs Hoeffding's lemma and moment generating function bounds to prove exponential decay of martingale-like sequences in the extremal regime.
• Establishes asymptotic independence by showing $\mathbb{P}(X_{0,1} > x^{1/\alpha_1} \mid X_{0,2} > x^{1/\alpha_2}) \to 0$ as $x \to \infty$ when $\alpha_1 > \alpha_2$.

Experimental results

Research questions

• RQ1 Can components of a multivariate SRE model with identical driving innovation exhibit asymptotic independence in extreme values?
• RQ2 How does the joint extremal behavior of diagonal BEKK-ARCH(1) processes depend on the relative values of $b_i, c_i$ and the tail index $\alpha_i$?
• RQ3 What is the role of vector scaling regular variation (VSRV) in characterizing extremal dependence when marginal tail indices differ?
• RQ4 Under what conditions does the spectral vector $\widetilde{\boldsymbol{\Theta}}_0$ concentrate on the axes, indicating asymptotic independence?
• RQ5 How can the spectral tail process be characterized for SREs with non-i.i.d. diagonal coefficients and distinct tail indices?

Key findings

• When $c_2/c_1 > b_2/b_1 \geq 1$ and $c_1 > 0$, the components $X_{0,1}$ and $X_{0,2}$ are asymptotically independent, with $\lim_{x\to\infty} \mathbb{P}(X_{0,1} > x^{1/\alpha_1} \mid X_{0,2} > x^{1/\alpha_2}) = 0$.
• The spectral vector $\widetilde{\boldsymbol{\Theta}}_0$ almost surely lies in $\{(1,0), (0,1)\}$, confirming asymptotic independence in the bivariate case.
• Even with identical $Q_{1,t} = Q_{2,t}$, the components remain asymptotically independent due to distinct tail indices $\alpha_1 \neq \alpha_2$.
• The model exhibits vector scaling regular variation (VSRV), allowing a non-standard extremal dependence structure despite identical innovation input.
• When tail indices are equal ($\alpha_i = \alpha_j$), the spectral vector may have non-degenerate support, indicating possible extremal dependence.
• The proof relies on bounding the moment generating function of $\Delta W_k = \log((b_2 + c_2 M_k)/(b_1 + c_1 M_k))$ under the $\alpha_1$-tilted measure, showing decay to zero.