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

Abstract

We consider multivariate stationary processes $({\mathbf{X}}_t)$ satisfying a stochastic recurrence equation of the form $${\mathbf{X}}_t={\mathbb{M}}_t{\mathbf{X}}_{t-1}+{\mathbf{Q}}_t,$$ where $({\mathbf{Q}}_t)$ 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 $({\mathbf{X}}_t)$, 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 $({\mathbf{X}}_t)$ in the framework of vector scaling regular variation. Our results are applied to some multivariate autoregressive conditional heteroskedasticity (BEKK-ARCH and CCC-GARCH) processes.

References in corpus (1)

• Tail indices for AX+B recursion with triangular matrices

Cited by in corpus (1)

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