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

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.