Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model - Sebastian Mentemeier; Olivier Wintenberger (Journal of Time Series Analysis, 2022)
We consider multivariate stationary processes (Xt) satisfying a stochastic recurrence equation of the form Xt=𝕄tXt-1+Qt, where (Qt) are i.i.d. random vectors and 𝕄t=Diag(b1+c1Mt,…,bd+cdMt) are i.i.d. diagonal matrices and (Mt) are i.i.d. random variables. We obtain a full characterization of the vector scaling regular variation properties of (Xt), proving that some coordinates Xt,0.167emi and Xt,0.167emj are asymptotically independent even though all coordinates rely on the same random input (Mt). We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK-ARCH and CCC-GARCH) processes and to log-returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.
Keywords: 519; Stochastic recurrence equations; multivariate ARCH; multivariate regular variation; non-standard regular variation