Componentwise different tail solutions for bivariate stochastic recurrence equations with application to ${\rm GARCH}(1,1)$ processes
Volume 155 / 2019
Colloquium Mathematicum 155 (2019), 227-254
MSC: Primary 60G70, 62M10; Secondary 60H25, 91B84.
DOI: 10.4064/cm7313A-5-2018
Published online: 7 December 2018
Abstract
We study bivariate stochastic recurrence equations (SREs) motivated by applications to ${\rm GARCH}(1,1)$ processes. If the coefficient matrices of the SREs have strictly positive entries, then Kesten’s result applies and it gives solutions with regularly varying tails. Moreover, the tail indices are the same for all coordinates. However, for applications, this framework is too restrictive. We study SREs whose coefficients are triangular matrices and prove that the coordinates of the solution may exhibit regularly varying tails with different indices. We also specify each tail index together with its constant. The results are used to characterize regular variation of bivariate stationary $\mathrm {GARCH}(1,1)$ processes.
