Componentwise different tail solutions for bivariate stochastic recurrence equations with application to ${\rm GARCH}(1,1)$ processes

Abstract

We study bivariate stochastic recurrence equations (SREs) motivated by applications to GARCH(1, 1) processes. If coefficient matrices of SREs have strictly positive entries, then the Kesten 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 when 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 variations of bivariate stationary GARCH(1, 1) processes.

Tail indices for $$\mathbf{A}\mathbf{X}+\mathbf{B}$$ Recursion with Triangular Matrices

“…Here and in what follows, the notation '∼' means that the quotient of the left-and right-hand sides tends to 1 as x → ∞. For d = 2, the result (1.3) was proved in [10] with indices α 1 = min(α 1 , α 2 ) and α 2 = α 2 . The dependency of α 1 on α 1 , α 2 comes from the SRE: W 1,t = A 11,t W 1,t−1 + A 12,t W 2,t−1 + B 1,t where W 1,t is influenced by W 2,t .…”

“…The tail of multivariate GARCH( p, q) has been investigated in [13] but with the setting of Goldie's condition. A bivariate GARCH(1, 1) series with a triangular setting has been studied in [23] and [10]. Particularly in [10], detailed analysis was presented including exact tail behaviors of both price and volatility processes.…”

Affine stochastic equation with triangular matrices

“…1 The statement in [10] is much more general than what we need here and the proof is quite advanced. If there is ε > 0 such that Ea ε < 1 and E(|y| ε + |b1| ε + |b2| ε ) < ∞, then negativity of the Lapunov exponent follows quite easily, see [23], Proposition 7.4.5 and e.g [9]. Finiteness of the above moments is assumed here anyway, see (2.6), (2.7) and (2.12) Let Π n = a 1 · · · a n and…”

“…Our results apply to the squared volatility sequence W t = (σ 2 1,t , σ 2 2,t ) of the bivariate GARCH(1,1) financial model, see [7], Section 4.4.5 and [9]. Then W t satisfies (1.1) with matrices A t having non-negative entries.…”

“…If all the entries of A t are strictly positive then the theorem of Kesten applies and both σ 2 1,t and σ 2 2,t are regularly varying with the same index, see [17], [19]. But if this is not the case then we have to go beyond Kesten's approach as it is done in [9] or in the present paper. From the point of view of applications it is reasonable to relax the assumptions on A t because it allows us to capture a larger class of financial models.…”