A generalized nonlinear model for long memory conditional heteroscedasticity. Issue 1 (2nd January 2017) - Grublytė, Ieva; Škarnulis, Andrius
ABSTRACT: We study the existence and properties of stationary solution of ARCH-type equationr t = ζ t σ t, whereζ t are standardized i.i.d. random variables and the conditional variance satisfies an AR(1) equationσ t 2 = Q 2 ( a + ∑ j = 1 ∞ b j r t − j ) + γ σ t − 1 2 with a Lipschitz functionQ ( x ) and real parametersa, γ, b j . The paper extends the model and the results in Doukhan, Grublytė, and Surgailis [A nonlinear model for long memory conditional heteroscedasticity. Lithuanian Math J. 2016;56:164–188] from the caseγ = 0 to the case0 < γ < 1 . We also obtain a new condition for the existence of higher moments ofr t which does not include the Rosenthal constant. In the particular case when Q is the square root of a quadratic polynomial, we prove thatr t can exhibit a leverage effect and long memory. We also present simulated trajectories and histograms of marginal density ofσ t for different values of γ .