Abstract
We discuss a class of conditionally heteroscedastic time series models satisfying the equation r t = ζ t σ t , where ζ t are standardized i.i.d. r.v.s, and the conditional standard deviation σ t is a nonlinear function Q of inhomogeneous linear combination of past values r s , s < t, with coefficients b j . The existence of stationary solution rt with finite pth moment, 0 < p < ∞ is obtained under some conditions on Q, b j and the pth moment of ζ 0. Weak dependence properties of r t are studied, including the invariance principle for partial sums of Lipschitz functions of r t . In the case where Q is the square root of a quadratic polynomial, we prove that r t can exhibit a leverage effect and long memory in the sense that the squared process r 2 t has long-memory autocorrelation and its normalized partial-sum process converges to a fractional Brownian motion.
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* This research was partly supported by grant No. MIP-063/2013 from the Research Council of Lithuania. This work has also been developed within the MME-DII center of excellence (ANR-11-LABEX-0023-01).
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Doukhan, P., Grublyt˙, I. & Surgailis, D. A nonlinear model for long-memory conditional heteroscedasticity* . Lith Math J 56, 164–188 (2016). https://doi.org/10.1007/s10986-016-9312-5
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DOI: https://doi.org/10.1007/s10986-016-9312-5