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A quadratic ARCH(∞) model with long memory and Lévy stable behavior of squares

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Donatas Surgailis
Donatas Surgailis*
Affiliation:
Institute of Mathematics and Informatics, Vilnius
*
∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, 08663 Vilnius, Lithuania. Email address: sdonatas@ktl.mii.lt
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Abstract

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We introduce a modification of the linear ARCH (LARCH) model (Giraitis, Robinson, and Surgailis (2000)) - a special case of Sentana's (1995) quadratic ARCH (QARCH) model - for which the conditional variance is a sum of a positive constant and the square of an inhomogeneous linear combination of past observations. Necessary and sufficient conditions for the existence of a stationary solution with finite variance are obtained. We give conditions under which the stationary solution with infinite fourth moment can exhibit long memory, the leverage effect, and a Lévy-stable limit behavior of partial sums of squares.

Keywords

Linear ARCH modelquadratic ARCH modellong memorypartial sumLévy stable processfractional Brownian motion

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G18: Self-similar processes 60G52: Stable processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1198 - 1222
Copyright
Copyright © Applied Probability Trust 2008 

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