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ADAPTIVE DENSITY ESTIMATION FOR GENERAL ARCH MODELS

Published online by Cambridge University Press:  17 July 2008

F. Comte ,
J. Dedecker  and
M.L. Taupin
F. Comte*
Affiliation:
Université Paris Descartes
J. Dedecker
Affiliation:
Université Paris 6
M.L. Taupin
Affiliation:
IUT de Paris 5 and Université Paris Sud
*
Address correspondence to F. Comte, Université Paris Descartes, MAP5, UMR CNRS 8145, 45, rue des Saints-Pères, 75006 Paris, France; e-mail: fabienne.comte@univ-paris5.fr
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Abstract

We consider a model Yt = σtηt in which (σt) is not independent of the noise process (ηt) but σt is independent of ηt for each t. We assume that (σt) is stationary, and we propose an adaptive estimator of the density of ln(σt2) based on the observations Yt. Under a new dependence structure, the τ-dependency defined by Dedecker and Prieur (2005, Probability Theory and Related Fields 132, 203–236), we prove that the rates of this nonparametric estimator coincide with the rates obtained in the independent and identically distributed (i.i.d.) case when (σt) and (ηt) are independent. The results apply to various linear and nonlinear general autoregressive conditionally heteroskedastic (ARCH) processes. They are illustrated by simulations applying the deconvolution algorithm of Comte, Rozenholc, and Taupin (2006, Canadian Journal of Statistics 34, 431–452) to a new noise density.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 6 , December 2008 , pp. 1628 - 1662
Copyright
Copyright © Cambridge University Press 2008

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