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ESTIMATION FOR A NONSTATIONARY SEMI-STRONG GARCH(1,1) MODEL WITH HEAVY-TAILED ERRORS

Published online by Cambridge University Press:  19 June 2009

Oliver Linton ,
Jiazhu Pan  and
Hui Wang
Oliver Linton*
Affiliation:
London School of Economics
Jiazhu Pan*
Affiliation:
University of Strathclyde and Peking University
Hui Wang
Affiliation:
Central University of Finance and Economics, China
*
†Oliver Linton, Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, U.K.; e-mail: [email protected].
*Address correspondence to Jiazhu Pan, Department of Mathematics and Statistics, University of Strathclyde, Richmond Street, Glasgow, G1 1XH, U.K.; e-mail: [email protected].
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Abstract

This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponent κ ∈ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponent κ ∈ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 1 , February 2010 , pp. 1 - 28
Copyright
Copyright © Cambridge University Press 2009

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