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Biometrika 2003 90(4):967-975; doi:10.1093/biomet/90.4.967
© 2003 by Biometrika Trust
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Least absolute deviations estimation for ARCH and GARCH models

Liang Peng1 and Qiwei Yao2

1 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, U.S.Apeng{at}math.gatech.edu 2 Department of Statistics, The London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, U.K.q.yao{at}lse.ac.uk

Hall & Yao (2003) showed that, for ARCH/GARCH, i.e.autoregressive conditional heteroscedastic/generalised autoregressive conditional heteroscedastic, models with heavy-tailed errors, the conventional maximum quasilikelihood estimator suffers from complex limit distributions and slow convergence rates. In this paper three types of absolute deviations estimator have been examined, and the one based on logarithmic transformation turns out to be particularly appealing. We have shown that this estimator is asymptotically normal and unbiased. Furthermore it enjoys the standard convergence rate of n1/2 regardless of whether the errors are heavy-tailed or not. Simulation lends further support to our theoretical results.

Key Words: ARCH; Asymptotic normality; GARCH; Gaussian likelihood; Heavy tail; Least absolute deviations estimator; Maximum quasilikelihood estimator; Time series

Received December 2001. Revised May 2003


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