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Estimation and Asymptotic Inference in the AR-ARCH Model

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  • Theis Lange
  • Anders Rahbek
  • S�ren Tolver Jensen

Abstract

This article studies asymptotic properties of the quasi-maximum likelihood estimator (QMLE) for the parameters in the autoregressive (AR) model with autoregressive conditional heteroskedastic (ARCH) errors. A modified QMLE (MQMLE) is also studied. This estimator is based on truncation of individual terms of the likelihood function and is related to the recent so-called self-weighted QMLE in Ling (2007b). We show that the MQMLE is asymptotically normal irrespectively of the existence of finite moments, as geometric ergodicity alone suffice. Moreover, our included simulations show that the MQMLE is remarkably well-behaved in small samples. On the other hand, the ordinary QMLE, as is well-known, requires finite fourth order moments for asymptotic normality. But based on our considerations and simulations, we conjecture that in fact only geometric ergodicity and finite second order moments are needed for the QMLE to be asymptotically normal. Finally, geometric ergodicity for AR-ARCH processes is shown to hold under mild and classic conditions on the AR and ARCH processes.

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File URL: http://www.tandfonline.com/doi/abs/10.1080/07474938.2011.534031
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Bibliographic Info

Article provided by Taylor & Francis Journals in its journal Econometric Reviews.

Volume (Year): 30 (2011)
Issue (Month): 2 ()
Pages: 129-153

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Handle: RePEc:taf:emetrv:v:30:y:2011:i:2:p:129-153

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Related research

Keywords: ARCH; Asymptotic theory; Geometric ergodicity; Modified QMLE; QMLE;

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Cited by:
  1. Emma M. Iglesias & Oliver Linton, 2009. "Estimation of tail thickness parameters from GJR-GARCH models," Economics Working Papers we094726, Universidad Carlos III, Departamento de Economía.
  2. Mika Meitz & Pentti Saikkonen, 2008. "Parameter Estimation in Nonlinear AR-GARCH Models," Economics Working Papers ECO2008/25, European University Institute.
  3. Zhu, Ke & Ling, Shiqing, 2013. "Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models," MPRA Paper 51509, University Library of Munich, Germany.
  4. M. Caivano & A. Harvey, 2013. "Time series models with an EGB2 conditional distribution," Cambridge Working Papers in Economics 1325, Faculty of Economics, University of Cambridge.
  5. Anders Rahbek & Heino Bohn Nielsen, 2012. "Unit Root Vector Autoregression with volatility Induced Stationarity," CREATES Research Papers 2012-29, School of Economics and Management, University of Aarhus.

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