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Journal of Time Series Econometrics

Editor-in-Chief: Hidalgo, Javier

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A Note on the QMLE Limit Theory in the Non-stationary ARCH(1) Model

Stelios Arvanitis
  • Department of Economics, Athens University of Economics and Business, P.O. Box 10434, Patision Str. 80, Athens, Greece
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/ Alexandros Louka
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  • Department of Economics, Athens University of Economics and Business, P.O. Box 10434, Patision Str. 80, Athens, Greece
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Published Online: 2015-05-21 | DOI: https://doi.org/10.1515/jtse-2014-0034


In this note we extend the standard results for the limit theory of the popular quasi-maximum likelihood estimator (QMLE) in the context of the non-stationary autoregressive conditional heteroskedastic ARCH(1) model by allowing the innovation process not to possess fourth moments. Depending on the value of the index of stability, we either derive α-stable weak limits with non-standard rates or inconsistency and non-tightness. We obtain the limit theory by the derivation of a limit theorem for multiplicative “martingale” transforms with limit mixtures of α-stable distributions for any α0,2.

Keywords: α-stable distribution; slow variation; domain of attraction; MLT with mixed limit; non-stationary ARCH(1); QMLE; inconsistency; non-tightness


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About the article

Published Online: 2015-05-21

Published in Print: 2016-01-01

Funding: Both authors acknowledge the financial support from the Greek General Secretariat for Research and Technology under the grant ARISTEIA II-08-5413.

Citation Information: Journal of Time Series Econometrics, Volume 8, Issue 1, Pages 21–39, ISSN (Online) 1941-1928, ISSN (Print) 2194-6507, DOI: https://doi.org/10.1515/jtse-2014-0034.

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