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# Journal of Time Series Econometrics

Editor-in-Chief: Hidalgo, Javier

2 Issues per year

CiteScore 2017: 0.25

SCImago Journal Rank (SJR) 2017: 0.236
Source Normalized Impact per Paper (SNIP) 2017: 0.682

Online
ISSN
1941-1928
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Volume 8, Issue 1

# 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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• Other articles by this author:
/ Alexandros Louka
• Corresponding author
• Department of Economics, Athens University of Economics and Business, P.O. Box 10434, Patision Str. 80, Athens, Greece
• Email
• Other articles by this author:
Published Online: 2015-05-21 | DOI: https://doi.org/10.1515/jtse-2014-0034

## Abstract

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 $\mathrm{\alpha }$-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 $\mathrm{\alpha }$-stable distributions for any $\mathrm{\alpha }\in \left(0,2\right]$.

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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,

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