IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v119y2016icp116-123.html
   My bibliography  Save this article

A CLT for martingale transforms with infinite variance

Author

Listed:
  • Arvanitis, Stelios
  • Louka, Alexandros

Abstract

We provide a CLT for martingale transforms that holds even when the second moments are infinite. Compared to an analogous result in Hall and Yao (2003) we impose minimal assumptions and utilize the Principle of Conditioning to verify a modified version of Lindeberg’s condition. When the variance is infinite, the rate of convergence, which we allow to be matrix valued, is slower than n and depends on the rate of divergence of the truncated second moments. In many cases it can be consistently estimated. A major application concerns the characterization of the rate and the limiting distribution of the Gaussian QMLE in the case of GARCH type models with infinite fourth moments for the innovation process. The results are particularly useful in the case of the EGARCH(1,1) model as we show that the usual limit theory is still valid without any further parameter restrictions when we relax the assumption for finite fourth moments of the innovation process.

Suggested Citation

  • Arvanitis, Stelios & Louka, Alexandros, 2016. "A CLT for martingale transforms with infinite variance," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 116-123.
    • Handle: RePEc:eee:stapro:v:119:y:2016:i:c:p:116-123
    DOI: 10.1016/j.spl.2016.07.015
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715215303990
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2016.07.015?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Hall, Peter & Yao, Qiwei, 2003. "Inference in ARCH and GARCH models with heavy-tailed errors," LSE Research Online Documents on Economics 5875, London School of Economics and Political Science, LSE Library.
    2. Wintenberger, Olivier & Cai, Sixiang, 2011. "Parametric inference and forecasting in continuously invertible volatility models," MPRA Paper 31767, University Library of Munich, Germany.
    3. Peter Hall & Qiwei Yao, 2003. "Inference in Arch and Garch Models with Heavy--Tailed Errors," Econometrica, Econometric Society, vol. 71(1), pages 285-317, January.
    4. Olivier Wintenberger, 2013. "Continuous Invertibility and Stable QML Estimation of the EGARCH(1,1) Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 846-867, December.
    5. Sung, Soo Hak, 1999. "Weak law of large numbers for arrays of random variables," Statistics & Probability Letters, Elsevier, vol. 42(3), pages 293-298, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stelios Arvanitis & Sofia Anyfantaki, 2020. "On the limit theory of the Gaussian SQMLE in the EGARCH(1,1) model," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(2), pages 341-350, March.
    2. Ngozi G. Emenogu & Monday Osagie Adenomon & Nwaze Obini Nweze, 2020. "On the volatility of daily stock returns of Total Nigeria Plc: evidence from GARCH models, value-at-risk and backtesting," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 6(1), pages 1-25, December.
    3. Conrad, Christian & Mammen, Enno, 2016. "Asymptotics for parametric GARCH-in-Mean models," Journal of Econometrics, Elsevier, vol. 194(2), pages 319-329.
    4. Oliver Linton & Dajing Shang & Yang Yan, 2012. "Efficient estimation of conditional risk measures in a semiparametric GARCH model," CeMMAP working papers 25/12, Institute for Fiscal Studies.
    5. Eric Beutner & Julia Schaumburg & Barend Spanjers, 2024. "Bootstrapping GARCH Models Under Dependent Innovations," Tinbergen Institute Discussion Papers 24-008/III, Tinbergen Institute.
    6. Stelios Arvanitis, 2017. "Non-Emptyness of Stochastic Dominance Effiicient Sets via Stochastic Spanning," Working Papers 201710, Athens University Of Economics and Business, Department of Economics.
    7. Luc Bauwens & Sébastien Laurent & Jeroen V. K. Rombouts, 2006. "Multivariate GARCH models: a survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(1), pages 79-109, January.
    8. M. Jiménez Gamero, 2014. "On the empirical characteristic function process of the residuals in GARCH models and applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 409-432, June.
    9. Cowell, Frank A. & Flachaire, Emmanuel, 2007. "Income distribution and inequality measurement: The problem of extreme values," Journal of Econometrics, Elsevier, vol. 141(2), pages 1044-1072, December.
    10. 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.
    11. Hu, Shuowen & Poskitt, D.S. & Zhang, Xibin, 2021. "Bayesian estimation for a semiparametric nonlinear volatility model," Economic Modelling, Elsevier, vol. 98(C), pages 361-370.
    12. Hill, Jonathan B. & Prokhorov, Artem, 2016. "GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference," Journal of Econometrics, Elsevier, vol. 190(1), pages 18-45.
    13. Bali, Rakesh & Guirguis, Hany, 2007. "Extreme observations and non-normality in ARCH and GARCH," International Review of Economics & Finance, Elsevier, vol. 16(3), pages 332-346.
    14. Feiyu Jiang & Dong Li & Ke Zhu, 2019. "Adaptive inference for a semiparametric generalized autoregressive conditional heteroskedasticity model," Papers 1907.04147, arXiv.org, revised Oct 2020.
    15. Mo Zhou & Liang Peng & Rongmao Zhang, 2021. "Empirical likelihood test for the application of swqmele in fitting an arma‐garch model," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(2), pages 222-239, March.
    16. Pan, Jiazhu & Wang, Hui & Tong, Howell, 2008. "Estimation and tests for power-transformed and threshold GARCH models," Journal of Econometrics, Elsevier, vol. 142(1), pages 352-378, January.
    17. Dennis Kristensen, 2009. "On stationarity and ergodicity of the bilinear model with applications to GARCH models," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(1), pages 125-144, January.
    18. Polonik, Wolfgang & Yao, Qiwei, 2008. "Testing for multivariate volatility functions using minimum volume sets and inverse regression," Journal of Econometrics, Elsevier, vol. 147(1), pages 151-162, November.
    19. Hoga, Yannick, 2021. "The uncertainty in extreme risk forecasts from covariate-augmented volatility models," International Journal of Forecasting, Elsevier, vol. 37(2), pages 675-686.
    20. Błażej Mazur & Mateusz Pipień, 2012. "On the Empirical Importance of Periodicity in the Volatility of Financial Returns - Time Varying GARCH as a Second Order APC(2) Process," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 4(2), pages 95-116, June.

    More about this item

    Keywords

    CLT; Generalized domain of attraction; Martingale transform; Matrix normalization; Self-normalized wald tests; QMLE;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:119:y:2016:i:c:p:116-123. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.