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A CLT for martingale transforms with infinite variance

Author

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  • 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
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    References listed on IDEAS

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    1. Wintenberger, Olivier & Cai, Sixiang, 2011. "Parametric inference and forecasting in continuously invertible volatility models," MPRA Paper 31767, University Library of Munich, Germany.
    2. 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.
    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. 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.
    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.
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    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

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