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Testing for (in)finite moments

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  • Trapani, Lorenzo

Abstract

This paper proposes a test to verify whether the kth moment of a random variable is finite. We use the fact that, under general assumptions, sample moments either converge to a finite number or diverge to infinity according as the corresponding population moment is finite or not. Building on this, we propose a test for the null that the kth moment does not exist. Since, by construction, our test statistic diverges under the null and converges under the alternative, we propose a randomised testing procedure to discern between the two cases. We study the application of the test to raw data, and to regression residuals. Monte Carlo evidence shows that the test has the correct size and good power; the results are further illustrated through an application to financial data.

Suggested Citation

  • Trapani, Lorenzo, 2016. "Testing for (in)finite moments," Journal of Econometrics, Elsevier, vol. 191(1), pages 57-68.
    • Handle: RePEc:eee:econom:v:191:y:2016:i:1:p:57-68
    DOI: 10.1016/j.jeconom.2015.08.006
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    References listed on IDEAS

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    1. Fedotenkov, Igor, 2015. "A simple nonparametric test for the existence of finite moments," MPRA Paper 66089, University Library of Munich, Germany.
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    5. Corradi, Valentina & Swanson, Norman R., 2006. "The effect of data transformation on common cycle, cointegration, and unit root tests: Monte Carlo results and a simple test," Journal of Econometrics, Elsevier, vol. 132(1), pages 195-229, May.
    6. Markowitz, Harry M & Usmen, Nilufer, 1996. "The Likelihood of Various Stock Market Return Distributions, Part 1: Principles of Inference," Journal of Risk and Uncertainty, Springer, vol. 13(3), pages 207-219, November.
    7. Drees, Holger & Kaufmann, Edgar, 1998. "Selecting the optimal sample fraction in univariate extreme value estimation," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 149-172, July.
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    9. Markowitz, Harry M & Usmen, Nilufer, 1996. "The Likelihood of Various Stock Market Return Distributions, Part 2: Empirical Results," Journal of Risk and Uncertainty, Springer, vol. 13(3), pages 221-247, November.
    10. Phillips, Peter C. B. & Loretan, Mico, 1991. "The Durbin-Watson ratio under infinite-variance errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 85-114, January.
    11. Hill, Jonathan B., 2010. "On Tail Index Estimation For Dependent, Heterogeneous Data," Econometric Theory, Cambridge University Press, vol. 26(5), pages 1398-1436, October.
    12. Linton, Oliver & Xiao, Zhijie, 2013. "Estimation Of And Inference About The Expected Shortfall For Time Series With Infinite Variance," Econometric Theory, Cambridge University Press, vol. 29(4), pages 771-807, August.
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    Cited by:

    1. Yang, Jangho & Heinrich, Torsten & Winkler, Julian & Lafond, François & Koutroumpis, Pantelis & Farmer, J. Doyne, 2019. "Measuring productivity dispersion: a parametric approach using the Lévy alpha-stable distribution," MPRA Paper 96474, University Library of Munich, Germany.
    2. Matteo Barigozzi & Giuseppe Cavaliere & Lorenzo Trapani, 2021. "Inference in heavy-tailed non-stationary multivariate time series," Papers 2107.13894, arXiv.org.
    3. Stavros Degiannakis & George Filis & Grigorios Siourounis & Lorenzo Trapani, 2023. "Superkurtosis," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 55(8), pages 2061-2091, December.
      • Degiannakis, Stavros & Filis, George & Siourounis, Grigorios & Trapani, Lorenzo, 2019. "Superkurtosis," MPRA Paper 94473, University Library of Munich, Germany.
      • Stavros Degiannakis & George Filis & Grigorios Siourounis & Lorenzo Trapani, 2023. "Superkurtosis," Working Papers 318, Bank of Greece.
      • Degiannakis, Stavros & Filis, George & Siourounis, Grigorios & Trapani, Lorenzo, 2019. "Superkurtosis," MPRA Paper 96563, University Library of Munich, Germany.
    4. Barigozzi, Matteo & Trapani, Lorenzo, 2020. "Sequential testing for structural stability in approximate factor models," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5149-5187.
    5. Francq, Christian & Zakoïan, Jean-Michel, 2022. "Testing the existence of moments for GARCH processes," Journal of Econometrics, Elsevier, vol. 227(1), pages 47-64.
    6. Lorenzo Trapani & Emily Whitehouse, 2020. "Sequential monitoring for cointegrating regressions," Papers 2003.12182, arXiv.org.
    7. Francq, Christian & Zakoian, Jean-Michel, 2021. "Testing the existence of moments and estimating the tail index of augmented garch processes," MPRA Paper 110511, University Library of Munich, Germany.
    8. Ahmadreza Marandi & Aharon Ben-Tal & Dick den Hertog & Bertrand Melenberg, 2022. "Extending the Scope of Robust Quadratic Optimization," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 211-226, January.
    9. Torsten Heinrich & Jangho Yang & Shuanping Dai, 2020. "Levels of structural change: An analysis of China's development push 1998-2014," Papers 2005.01882, arXiv.org, revised Sep 2020.
    10. Torsten Heinrich & Jangho Yang & Shuanping Dai, 2022. "Levels of structural change," Journal of Evolutionary Economics, Springer, vol. 32(1), pages 35-86, January.
    11. Li, Dong & Zhang, Xingfa & Zhu, Ke & Ling, Shiqing, 2018. "The ZD-GARCH model: A new way to study heteroscedasticity," Journal of Econometrics, Elsevier, vol. 202(1), pages 1-17.
    12. Yang, Jangho & Heinrich, Torsten & Winkler, Julian & Lafond, François & Koutroumpis, Pantelis & Farmer, J. Doyne, 2019. "Measuring productivity dispersion: a parametric approach using the Lévy alpha-stable distribution," MPRA Paper 96474, University Library of Munich, Germany.
    13. Dewitte, Ruben, 2020. "From Heavy-Tailed Micro to Macro: on the characterization of firm-level heterogeneity and its aggregation properties," MPRA Paper 103170, University Library of Munich, Germany.

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    More about this item

    Keywords

    Finite moments; Randomised tests; Chover-type Law of the Iterated Logarithm; Strong Law of Large Numbers;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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