IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v6y2018i3p93-d168825.html
   My bibliography  Save this article

Linear Regression for Heavy Tails

Author

Listed:
  • Guus Balkema

    (Department of Mathematics, Universiteit van Amsterdam, 1098xh Amsterdam, The Netherlands)

  • Paul Embrechts

    (RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland)

Abstract

There exist several estimators of the regression line in the simple linear regression: Least Squares, Least Absolute Deviation, Right Median, Theil–Sen, Weighted Balance, and Least Trimmed Squares. Their performance for heavy tails is compared below on the basis of a quadratic loss function. The case where the explanatory variable is the inverse of a standard uniform variable and where the error has a Cauchy distribution plays a central role, but heavier and lighter tails are also considered. Tables list the empirical sd and bias for ten batches of one hundred thousand simulations when the explanatory variable has a Pareto distribution and the error has a symmetric Student distribution or a one-sided Pareto distribution for various tail indices. The results in the tables may be used as benchmarks. The sample size is n = 100 but results for n = ∞ are also presented. The error in the estimate of the slope tneed not be asymptotically normal. For symmetric errors, the symmetric generalized beta prime densities often give a good fit.

Suggested Citation

  • Guus Balkema & Paul Embrechts, 2018. "Linear Regression for Heavy Tails," Risks, MDPI, vol. 6(3), pages 1-70, September.
    • Handle: RePEc:gam:jrisks:v:6:y:2018:i:3:p:93-:d:168825
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/6/3/93/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/6/3/93/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Nolan, John P. & Ojeda-Revah, Diana, 2013. "Linear and nonlinear regression with stable errors," Journal of Econometrics, Elsevier, vol. 172(2), pages 186-194.
    2. Postnikov, Eugene B. & Sokolov, Igor M., 2015. "Robust linear regression with broad distributions of errors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 434(C), pages 257-267.
    3. DRYGAS, Hilmar, 1976. "Weak and strong consistency of the least squares estimators in regression models," LIDAM Reprints CORE 236, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Janet E. Heffernan & Jonathan A. Tawn, 2004. "A conditional approach for multivariate extreme values (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 497-546, August.
    5. Smith, V Kerry, 1973. "Least Squares Regression with Cauchy Errors," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 35(3), pages 223-231, August.
    6. Mikosch, Thomas & de Vries, Casper G., 2013. "Heavy tails of OLS," Journal of Econometrics, Elsevier, vol. 172(2), pages 205-221.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Neil Shephard, 2020. "An estimator for predictive regression: reliable inference for financial economics," Papers 2008.06130, arXiv.org.

    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. Neil Shephard, 2020. "An estimator for predictive regression: reliable inference for financial economics," Papers 2008.06130, arXiv.org.
    2. Refk Selmi & Christos Kollias & Stephanos Papadamou & Rangan Gupta, 2017. "A Copula-Based Quantile-on-Quantile Regression Approach to Modeling Dependence Structure between Stock and Bond Returns: Evidence from Historical Data of India, South Africa, UK and US," Working Papers 201747, University of Pretoria, Department of Economics.
    3. Maarten R C van Oordt & Chen Zhou, 2019. "Estimating Systematic Risk under Extremely Adverse Market Conditions," Journal of Financial Econometrics, Oxford University Press, vol. 17(3), pages 432-461.
    4. Anne‐Laure Fougères & John P. Nolan & Holger Rootzén, 2009. "Models for Dependent Extremes Using Stable Mixtures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 42-59, March.
    5. Alexandra Ramos & Anthony Ledford, 2009. "A new class of models for bivariate joint tails," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 219-241, January.
    6. Lee Fawcett & David Walshaw, 2014. "Estimating the probability of simultaneous rainfall extremes within a region: a spatial approach," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(5), pages 959-976, May.
    7. Vijverberg, Wim P. & Hasebe, Takuya, 2015. "GTL Regression: A Linear Model with Skewed and Thick-Tailed Disturbances," IZA Discussion Papers 8898, Institute of Labor Economics (IZA).
    8. Maarten van Oordt & Chen Zhou, 2015. "Systemic risk of European banks: Regulators and markets," DNB Working Papers 478, Netherlands Central Bank, Research Department.
    9. Norbert Christopeit & Michael Massmann, 2013. "A Note on an Estimation Problem in Models with Adaptive Learning," Tinbergen Institute Discussion Papers 13-151/III, Tinbergen Institute.
    10. Goodell, John W. & Ben Jabeur, Sami & Saâdaoui, Foued & Nasir, Muhammad Ali, 2023. "Explainable artificial intelligence modeling to forecast bitcoin prices," International Review of Financial Analysis, Elsevier, vol. 88(C).
    11. Barme-Delcroix, Marie-Francoise & Gather, Ursula, 2007. "Limit laws for multidimensional extremes," Statistics & Probability Letters, Elsevier, vol. 77(18), pages 1750-1755, December.
    12. Yannis Yatracos, 2006. "On Consistent Statistical Procedures in Regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(2), pages 379-387, June.
    13. Marmai, Nadin & Franco Villoria, Maria & Guerzoni, Marco, 2016. "How the Black Swan damages the harvest: statistical modelling of extreme events in weather and crop production in Africa, Asia, and Latin America," Department of Economics and Statistics Cognetti de Martiis LEI & BRICK - Laboratory of Economics of Innovation "Franco Momigliano", Bureau of Research in Innovation, Complexity and Knowledge, Collegio 201605, University of Turin.
    14. Daniela Castro Camilo & Miguel de Carvalho & Jennifer Wadsworth, 2017. "Time-Varying Extreme Value Dependence with Application to Leading European Stock Markets," Papers 1709.01198, arXiv.org.
    15. Balakrishnan, N. & Hashorva, E., 2011. "On Pearson-Kotz Dirichlet distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 948-957, May.
    16. Noureddine Kouaissah & Sergio Ortobelli Lozza & Ikram Jebabli, 2022. "Portfolio Selection Using Multivariate Semiparametric Estimators and a Copula PCA-Based Approach," Computational Economics, Springer;Society for Computational Economics, vol. 60(3), pages 833-859, October.
    17. Mahdi Teimouri & Saralees Nadarajah, 2022. "Maximum Likelihood Estimation for the Asymmetric Exponential Power Distribution," Computational Economics, Springer;Society for Computational Economics, vol. 60(2), pages 665-692, August.
    18. de Valk, Cees, 2016. "A large deviations approach to the statistics of extreme events," Other publications TiSEM 117b3ba0-0e40-4277-b25e-d, Tilburg University, School of Economics and Management.
    19. Panagiota Galiatsatou & Christos Makris & Panayotis Prinos & Dimitrios Kokkinos, 2019. "Nonstationary joint probability analysis of extreme marine variables to assess design water levels at the shoreline in a changing climate," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 98(3), pages 1051-1089, September.
    20. Hentschel, Manuel & Engelke, Sebastian & Segers, Johan, 2022. "Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions," LIDAM Discussion Papers ISBA 2022032, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

    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:gam:jrisks:v:6:y:2018:i:3:p:93-:d:168825. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.