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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
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    References listed on IDEAS

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    Cited by:

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

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