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The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance

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  • Davide Ferrari

    ()

  • Sandra Paterlini

    ()

Abstract

Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on Extreme Value Theory (EVT) has found a successful domain of application in such a context, outperforming other approaches. Given a parametric model provided by EVT, a natural approach is Maximum Likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small in order to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the Maximum Lq-Likelihood estimator (MLqE), introduced by Ferrari and Yang (2007). We show that the MLqE can outperform the standard MLE, when estimating tail probabilities and quantiles of the Generalized Extreme Value (GEV) and the Generalized Pareto (GP) distributions. First, we assess the relative efficiency between the the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q!1, the new estimator approaches the traditional Maximum Likelihood Estimator (MLE), recovering its desirable asymptotic properties; when q 6= 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (Mean Squared Error).

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Bibliographic Info

Paper provided by Universita di Modena e Reggio Emilia, Facoltà di Economia "Marco Biagi" in its series Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) with number 07071.

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Length: pages 20
Date of creation: Jul 2007
Date of revision:
Handle: RePEc:mod:wcefin:07071

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Web page: http://www.economia.unimore.it
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Keywords: Maximum Likelihood; Extreme Value Theory; q-Entropy; Tail-related risk measures;

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References

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  1. Keith Kuester & Stefan Mittnik & Marc S. Paolella, 2006. "Value-at-Risk Prediction: A Comparison of Alternative Strategies," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 4(1), pages 53-89.
  2. repec:cup:cbooks:9780521496032 is not listed on IDEAS
  3. Manfred Gilli & Evis këllezi, 2006. "An Application of Extreme Value Theory for Measuring Financial Risk," Computational Economics, Society for Computational Economics, vol. 27(2), pages 207-228, May.
  4. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
  5. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
  6. Brooks, C. & Clare, A.D. & Dalle Molle, J.W. & Persand, G., 2005. "A comparison of extreme value theory approaches for determining value at risk," Journal of Empirical Finance, Elsevier, vol. 12(2), pages 339-352, March.
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Cited by:
  1. Stefano Cosma & Elisabetta Gualandri, 2014. "The sovereign debt crisis: the impact on the intermediation model of Italian banks," BANCARIA, Bancaria Editrice, vol. 2, pages 48-60, February.
  2. Elisabetta Gualandri, 2011. "Basel 3, Pillar 2: the role of banks’ internal governance and control function," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 11091, Universita di Modena e Reggio Emilia, Facoltà di Economia "Marco Biagi".
  3. Elisabetta Gualandri & Valeria Venturelli, 2013. "The financing of Italian firms and the credit crunch: findings and exit strategies," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 13101, Universita di Modena e Reggio Emilia, Facoltà di Economia "Marco Biagi".
  4. Chiara Pederzoli & Costanza Torricelli, 2013. "Efficiency and unbiasedness of corn futures markets: New evidence across the financial crisis," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 13091, Universita di Modena e Reggio Emilia, Facoltà di Economia "Marco Biagi".
  5. Chiara Pederzoli & Costanza Torricelli, 2010. "A parsimonious default prediction model for Italian SMEs," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 10061, Universita di Modena e Reggio Emilia, Facoltà di Economia "Marco Biagi".
  6. Lisa Mattioli & Riccardo Ferretti, 2013. "La regolamentazione dello short selling: effetti sul mercato azionario italiano (Short selling ban: effects on the Italian stock market)," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 13081, Universita di Modena e Reggio Emilia, Facoltà di Economia "Marco Biagi".
  7. Chao Huang & Jin-Guan Lin & Yan-Yan Ren, 2013. "Testing for the shape parameter of generalized extreme value distribution based on the $$L_q$$ -likelihood ratio statistic," Metrika, Springer, vol. 76(5), pages 641-671, July.

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