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

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  • Davide Ferrari

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  • Sandra Paterlini

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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 traditionalMaximum 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).

Suggested Citation

  • Davide Ferrari & Sandra Paterlini, 2007. "The Maximum Lq-Likelihood Method: an Application to Extreme Quantile Estimation in Finance," Center for Economic Research (RECent) 001, University of Modena and Reggio E., Dept. of Economics "Marco Biagi".
  • Handle: RePEc:mod:recent:001
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Manfred Gilli & Evis këllezi, 2006. "An Application of Extreme Value Theory for Measuring Financial Risk," Computational Economics, Springer;Society for Computational Economics, vol. 27(2), pages 207-228, May.
    4. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    5. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
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    Citations

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    Cited by:

    1. Li, Ling-Wei & Lee, Loo-Hay & Chen, Chun-Hung & Guo, Bo, 2012. "On unbiased optimal L-statistics quantile estimators," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1891-1897.
    2. 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, Dipartimento di Economia "Marco Biagi".
    3. C. Pederzoli & C. Torricelli, 2013. "Efficiency and unbiasedness of corn futures markets: new evidence across the financial crisis," Applied Financial Economics, Taylor & Francis Journals, pages 1853-1863.
    4. 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, Dipartimento di Economia "Marco Biagi".
    5. C. Pederzoli & C. Torricelli, 2013. "Efficiency and unbiasedness of corn futures markets: new evidence across the financial crisis," Applied Financial Economics, Taylor & Francis Journals, pages 1853-1863.
    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, Dipartimento 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: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(5), pages 641-671, July.
    8. 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.
    9. Chao Huang & Jin-Guan Lin, 2014. "Modified maximum spacings method for generalized extreme value distribution and applications in real data analysis," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(7), pages 867-894, October.
    10. Carlo Alberto Magni, 2015. "Pseudo-naïve approaches to investment performance measurement," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 15021, Universita di Modena e Reggio Emilia, Dipartimento di Economia "Marco Biagi".
    11. Enrico Rubaltelli & Sergio Agnoli & Michela Rancan & Tiziana Pozzoli, 2015. "Emotional Intelligence and risk taking in investment decision-making," Centro Studi di Banca e Finanza (CEFIN) (Center for Studies in Banking and Finance) 15107, Universita di Modena e Reggio Emilia, Dipartimento di Economia "Marco Biagi".
    12. 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, Dipartimento di Economia "Marco Biagi".

    More about this item

    Keywords

    Maximum Likelihood; Extreme Value Theory; q-Entropy; Tail-related Risk Measures;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation

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