코스피 지수 자료의 베이지안 극단값 분석

Yun, Seok-Hoon
윤석훈

• Accepted : 20111000
• Published : 2011.10.31
• 37 88

Abstract

This paper conducts a statistical analysis of extreme values for both daily log-returns and daily negative log-returns, which are computed using a collection of KOSPI data from January 3, 1998 to August 31, 2011. The Poisson-GPD model is used as a statistical analysis model for extreme values and the maximum likelihood method is applied for the estimation of parameters and extreme quantiles. To the Poisson-GPD model is also added the Bayesian method that assumes the usual noninformative prior distribution for the parameters, where the Markov chain Monte Carlo method is applied for the estimation of parameters and extreme quantiles. According to this analysis, both the maximum likelihood method and the Bayesian method form the same conclusion that the distribution of the log-returns has a shorter right tail than the normal distribution, but that the distribution of the negative log-returns has a heavier right tail than the normal distribution. An advantage of using the Bayesian method in extreme value analysis is that there is nothing to worry about the classical asymptotic properties of the maximum likelihood estimators even when the regularity conditions are not satisfied, and that in prediction it is effective to reflect the uncertainties from both the parameters and a future observation.

Keywords

KOSPI;extreme value theory;Poisson-GPD model;Bayesian method;noninformative prior distribution;Markov chain Monte Carlo method

References

1. 윤석훈 (2009). 원/달러 환율 투자 손실률에 대한 극단분위수 추정, <한국통계학회논문집>, 16, 803-812. https://doi.org/10.5351/CKSS.2009.16.5.803
2. 윤석훈 (2010). 국제현물원유가의 일일 상승 및 하락률의 극단값 분석, <응용통계연구>, 23, 835-844. https://doi.org/10.5351/KJAS.2010.23.5.835
3. Albert, J. (2009). Bayesian Computation with R, 2nd ed., Springer, New York.
4. Coles, S. G. and Powell, E. A. (1996). Bayesian methods in extreme value modelling: A review and new developments, International Statistical Review, 64, 119-136. https://doi.org/10.2307/1403426
5. Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds (with discussion), Journal of the Royal Statistical Society, Series B, 52, 393-442.
6. Fisher, R. A. and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proceedings of the Cambridge Philosophical Society, 24, 180-190. https://doi.org/10.1017/S0305004100015681
7. Gnedenko, B. V. (1943). Sur la distribution limite du terme maximum d'une serie aleatoire, Annals of Mathematics, 44, 423-453. https://doi.org/10.2307/1968974
8. Gumbel, E. J. (1958). Statistics of Extremes, Columbia University Press, New York.
9. Pickands, J. (1975). Statistical inference using extreme order statistics, Annals of Statistics, 3, 119-131. https://doi.org/10.1214/aos/1176343003
10. Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods, Springer, New York.
11. Smith, R. L. (1985). Maximum likelihood estimation in a class of nonregular cases, Biometrika, 72, 67-90. https://doi.org/10.1093/biomet/72.1.67
12. Smith, R. L. (1989). Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone (with discussion), Statistical Science, 4, 367-393. https://doi.org/10.1214/ss/1177012400
13. von Mises, R. (1936). La distribution de la plus grande de n valeurs. Reprinted in Selected Papers II, American Mathematical Society, Providence, R.I. (1954), 271-294.