Optimal choice of sample fraction in univariate financial tail index estimation

Abstract

This study introduces a technique to estimate the Pareto distribution of the stock exchange index by using the maximum-likelihood Hill estimator. Recursive procedures based on the goodness-of-fit statistics are used to determine the optimal threshold fraction of extreme values to be included in tail estimation. These procedures are applied to three indices in the Malaysian stock market which included the consideration of a drastic economic event such as the Asian financial crisis. The empirical results evidenced alternating varying behavior of heavy-tailed distributions in the regimes for both upper and lower tails.

Keywords:

• heavy-tailed distribution
• Hill estimator
• goodness-of-fit test
• structural change

Acknowledgements

The author would like to thank the anonymous referees for their constructive comments. The author also would like to gratefully acknowledge the financial support from the e-Science Grant (IP2008-102-1002) funded by the MOSTI.

Notes

This study mainly focused on the tail estimations with the optimal samples selection. Thus, the application such as measuring the market risk is not included in this study. However, the results from the tail estimations provided useful information and implications in the market risk analysis such as the determination of value-at-risk.

Alternative initial threshold such as R ^0.6 24 or the ratio n/R ∼ 0.5%–1.0% can be used as well.