Bayesian estimation of stochastic tail index from high-frequency financial data

Abstract

Tails of the return distribution of an asset are informative about the (financial) risk behavior of that asset. Stochastic tail index (STI) models are designed to quantify riskiness by estimating a time-varying tail index from the distribution of extreme values using high-frequency financial data. In this paper, we propose a computationally efficient Bayesian estimation method for STI models based on the recent advances in band and sparse matrix algorithms. We then show how the deviance information criterion (DIC) can be calculated from the integrated likelihood function for model comparison exercises. In a Monte Carlo study, we investigate the finite sample properties of the Bayesian estimator as well as the performance of two DIC measures. Our results show that the Bayesian estimator performs sufficiently well and the DIC measures based on the integrated likelihood function are useful for model selection exercises. In an empirical application, we illustrate calculation of the tail index using high-frequency data on IBM stock returns. Our estimation results indicate that the daily tail index of the return distribution of IBM stock has a time-varying feature: It tends to decline when there are large losses, whereas it tends to increase when there are small losses.