Efficient filtering of financial time series and extreme value theory

Abstract

We describe an efficient filtering process for a univariate time series, yt, for subsequent application of extreme value theory and the estimation of extreme quantiles, ie, value-at-risk. Our filtering model is based on a stationary ARMA-asymmetric Garch (1, 1) process. After filtering, the sequence of conditional residuals is approximately independent and identically distributed and therefore appropriate for the application of extreme value theory. The efficiency of the filtering process is of importance in this approach, and we apply an efficient estimator based on the generalized method of moments. The estimator is simple to compute but - more importantly - it is, in case of skewness or excess kurtosis of the conditional residuals, asymptotically efficient relative to the commonly applied quasi-maximum likelihood estimator. We further evaluate the robustness and efficiency of extreme value theory tail index estimators through a Monte Carlo study. An application to the daily returns of the ABB (Asea Brown Boveri) equity illustrates the methods, and a comparison is made with methods based on conditional normality, the conditional t-distribution and the empirical distribution function.