Quantile Estimation for Heavy-Tailed Data 23/03/2000 J. Beirlant G. Matthys

Contents MODEL SPECIFICATION EXAMPLE : S&P500’s daily %returns EXTREME QUANTILE ESTIMATION : review of 3 classes of methods Blocks method Peaks-over-treshold (POT) method Q(uantile)-based methods IMPROVING Q-BASED METHODS : ML-estimator SIMULATION RESULTS CONCLUSIONS Introduction & Notation

Notation Sample X 1, X 2,…,X n from Order statistics : Pareto-type / heavy-tailed model –Tail decays essentially as a power function : with l slowly varying, i.e. –Terminology :tail index  extreme value index (EVI) –Equivalently : for some other slowly varying function l *

Examples of heavy-tailed distributions : –Pareto : –Student’s t with degrees of freedom : –Loggamma : –Fréchet : !Not: normal, exponential, lognormal, Weibull !

Heavy-tailed distributions in finance log-returns and exchange rates mostly do exhibit heavy tails (often for log-return series) often one is interested in a certain high level, which will be exceeded with only a (very) small probability search for an extreme quantile of the distribution Example : 16/10/1987: Risk manager wants to assess the risks his investment is exposed to and investigates some “worst case scenarios” estimate %fall of the S&P500 index that will happen only once every days ( 40 years) on average

Formally: Consider daily %falls as random variables X 1, X 2,…, X n,… and denote their stationary distribution (assuming it exists) with F then we look for quantile x of F that satisfies : F(x) = 1 - 1/ Problem of estimating extreme quantiles x p with tail probability p  (0,1), where p will be usually very small : 3 classes of methods : method of block maxima POT-method Q(uantile)-based methods

Peaks Over Treshold (POT) Method Limit law for excesses over a high treshold (Pickands— Balkema—de Haan, 1974, 1975): If F is a heavy-tailed distribution with EVI  >0 and excess distribution then where G , denotes the Generalised Pareto Distribution: (GPD)

Thus, for large tresholds u, for  >0 and some  > 0. Description of the POT method: 1 given a sample X 1, X 2,…, X n, select a high treshold u let N u be the number of exceedances denote the excesses for j=1,…,N u 2 fit a GPD G , to the excesses Y 1,…,Y n, e.g. with MLE parameter estimates

3 as estimate F(x) for x > u with where F n is the empirical distribution function; F n (u) = 1-N u /n 4 obtain quantile estimates by inverting (*) Note: in general for each different choice of treshold other parameter and quantile estimates ! (*)

MLE’s for GPD-fit of the excesses above treshold u = 1.5%: (thus ) 95%CI: (4.9, 10) compare the fitted excess distribution with the empirical one:

Q(uantile)-based Methods Hill estimator Pickands estimator Moment estimator MLE for exponential regression model

Hill estimator Heavy-tailed distribution: distribution of log(X): Quantile function: Order statistics of X 1, X 2,…,X n : is a consistent estimator for plot

slope estimator for k large, denominator 1 Hill estimator (1975) for each k (number of order statistics): different estimator –k smallbias small, variance large –k largevariance small, bias large bias-variance trade-off to select ‘optimal’ k

estimating a quantile Q(1-p) with Hill estimator: extrapolate along fitted line on Pareto quantile plot again different estimator for each k bias-variance trade-off needed to select ‘optimal’ k (Weissman, 1978)

Pickands estimator EVI estimator (1975): derived quantile estimator:

Moment estimator EVI estimator (Dekkers—Einmahl—de Haan, 1989): where derived quantile estimator:

k Q-based estimators for EVI of S&P500's %falls

k Q-based estimators for quantile of S&P500's %falls

ML Method improving Q-based methods (Beirlant, Dierckx, Goegebeur & Matthys, Extremes, 1999) Hill estimator : for 0 < k < n, consider log-spacings Assumption on second-order slow variation : for x   with  < 0 and b(x)  0 as x  

exponential regression model for log-spacings : with f 1,k, f 2,k,…,f k,k i.i.d. exponential (1) with uniform (0,1) order statistics with f 1, f 2,…,f k i.i.d. exponential (1)

Burr(1,1,2), k = 200, n = 500

k Q-based estimators for EVI of S&P500's %falls

quantile estimation : try to preserve stability –first idea, by analogy to Weissman estimator : –improvement, including estimated information on l* :

k Q-based estimators for quantile of S&P500's %falls

k Loggamma(1,2), p =

k Loggamma(2,2), p =

k Student's t2, p =

k Student's t4, p =

k Burr(1,1,1), p =

k Burr(1,0.5,2), p =

k Arch(1), p = 0.001, gamma = 0.466

k Arch(1), p = 0.001, gamma = 0.094

k Garch(1,1), p = 0.001, gamma = 0.419

k Garch(1,1), p = 0.001, gamma = 0.096

How to detect heavy-tailedness Extension of the Q-based ML method to all classes of distributions (not only heavy-tailed ones) Further investigation of asymptotic properties of the Q-based ML estimators + construction of CI’s for (financial) time series (dependent data) Related Problems and Questions