Estimation Of Tail Index Research Articles

Efficient filtering of financial time series and extreme value theory

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.

Unbiased Tail Estimation by an Extension of the Generalized Pareto Distribution

The generalized Pareto distribution (GPD) is probably the most popular model for inference on the tail of a distribution. The peaks-over-threshold methodology postulates the GPD as the natural model for excesses over a high threshold. However, for the GPD to fit such excesses well, the threshold should often be rather large, thereby restricting the model to only a small upper fraction of the data. In case of heavy-tailed distributions, we propose an extension of the GPD with a single parameter, motivated by a second-order refinement of the underlying Pareto-type model. Not only can the extended model be fitted to a larger fraction of the data, but in addition is the resulting maximum likelihood for the tail index asymptotically unbiased. In practice, sample paths of the new tail index estimator as a function of the chosen threshold exhibit much larger regions of stability around the true value. We apply the method to daily log-returns of the euro-UK pound exchange rate. Some simulation results are presented as well.

Revisiting the Role of the Jackknife Methodology in the Estimation of a Positive Tail Index

In this article, and in a context of regularly varying tails, we analyze a generalization of the classical Hill estimator of a positive tail index. The members of this general class of estimators are not asymptotically more efficient than the original one, the Hill estimator. We thus propose a class of generalized Jackknife estimators associated with any two members of the first class. They enable the reduction of the main component of bias of the Hill estimator and are dependent of a tuning parameter, which is adequately chosen through an asymptotic variance minimization criterion. These new estimators are compared with the Hill estimator, both asymptotically and for finite samples and, when the underlying distribution is in Hall's class of models, they really improve on the well-known, bias-variance, trade-off characteristic of the Hill estimator.

Tail index estimation in small smaples Simulation results for independent and ARCH-type financial return models

Estimation of the tail index of stationary, fat-tailed return distributions is non-trivial since the well-known Hill estimator is optimal only under iid draws from an exact Pareto model. We provide a small sample simulation study of recently suggested adaptive estimators under ARCH-type dependence. The Hill estimator’s performance is found to be dominated by a ratio estimator. Dependence increases estimation error which can remain substantial even in larger data sets. As small sample bias is related to the magnitude of the tail index, recent standard applications may have overestimated (underestimated) the risk of assets with low (high) degrees of fat-tailedness.

A functional law of the iterated logarithm for kernel-type estimators of the tail index

Under von Mises's condition, the kernel tail index estimators of Csörgő et al. (Ann. Statist. 13 (1985) 1050) have been generalized by Groeneboon et al. (Ann. Statist. 31 (2003) 1956) for all real tail indices. Weak consistency and asymptotic normality of their kernel estimators have been established. In this paper, we present a characterization of the almost sure behavior of these estimators and we show their strong consistency.

Autoregressive conditional tail behavior and results on Government bond yield spreads

Previous evidence in empirical finance indicates the potential usefulness of modeling time variation particularly in the tails of speculative return distributions. Based on results from extreme value theory, the present paper proposes a fixed changepoint Pareto-type autoregressive conditional tail (ARCT) model. Regression-based parameter estimation of the unobservable time-varying tail index is carried out via classical Kalman filtering. A model application highlights the tail index dynamics for daily changes in Government bond yield spreads between the U.S. Dollar and the Euro zone.

Extreme quantiles estimation for actuarial applications

This paper is devoted to the estimation of tail index and extreme quantiles in actuarial applications. In this domain, the observations are often censored. Nevertheless, conversely to the classical randomly right-censored model, the censoring variables are always observed. Therefore, under this assumption, we introduce new estimators and we study their asymptotic properties. Their behaviour are illustrated in a small simulation study. To cite this article: E. Delafosse, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Measuring tail thickness under GARCH and an application to extreme exchange rate changes

Accurate modeling of extreme price changes is vital to financial risk management. We examine the small sample properties of adaptive tail index estimators under the class of student-t marginal distribution functions including generalized autoregressive conditional heteroskedastic (GARCH) models and propose a model-based bias-corrected estimation approach. Our simulation results indicate that bias relates to the underlying model and may be positively as well as negatively signed. The empirical study of daily exchange rate changes reveals substantial differences in measured tail thickness due to small sample bias. Thus, high quantile estimation may lead to a substantial underestimation of tail risk.

Pasiskirstymo „uodegos“ indekso pasikeitimo nustatymas

This article deals with a new estimator for tail index, introduced by proof V. Paulauskas in [1]. Hypothesis of a change-point at beforehand-unknown place is examined. Besides simple theoretical calculations, simulation results are presented, which enable to expect successful practical applications of the estimator.

Power Transformations to Induce Normality and their Applications

SummaryRandom variables which are positive linear combinations of positive independent random variables can have heavily right-skewed finite sample distributions even though they might be asymptotically normally distributed. We provide a simple method of determining an appropriate power transformation to improve the normal approximation in small samples. Our method contains the Wilson–Hilferty cube root transformation for χ2 random variables as a special case. We also provide some important examples, including test statistics of goodness-of-fit and tail index estimators, where such power transformations can be applied. In particular, we study the small sample behaviour of two goodness-of-fit tests for time series models which have been proposed recently in the literature. Both tests are generalizations of the popular Box–Ljung–Pierce portmanteau test, one in the time domain and the other in the frequency domain. A power transformation with a finite sample mean and variance correction is proposed, which ameliorates the small sample effect. It is found that the corrected versions of the tests have markedly better size properties. The correction is also found to result in an overall increase in power which can be significant under certain alternatives. Furthermore, the corrected tests also have better power than the Box–Ljung–Pierce portmanteau test, unlike the uncorrected versions.

Measuring Credit Spread Risk

It is widely known that the possibility of default makes the expected return distribution for financial products that are subject to credit risk highly skewed and fat-tailed. Recent development of an unbiased tail index estimator enables modeling of the additional risk presented by changes in swap spreads. Tests on data from the U.S., the U.K., Germany, and Japan indicate that we tend to grossly underestimate the risk of large spread widenings and tightenings. Estimation of swap spread risk is dramatically improved when the severity of the additional downside risk is measured and incorporated into current estimation techniques. These results are crucial in improving credit risk management and pricing out-of-the-money credit derivatives.

Measuring Tail Thickness under GARCH and an Application to Extreme Exchange Rate Changes

Accurate modeling of extreme price changes is vital to financial risk management. We examine the small sample properties of adaptive tail index estimators under the class of student-t marginal distribution functions including GARCH and propose a model-based bias-corrected estimation approach. Our simulation results indicate that bias strongly relates to the underlying model and may be positively as well as negatively signed. The empirical study of daily exchange rate changes reveals substantial differences in measured tail-thickness due to small sample bias. As a consequence, high quantile estimation may lead to a substantial underestimation of tail risk.

Tail Estimation Based on Numbers of Near m-Extremes

Let \(\left\{ {{\text{X}}_n ,n \geqslant 1} \right\}\) be a sequence of independent random variables with common continuous distribution function F having finite upper endpoint. A new tail index estimator γ^n is defined based on only two numbers of near m-extremes \(K_n \left( {a_i ,m} \right) = {\text{\# }}\left\{ {{\text{j:X}}_{\left( {n - m + :1} \right)} - a_i a1 > 0. The weak and almost sure convergence of γ^n to the tail index γ, as well as the asymptotic distribution is given. Moreover, the asymptotic distribution of Kn(an, m) for an→ 0 is derived.

A class of asymptotically unbiased semi-parametric estimators of the tail index

In this paper we consider a class of consistent semi-parametric estimators of a positive tail index γ, parameterized in atuning orcontrol parameter α. Such a control parameter enables us to have access, for any available sample, to an estimator of γ with a null dominant component of asymptotic bias, and with a reasonably flatMean Squared Error pattern, as a functional ofk, the number of top order statistics considered. Moreover, we are able to achieve a high efficiency relatively to the classical Hill estimator, provided we may have access to a larger number of top order statistics than the number needed for optimal estimation through the Hill estimator.

Transformed Estimates of Densities of Heavy-Tailed Distributions and Classification

Nonparametric estimation of the density of a heavy-tailed probability distribution is investigated. The initial data are transformed to a bounded interval and the distribution density is determined by an inverse transformation of the distribution density estimate of transformed data. An adaptive data transformation is studied, in which the order of decay of the tail of the true distribution density is preserved and stable estimation of the deviation in tail index estimates is guaranteed. In classification, the empirical risk of erroneous classification by the Bayes empirical classifier is used as a measure for the quality of distribution density estimates.

A new approach on estimation of the tail index

A new approach on tail index estimation is proposed based on studying the in-sample evolution of appropriately chosen diverging statistics. The resulting estimators are simple to construct, and they can be generalized to address other rate estimation problems as well. To cite this article: D.N. Politis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 279–282.

Almost sure convergence of a tail index estimator in the presence of censoring

In Beirlant and Guillou [1] an exponential regression model was introduced on the basis of scaled log-spacing between subsequent extreme order statistics from a Pareto-type distribution in the presence of censoring. From this representation, they derived an estimator for the Pareto index. In this note, we revisit this adaptation of the popular Hill [5] estimator for heavy-tailed distributions, generalizing the almost sure convergence of this estimator under very general conditions on N r , the number of non-censored observations. To cite this article: E. Delafosse, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 375–380.

On exponential representations of log-spacings of extreme order statistics

In Beirlant et al. (1999) and Feuerverger and Hall (1999) an exponential regression model (ERM) was introduced on the basis of scaled log-spacings between subsequent extreme order statistics from a Pareto-type distribution. This lead to the construction of new bias-corrected estimators for the tail index. In this note, under quite general conditions, asymptotic justification for this regression model is given as well as for resulting tail index estimators. Also, we discuss diagnostic methods for adaptive selection of the threshold when using the Hill (1975) estimator which follow from the ERM approach. We show how the diagnostic presented in Guillou and Hall (2001) is linked to the ERM, while a new proposal is suggested. We also provide some small sample comparisons with other existing methods.

Non-Parametric Heavy-Tailed Density Estimation and Classification Problem

This report is devoted to the non-parametric probability density function (PDF) estimation for distributions with heavy tails. It is proposed to transform the data to a compact interval. The PDF of the original data is obtained by the “back-transform” of the PDF estimate of the transformed data. The transformation adapted to the data is explored to keep the tail decay rate of the true PDF and to provide a stable estimation regarding the deviations of the tail index estimates. The relative risk of the misclassification is proposed as a measure of quality for the PDF estimates

The limiting extremal behaviour of speculative returns: an analysis of intra-daily data from the Frankfurt Stock Exchange

This paper provides a statistical analysis of high-frequency recordings of the German share price index DAX. The data set extends from November 1988 to the end of the year 1995 and includes all minute-to-minute changes during trading hours at the Frankfurt Stock Exchange. The focus of this study is on the limiting behaviour characterizing the tail regions of the empirical distribution. Application of the popular Hill estimator for the tail shape yields results very similar to those of other analyses of speculative returns. However, since the reliability of tail index estimation rests on the appropriateness of the tail regions, the question of optimally choosing the sample fraction emerges. Exploiting recent advances in extreme value theory a couple of novel approaches are applied for determining the optimum cut-off value for the ‘tail’ of the empirical distribution. As it turns out, most algorithms suggest that one has to go out quite far into the tails for estimation of the extremal index. The findings obtained at the highest frequency (minute-to-minute returns) are confirmed when considering data at various levels of time-aggregation. A test for stability of extreme value behaviour over time gives no clear indication of changes of the limiting distribution. It is also illustrated how the approximation of the tails can be used to estimate the likelihood of large returns.