Tail Dependence Coefficients Research Articles

On the worst and least possible asymptotic dependence

Multivariate extremes behave very differently under asymptotic dependence as compared to asymptotic independence. In the bivariate setting, we are able to characterise the extreme behaviour of the asymptotic dependent case by using the concept of the copula. As a result, we are able to identify the properties of the boundary cases, that are asymptotic independent but still have some asymptotic dependent features. These situations are the most problematic in statistical extreme, and, for this reason, distinguishing between asymptotic dependence and asymptotic independence represents a difficult problem. We propose a simple test to resolve this issue which is an alternative to the procedure based on the classical coefficient of tail dependence. In addition, we are able to identify the worst/least asymptotic dependence (in the presence of asymptotic dependence) that maximises/minimises the probability of a given extreme region if tail dependence parameter is fixed. It is found that the perfect extreme association is not the worst asymptotic dependence, which is consistent with the existing literature. We are able to find lower and upper bounds for some risk measures of functions of random variables. A particular example is the sum of random variables, for which a vivid academic effort has been noticed in the last decade, where bounds for a sum of random variables are sought. It is numerically shown that our approach provides a great improvement of the existing methods, which reiterates the sensible conclusion that any additional piece of information on dependence would help to reduce the spread of these bounds.

Characterizing Asymptotic Dependence via Conditional Kendall's Tau

Modeling and forecasting extreme co-movements in financial market is important for conducting stress test in risk management. Asymptotic independence and asymptotic dependence behave drastically different in modeling such co-movements. For example, the impact of extreme events is usually overestimated whenever asymptotic dependence is wrongly assumed. On the other hand, the impact is seriously underestimated whenever the data is misspecified as asymptotic independent. Therefore, distinguishing between asymptotic independence/dependence scenarios is very informative for any decision-making and especially in risk management. We investigate the properties of the limiting conditional Kendall's tau which can be used to detect the presence of asymptotic independence/dependence. We also propose nonparametric estimation for this new measure and derive its asymptotic limit. A simulation study shows good performances of the new measure and its combination with the coefficient of tail dependence proposed by Ledford and Tawn (1996, 1997). Finally, applications to financial and insurance data are provided.

Forecasting portfolio-Value-at-Risk with nonparametric lower tail dependence estimates

Abstract We propose to forecast the Value-at-Risk of bivariate portfolios using copulas which are calibrated on the basis of nonparametric sample estimates of the coefficient of lower tail dependence. We compare our proposed method to a conventional copula-GARCH model where the parameter of a Clayton copula is estimated via Canonical Maximum-Likelihood. The superiority of our proposed model is exemplified by analyzing a data sample of nine different bivariate and one nine-dimensional financial portfolio. A comparison of the out-of-sample forecasting accuracy of both models confirms that our model yields economically significantly better Value-at-Risk forecasts than the competing parametric calibration strategy.

Coastal flooding: A copula based approach for estimating the joint probability of water levels and waves

An accurate joint probability assessment of water levels and waves is of primary importance for effective coastal flooding management even in microtidal environments subjected to severe storm surge events. A copula based approach is presented for modeling the joint distribution derived from almost six years of sea levels and waves at a site suffering from coastal flooding. The evaluation of the upper tail dependence coefficient represents an unavoidable step in the copula selection process since it provides indications on extreme dependence that cannot be neglected to reliably estimate the probability of marine inundation. Based on the results of various statistical tests and estimation of the upper tail dependence coefficient, a one-parameter extreme value copula is selected to model the dependence structure of events representing conditions at peak water levels, including wave height, incoming wave direction and season of occurrence. The joint distribution obtained is subsequently used for reliability analysis. A particular simplified application case is described for the Ravenna coast (Italy) and the probability of failure/inundation is estimated through the direct integration method. Since the failure function employed involves the wave runup depending on wave period, the joint distribution of wave height and wave period is also assessed. The study highlights the importance of taking into account all the variables involved in the flooding phenomenon for a reliable flood probability estimate. The presented methodology can be applied to the assessment of flood probability at coastal sites at risk of inundation due to the combined impact of waves and water levels.

On the Worst and Least Possible Asymptotic Dependence

Multivariate extremes behave very differently under asymptotic dependence as compared to asymptotic independence. In the bivariate setting, we are able to characterise the extreme behaviour of the asymptotic dependent case by using the concept of the copula. As a result, we are able to identify the properties of the boundary cases, that are asymptotic independent but still have some asymptotic dependent features. These situations are the most problematic in statistical extreme, and, for this reason, distinguishing between asymptotic dependence and asymptotic independence represents a difficult problem. We propose a simple test to resolve this issue which is an alternative to the procedure based on the classical coefficient of tail dependence. In addition, we are able to identify the worst/least asymptotic dependence (in the presence of asymptotic dependence) that maximises/minimises the probability of a given extreme region if tail dependence parameter is fixed. It is found that the perfect extreme association is not the worst asymptotic dependence, which is consistent with the existing literature. We are able to find lower and upper bounds for some risk measures of functions of random variables. A particular example is the sum of random variables, for which a vivid academic effort has been noticed in the last decade, where bounds for a sum of random variables are sought. It is numerically shown that our approach provides a great improvement of the existing methods, which reiterates the sensible conclusion that any additional piece of information on dependence would help to reduce the spread of these bounds.

Bivariate Tail Dependence and the Generation of Multivariate Extreme Value Distributions

We define, in a probabilistic way, a parametric family of multivariate extreme value distributions. We derive its copula, which is a mixture of several complete dependent copulas and total independent copulas, and the bivariate tail dependence and extremal coefficients. Based on the obtained results for these coefficients, we propose a method to build multivariate extreme value distributions with prescribed tail/extremal coefficients. We illustrate the results with examples.

LATIN AMERICAN EXCHANGE RATE DEPENDENCIES: A REGULAR VINE COPULA APPROACH

This study implements a regular vine copula methodology to evaluate the level of contagion among the exchange rates of six Latin American countries (Argentina, Brazil, Chile, Colombia, Mexico, and Peru) from June 2005 to April 2012. We measure contagion in terms of tail dependence coefficients, following Fratzscher's (1999) definition of contagion as interdependence. Our results indicate that these countries are divided into two blocks. The first block consists of Brazil, Colombia, Chile, and Mexico, whose exchange rates exhibit the largest dependence coefficients, and the second block consists of Argentina and Peru, whose exchange rate dependence coefficients with other Latin American countries are low. We also found that most of the Latin American exchange rate pairs exhibit asymmetric behaviors characterized by nonsignificant upper tail dependence and significant lower tail dependence. These results imply that there exists contagion in Latin American exchange rates in periods of large appreciations, whereas there is no evidence of contagion during periods of currency depreciation. This empirical regularity may reflect the “fear of appreciation” in emerging economies identified by Levy‐Yeyati, Sturzenegger, and Gluzmann (2013). (JEL C32, C51, E42)

Copula-based risk assessment of drought in Yunnan province, China

Yunnan is one of the provinces which had been frequently and heavily affected by drought disasters in China. Recently, large severe droughts struck Yunnan, caused considerable social, economic and ecological losses. A risk assessment of meteorological drought for Yunnan province is provided in this study. Based on the daily meteorological data of 29 stations during 1960–2010, duration and severity as two major drought characteristics, defined by the runs and the composite meteorological drought index, are abstracted from the observed drought events. Three bivariate Archimedean copulas are employed to construct the joint distributions of the drought characteristics. Based on the error analysis and tail dependence coefficient, the Gumbel–Hougaard copula is selected to analyze spatial distributions of the joint return periods of drought. The results indicate that a high risk is observed in the middle parts and the northeast parts of Yunnan province, while a relative lower drought risk is observed in the northwest of Yunnan province. The probabilistic properties can provide useful information for water resources planning and management.

Upper tail dependence in rainfall extremes: would we know it if we saw it?

The simultaneous occurrence of extreme events, such as simultaneous storms and floods at different locations, has a serious impact on risk assessment and mitigation strategies. The joint occurrence of extreme events can be measured by the so-called upper tail dependence (UTD) coefficient λ U. In this study, we reconsider the properties of the most popular λ U estimators and show that their strong bias and uncertainty make most of the empirical results reported in the hydrological literature questionable. In order to overcome the limits of λ U analysis, we test several alternative tools such as a pool of formal statistical tests devised for recognizing upper tail independence and graphical diagnostics based on binary correlation and binary entropy. The reliability of all the methods is preliminarily checked by Monte Carlo experiments. Statistical tests and graphical diagnostics are therefore applied to three different rainfall data sets that allow us to explore the properties of the spatial dependence structure of rainfall extremes over a wide range of spatio-temporal scales ranging from 30 min and 1 km to 30 days and $$\approx $$ 3000 km. Results highlight that (1) classical estimators provide non zero tail dependence even for cases where it should be zero; (2) formal tests and binary correlation highlight that the pairwise spatial dependence structure can be weaker than Gaussian, thus excluding UTD calculated in a pairwise manner; (3) the binary entropy computed on triples of locations shows that the pairwise UTD is not enough to explain the spatial dependence structure of extreme rainfall, whose complexity becomes evident only after resorting to higher order correlation measures. The results concerning the bias and uncertainty of λ U estimators are fully general and suggest avoiding their use especially for the short time series usually available in hydrology.

Riesgo sistémico en el mercado de acciones colombiano: alternativas de diversificación bajo eventos extremos

The coefficients of tail dependence, based on copulas, are proposed as a measure for portfolio risk management. Some aspects of the macro and microstructures of the Colombian stock market are described. Such features encourage the use of measures such as the one proposed herein. The application of the proposed methodology to the fourteen most liquid shares in the study period (2007-2012) derived in some diversification recommendations under extreme events. The main conclusion, based on the starting hypothesis, is related to the evidence of high vulnerability in the Colombian stock market to an event of systemic risk.

Nonparametric estimation of general multivariate tail dependence and applications to financial time series

In order to analyse the entire tail dependence structure among random variables in a multidimensional setting, we present and study several nonparametric estimators of general tail dependence functions. These estimators measure tail dependence in different orthants, complementing the commonly studied positive (lower and upper) tail dependence. This approach is in line with the parametric analysis of general tail dependence. Under this unifying approach the different dependencies are analysed using the associated copulas. We generalise estimators of the lower and upper tail dependence coefficient to the general multivariate tail dependence function and study their statistical properties. Tail dependence measures come as a response to the incapability of the correlation coefficient as an extreme dependence measure. We run a Monte Carlo simulation study to assess the performance of the nonparametric estimators. We also employ selected estimators in two empirical applications to detect and measure the general multivariate non-positive tail dependence in financial data, which popular parametric copula models commonly applied in the financial literature fail to capture.

Extremal behavior of pMAX processes

The well-known M4 processes of Smith and Weissman are very flexible models for asymptotically dependent multivariate data. Extended M4 of Heffernan et al. allows to also account for asymptotic independence. In this paper we introduce a more general multivariate model comprising asymptotic dependence and independence, which has the extended M4 class as a particular case. We study properties of the proposed model. In particular, we compute the multivariate extremal index, tail dependence and extremal coefficients.

Instability and dependence structure between oil prices and GCC stock markets

This paper investigates the dependence structure between daily oil price changes and stock market returns in six GCC countries (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and United Arab Emirates) during the period from June 1, 2005 to February 11, 2013. For that, we apply copula functions that capture several dependence structures. The empirical results provide evidence of positive and asymmetric dependence between oil price changes and stock market returns. All countries (except Oman) exhibit left tail dependence while Oman shows right tail dependence. Moreover, we check whether the dependence structure is constant over time using change point testing method. The empirical results indicate significant change in the dependence structure. For all countries, the copula parameters and tail dependence coefficients are greater during financial period than tranquil implying a presence of contagion effect.

Distorted Mix Method for constructing copulas with tail dependence

This paper introduces a method for constructing copula functions by combining the ideas of distortion and convex sum, named Distorted Mix Method. The method mixes different copulas with distorted margins to construct new copula functions, and it enables us to model the dependence structure of risks by handling the central and tail parts separately. By applying the method we can modify the tail dependence of a given copula to any desired level measured by tail dependence function and tail dependence coefficients of marginal distributions. As an application, a tight bound for asymptotic Value-at-Risk of order statistics is obtained by using the method. An empirical study shows that copulas constructed by this method fit the empirical data of SPX 500 Index and FTSE 100 Index very well in both central and tail parts.

Robust and bias-corrected estimation of the coefficient of tail dependence

We introduce a robust and asymptotically unbiased estimator for the coefficient of tail dependence in multivariate extreme value statistics. The estimator is obtained by fitting a second order model to the data by means of the minimum density power divergence criterion. The asymptotic properties of the estimator are investigated. The efficiency of our methodology is illustrated on a small simulation study and by a real dataset from the actuarial context.

Convergence rate to a lower tail dependence coefficient of a skew-tdistribution

Abstract We examine the rate of decay to the limit of the tail dependence coefficient of a bivariate skew- t distribution. This distribution always displays asymptotic tail dependence. It contains as a special case the usual bivariate symmetric t distribution, and hence is an appropriate (skew) extension. The rate is asymptotically a power-law. The second-order structure of the univariate quantile function for such a skew- t distribution is a central issue. Our results generalise those for the bivariate symmetric t .

Extremal properties of M4 processes

The existence of data with different dependence structures motivates the development of models which can capture several types of dependence. In this paper we consider a stationary sequence of moving maxima vectors $$\{{\mathbf {X}}_n=(X_{n1},\ldots ,X_{nd})\}_{n\ge 1}$$ having innovations $${\mathbf {Z}}_{l,n}=(Z_{l,n,1},\ldots ,Z_{l,n,d})$$ with totally dependent margins for certain values of $$l,$$ $$l\in I_1,$$ and independent margins for the remaining values of $$l,$$ $$l\in I_2.$$ We obtain in this way a $$d$$ -dimensional process $$\{{\mathbf {X}}_n\}_{n\ge 1}$$ whose extremal dependence, measured by the tail dependence coefficients, lies between asymptotic independence and total dependence. The extremal properties of these M4 processes are studied and examined both theoretically and through simulation studies: we derive the multivariate extremal index, the tail dependence coefficients and co-movement indices.

Self-consistent estimation of conditional multivariate extreme value distributions

Analysing the extremes of multi-dimensional data is a difficult task for many reasons, e.g. the wide range of extremal dependence structures and the scarcity of the data. Some popular approaches that account for various extremal dependence types are based on asymptotically motivated models so that there is a probabilistic underpinning basis for extrapolating beyond observed levels. Among these efforts, Heffernan and Tawn developed a methodology for modelling the distribution of a d-dimensional variable when at least one of its components is extreme. Their approach is based on a series (i=1,…,d) of conditional distributions, in which the distribution of the rest of the vector is modelled given that the ith component is large. This model captures a wide range of dependence structures and is applicable to cases of large d. However their model suffers from a lack of self-consistency between these conditional distributions and so does not uniquely determine probabilities when more than one component is large. This paper looks at these unsolved issues and makes proposals which aim to improve the efficiency of the Heffernan–Tawn model in practice. Tests based on simulated and financial data suggest that the proposed estimation method increases the self-consistency and reduces the RMSE of the estimated coefficient of tail dependence.

An Analysis of a Heuristic Procedure to Evaluate Tail (in)dependence

Measuring tail dependence is an important issue in many applied sciences in order to quantify the risk of simultaneous extreme events. A usual measure is given by the tail dependence coefficient. The characteristics of events behave quite differently as these become more extreme, whereas we are in the class of asymptotic dependence or in the class of asymptotic independence. The literature has emphasized the asymptotic dependent class but wrongly infers that tail dependence will result in the overestimation of extreme value dependence and consequently of the risk. In this paper we analyze this issue through simulation based on a heuristic procedure.

Modeling Dependence of Price Spikes in Australian Electricity Markets

Price spikes are of particular importance due to their severe impacts on consumers, businesses and industry. They constitute a major source of price risk to market participants, e.g., electricity retailers with commitments to meet customers' daily electricity demands. To those trading in several electricity markets simultaneously, the probability of simultaneous price spikes termed as tail dependence is of great importance when computing risks. For this purpose, the problem of modeling joint price spikes in the Australian electricity market is considered. A common measure of tail dependence measure is the so-called tail dependence coefficient (TDC). We present a nonparametric estimator of the tail dependence and further, the point estimation is complemented with an hypotheses test. We find significant tail dependence in electricity prices that cannot be ignored. Accurate characterization of this tail dependence is important for a variety of risk management purposes. These include hedging activities of market participants.