The weak tail dependence coefficient of the elliptical generalized hyperbolic distribution

This article defines coefficient of upper-lower tail dependence and coefficient of lower-upper tail dependence based on coefficient of upper tail dependence and coefficient of lower tail dependence and does an empirical research on tail dependence in China stock market. A mixed Copula that can well describe dependence structure between Shanghai Exchange Composite Index(SHECI) and Shenzhen Exchange Component Index(SZECI) is found by empirical analysis firstly, and is used to calculate coefficients of tail dependence. The results show that there are strong upper tail dependence and lower tail dependence, but no upper-lower tail dependence and lower-upper tail dependence between daily returns of SHECI and SZECI.

This thesis presents the concept of tail dependence in a financial context as one tool to measure dependence in the extremes of a bivariate distribution. Copulae can separate the problem of estimating a multidimensional distribution into the estimation of the marginal distributions and the dependence between the onedimensional random variables. Therefore, copulae are used in order to carry out the estimation of the tail dependence coefficient (TDC). Four estimators of the TDC are presented and compared in a simulation study for various distributions and copulae. Furthermore, an introduction into bivariate Extreme Value Theory (EVT) is given, which tries precisely to analyze the behavior at the tail of a bivariate distribution. EVT allows to construct estimators of the TDC and to derive a test for tail independence, which is recognized to be indispensable but rarely utilized in a financial context. As an application to nine different financial data sets shows, the phenomenon of tail dependence is less common than often argued in the literature: the periods where indeed tail independence can be rejected are few.

Characterization and quantification of climate extremes and their dependencies are fundamental to the studying of natural hazards. This chapter reviews various parametric and nonparametric tail dependence coefficient estimators. The tail dependence coefficient describes the dependence (degree of association) between concurrent extremes at different locations. Accurate and reliable knowledge of the spatial characteristics of extremes can help improve the existing methods of modeling the occurrence probabilities of extreme events. This chapter will review these methods and use two case studies to demonstrate the application of tail dependence analysis.

Models characterizing the asymptotic dependence structures of bivariate distributions have been introduced by Ledford and Tawn (1996), among others, and diagnostics for such dependence behavior are presented in Coles et al. (1999). The following pages are intended as a supplement to the papers of Ledford and Tawn and Coles et al. In particular we focus on the coefficient of tail dependence, which we evaluate for a wide range of bivariate distributions. We find that for many commonly employed bivariate distributions there is little flexibility in the range of limiting dependence structure accommodated. Many distributions studied have coefficients of tail dependence corresponding to near independence or a strong form of dependence known as asymptotic dependence.

Interval estimation for a measure of tail dependence

Operational forest fire danger rating systems rely on the recent evolution of meteorological variables to estimate dead fuel condition. Further combining the latter with meteorological and environmental variables, they predict fire occurrence and spread. In this study we retrieved live fuel condition from MODIS multispectral measurements in the near infrared and shortwave infrared. Next, we combined these retrievals with an extensive dataset on actual forest fires in Campania (13,595 km2), Italy, to determine how live fuel condition affects the probability distribution functions of fire characteristics. Accordingly, the specific objective of this study was to develop and evaluate a new approach to estimate the probability distribution functions of fire burned area, duration and rate of spread as a function of the Perpendicular Moisture Index (PMI), whose value decreases with decreasing live fuel moisture content (LFMC). To this purpose, available fire data was intersected with MODIS 8-day composited reflectance data so to associate each fire event with the corresponding pre-fire PMI observation. Fires were then grouped in ten decile bins of PMI, and the conditional probability distribution functions of burned area, fire duration and rate of spread were determined in each bin. Distributions of burned area and rate of spread vary across PMI decile bins, while no significant difference was observed for fire duration. Further testing this result with a likelihood ratio test confirmed that PMI is a covariate of burned area and rate of spread, but not of fire duration. We defined an extreme event as a fire whose burned area (respectively rate of spread) exceeds the 95th percentile of the frequency distribution of all observed fire events. The probability distribution functions in the ten decile bins of PMI were combined to obtain a conditional probability distribution function, which was then used to predict the probability of extreme fires, as defined. It was found that the probability of extreme events steadily increases with decreasing PMI. Overall, at the end of the dry season the probability of extreme events is about the double than at the beginning. These results may be used to produce frequently (e.g. daily) updated maps of the probability of extreme events given a PMI map retrieved from e.g. MODIS reflectance data.

This paper is concerned with a procedure for financial time series clustering, aimed at creating groups of time series characterized by similar behavior with regard to extreme events. The core of our proposal is a double clustering procedure: the former is based on the lower tail dependence of all the possible pairs of time series, the latter on the upper tail dependence. Tail dependence coefficients are estimated with copula functions. The final goal is to exploit the two clustering solutions in an algorithm designed to create a portfolio that maximizes the probability of joint positive extreme returns while minimizing the risk of joint negative extreme returns. In financial crisis scenarios, such a portfolio is expected to outperform portfolios generated by the traditional methods. We describe the results of a simulation study and, finally, we apply the procedure to a dataset composed of the 50 assets included in the EUROSTOXX index.

We explore extreme return-volumes dependence among different cryptocurrencies such as Bitcoin, Ethereum, Ripple, and Litecoin by using the Copula approach. We use Student-t, Frank, Clayton, Survival Clayton, Gumbel, and SJC copulas. We filter out margins by using the EGARCH model for return series and GARCH model for volume series. Evidence of significant symmetric dependence between return-volume is not found due to insignificance of student-t and Frank copula parameters. In a return-volume relationship, coefficients of lower tail dependence are significant for Bitcoin, Ripple, and Litecoin which means that low returns are followed by low volumes. Lower tail dependence for the return-volume relationship is stronger than the upper tail dependence for Bitcoin, Ripple, and Litecoin. Moreover, for negative return-volume, left tail dependence coefficients are significant for Ripple and Litecoin, which means that high returns are followed by low volumes for Ripple and Litecoin. Our investigation shows that investors (buyer or seller) are very careful in extreme market conditions for both Ripple and Litecoin. Extreme upper tail and lower tail dependence coefficients are insignificant for Ethereum.

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

Examples for the coefficient of tail dependence and the domain of attraction of a bivariate extreme value distribution

Wind speed scenario generation based on dependency structure analysis

Tail dependence is understood as the dependence between the variables assuming that these variables take the values from the tails of univariate distributions. In the paper two approaches of tail dependence determination are discussed: conditional correlation coefficient and tail dependence coefficients. It can be argued that both approaches are the generalizations of the well-known univariate approach, based on conditional excess distribution. In the paper the proposal is also given to extend tail dependence coefficients to the general multivariate case and to represent these coefficients through copula function.

Among bivariate tail dependence measures, the tail dependence coefficient has emerged as the popular choice. Akin to the correlation matrix, a multivariate dependence measure is constructed using these bivariate measures, and this is referred to in the literature as the tail dependence matrix (TDM). While the problem of determining whether a given d × d matrix is a correlation matrix is of the order O(d3) in complexity, determining if a matrix is a TDM (the realization problem) is believed to be non-polynomial in complexity. Using a linear programming (LP) formulation, we show that the combinatorial structure of the constraints is related to the intractable max-cut problem in a weighted graph. This connection provides an avenue for constructing parametric classes admitting a polynomial in d algorithm for determining membership in its constraint polytope. The complexity of the general realization problem is justifiably of much theoretical interest. Since in practice one resorts to lower dimensional parametrization of the TDMs, we posit that it is rather the complexity of the realization problem restricted to parametric classes of TDMs, and algorithms for it, that are more practically relevant. In this paper, we show how the inherent symmetry and sparsity in a parametrization can be exploited to achieve a significant reduction in the LP formulation, which can lead to polynomial complexity of such realization problems - some parametrizations even resulting in the constraint polytope being independent of d. We also explore the use of a probabilistic viewpoint on TDMs to derive the constraint polytopes.

Mid-latitude extreme weather events are responsible for a large part of climate-related damage. Yet large uncertainties remain in climate model projections of heat waves, droughts, and heavy rain/snow events on regional scales, limiting our ability to effectively use these projections for climate adaptation and mitigation. These uncertainties can be attributed to both the lack of spatial resolution in the models, and to the lack of a dynamical understanding of these extremes. The objectives of this proposal were to first to develop empirical and theoretical understanding of how fine-scale features of the atmospheric circulation that are responsible for extreme events (such as blocking, atmospheric rivers, Rossby wave breaking) are related to more easily quantifiable larger scale circulation patterns, and on the basis of this develop specific metrics that can be applied to a range of atmospheric and coupled atmosphere-ocean models to diagnose their prediction of extremes. These metrics were used to understand what model resolution is necessary to describe the probability of extreme events realistically, and will be applied to CMIP5 models and high resolution time-slice projections of future climate to better understand the potential for the shifts and intensification of extremes over North America in a changing climate. The approach was to relate the fine-scale features to the large scales in current climate simulations, seasonal re-forecasts, and climate change projections in a very wide range of models, including the atmospheric and coupled models of ECMWF over a range of horizontal resolutions (125 to 10 km), aqua-planet configuration of the Model for Prediction Across Scales and High Order Method Modeling Environments (resolutions ranging from 240 km – 7.5 km) with various physics suites, and selected CMIP5 model simulations. The large scale circulation will be quantified both on the basis of the well-tested preferred circulation regime approach, and very recently developed measures, the finite amplitude Wave Activity and its spectrum. The fine scale structures related to extremes were diagnosed following the latest approaches in the literature. The goal is to use the large scale measures as indicators of the probability of occurrence of the finer scale structures, and hence extreme events. These indicators will then be applied to the CMIP5 models and time-slice projections of a future climate. The expected outcomes of this research were: (i) a unified methodology and set of metrics that will enable modelers to assess the probability of extreme events in various current and future climate simulations, (ii) an estimate of the model resolution needed to capture extreme events in a realistic way, and (iii) a better understanding of the extreme events over North America in a changing climate.

Motivation: The motivation behind the study was to analyze the risk of sustainable stock indices (SSIs) and their conventional peers during the COVID-19 health crisis. We wanted to check if SSIs were more resistant to market risk in the times of huge volatility as well as what the correlation between particular SSIs was and, similarly, between their conventional peers. Aim: The main objective of the study was to analyze volatility spillover among sustainability stock indices (SSIs) and their conventional peers during the COVID-19 health crisis. The authors analyze conditional volatility among SSIs, which was obtained from univariate GARCH-type models, and the tail dependence coefficient, which was derived from the Copula-GARCH models. The indices from FTSE4Good family for the USA, Europe and Japan markets together with their corresponding conventional indices have been chosen. Results: The research shows that during the COVID-19 health crisis SSIs were less volatile than their conventional peers. Moreover the relations between particular SSIs in most cases were weaker, whereas extreme observations which occurred were less concordant for conventional indices. It implies that the stability of sustainable stock indices from FTSE4good family is greater than for their conventional peers.