Tail Dependence Research Articles

Probabilistic analysis of a sustainable landfill cover considering stress-dependent water retention model and copula-based random fields

Material properties, crucial design parameters in landfill cover systems, can exhibit spatial variability due to non-uniform particle size and uneven compaction, affecting cover performance. Additionally, cover thickness, another key design parameter, can induce stress variation and influence soil-water retention capability. Previous designs always ignored these uncertainties and stress effects, potentially leading to unreasonable design schemes. This study aims to provide design recommendations for a three-layer landfill cover system under heavy rainfall, considering these factors simultaneously by the random finite element method (RFEM). The stress effects are captured using a stress-dependent soil-water retention curve (SDSWRC) and the hysteresis in the SDSWRC is considered in probabilistic analyses. For practical applications, uncertainties in two easily controllable particle size parameters, D10 and D60, are depicted by copula-based cross-correlated random fields. Based on empirical equations, the spatial variability of SDSWRC and water permeability function (WPF) can be further characterised by D10 and D60 random fields. It is found that considering SDSWRC and its spatial variability (e.g., shape) can overestimate percolation by at least 1.7 times compared to deterministic analysis using SWRC. The spatial variability of SDSWRC shape has a more significant impact than spatially variable saturated water permeability on cover system performance. Among candidate copulas, the CClayton copula (survival copula of Clayton copula) yields the most conservative results due to its upper tail dependence. The particle size and thickness of the bottom layer influence the cover system performance more than the upper two layers. A bottom layer with D10≤ 0.009 mm and D60/D10≥ 7.54, and thickness ≥ 0.8 m is recommended to achieve a satisfactory performance level with a 100-year design life.

Consideration of the tail dependence of copula functions in the selection of combinations of natural hazards

In accordance with the Federal rules and regulations of the Russian Federation (paragraphs 2.8, 2.9 of RR-064-17) and international standards (IAEA SSG-3), the OUAE (the object of the use of atomic energy) project should provide information on interrelated processes, phenomena and factors of natural origin identified during engineering surveys and studies, and the parameters of relevant combinations of external hazards are determined, for which an analysis of their impact on the safety of the NPP as part of the NPP design and beyond-design accidents should be performed. Combinations of hazards can have a stronger impact on the safety of the station than each impact considered separately, and the frequency of occurrence of the combination can be comparable to the frequency of occurrence of individual impacts. Currently, probabilistic approaches are not presented in the regulatory documents of the Russian Federation and international norms. The paper [1] presents a methodology based on the theory of copula functions, the practice of which has shown that the determination of the frequency and intensity of combinations is influenced by taking into account the empirical parameters of the distribution function: correlation coefficients, tail dependencies. The analysis of the effect of taking into account the value of tail dependencies on the result of choosing a model copula function within the selection of combinations of external natural factors that should be taken into account in the project of the OUAE is carried out. It is shown that the approach presented in [1] needs to be supplemented with an estimate of the value of the upper tail dependence based on statistical data, since due to a more complete application of the properties of copula functions, the depth of justification of the results increases.

The dependence structure and causality detection in crude oil markets

ABSTRACT It is essential to capture the dependence structure among the world’s dominant crude oil markets for understanding the world’s crude oil pricing mechanism. The dependence structure and interaction between Shanghai International Energy Exchange (INE) market and other major crude oil markets in the world are investigated to determine whether the world crude oil market is efficient. Using the vine copula model based on GARCH, we analyze the dependence structure and interactions among these markets. The highest dependence is found between West Texas Intermediate (WTI) and BRENT crude oil markets, displaying strong symmetric tail dependence. DUBAI market exhibits unidirectional Granger causality to WTI and INE markets. Empirical results suggest that the world crude oil markets are not efficient and are far from One Great Pool. DUBAI and Brent markets emerge as leaders in the current international crude oil pricing system, while INE markets do not significantly influence DUBAI and BRENT markets.

Exploring Two Modified Ali-Mikhail-Haq Copulas and New Bivariate Logistic Distributions

The Ali-Mikhail-Haq copula is a bivariate ratio-type Archimedean copula known for its simplicity and flexibility in modeling moderate negative and positive dependence structures, making it widely used in various fields. However, it is limited to capturing asymmetric dependence features, significant negative correlations, or versatile tail dependence properties. This research paper proposes two modifications of the Ali-Mikhail-Haq copula that overcome these limitations, but at the price of the loss of the positive dependence nature. Contrary to the common approach that focuses on modifying the corresponding generator function, we apply direct functional changes to the Ali-Mikhail-Haq copula. We thus perturb its Archimedean identity. The first copula has the particularity of being non-exchangeable, capable of reaching an interesting level of negative dependence correlations, and possessing flexible tail dependence properties. The second copula offers another modeling option; it is exchangeable like the Ali-Mikhail-Haq copula, but it benefits from a broader range of negative dependence correlations and more adaptable tail dependence properties. For the two proposed copulas, we investigate their main characteristics, including quadrant dependence, copula density function, conditional copulas, couples of value generation, extended variants via standard copula schemes, comprehensive copula orders, and weighted harmonic mean copula transformations. An application on two new bivariate logistic distributions in a two-component system context is given. When possible, numerical and graphical studies are given to strengthen the theory.

Volatility and dependence in crude oil and agricultural commodity markets

ABSTRACT This paper studies the evolution, volatility dynamics, and nonlinear dynamics between crude oil prices and major agricultural commodity prices. Semiparametric GARCH-in-Mean copula models are applied to monthly crude oil, corn, soybean, palm oil, rice, sugar, and wheat prices from January 1990 to May 2023. We find that the Clayton copula is the best copula to describe the (bivariate) dependence structures between the crude oil and soybean, crude oil and palm oil, crude oil and sugar, and crude oil and wheat markets, suggesting the statistically significant lower tail dependence between crude oil and major agricultural commodity returns. Moreover, the tail dependence is strongest between crude oil and palm oil.

Sharp bounds on the survival function of exchangeable min-stable multivariate exponential sequences

Abstract We derive lower and upper bounds for the survival function of an exchangeable sequence of random variables, for which the scaled minimum of each finite subgroup has a univariate exponential distribution. These bounds are sharp in the sense that both bounds themselves are attained by exchangeable sequences of the same kind, for which the (non-scaled) minimum of each subgroup has the same univariate exponential distribution as the original sequence. This result is equivalent to inequalities between infinite-dimensional stable tail dependence functions, which leads to inequalities between multivariate extreme-value copulas. In addition, it is explained how an infinite-dimensional symmetric stable tail dependence function can be obtained from its upper bound by censoring certain distributional information. This technique is applied to derive new parametric families.

Transformed‐Linear Models for Time Series Extremes

To capture the dependence in the upper tail of a time series, we develop non‐negative regularly varying time series models that are constructed similarly to classical non‐extreme ARMA models. Rather than fully characterizing tail dependence of the time series, we define the concept of weak tail stationarity which allows us to describe a regularly varying time series via a measure of pairwise extremal dependencies, the tail pairwise dependence function (TPDF). We state consistency requirements among the finite‐dimensional collections of the elements of a regularly varying time series and show that the TPDF's value does not depend on the dimension of the random vector being considered. So that our models take non‐negative values, we use transformed‐linear operations. We show existence and stationarity of these models, and develop their properties such as the model TPDFs. We fit models to hourly windspeed and daily fire weather index data, and we find that the fitted transformed‐linear models produce better estimates of upper tail quantities than a traditional ARMA model, classical linear regularly varying models, a max‐ARMA model, and a Markov model.

Do shipping freight markets impact commodity markets?

The Baltic Dry Index (BDI) is mostly perceived as the leading indicator of shipping freight market and economic activity. In this study, we investigate the dependence structure and distributional predictability between BDI and nineteen major commodity investable indices to offer insights on the predictive ability of BDI for commodity assets. In addition, we explore the effects of policy uncertainty and geopolitical risk on the dynamic dependence structure between BDI and commodities futures. To this end, we apply time-varying Markov switching copula models and non-parametric causality in quantile approach to daily closed price indices of commodities futures running from 23rd May 2006 to 20th November 2020. We provide for the first-time empirical evidence of time-varying Markov tail dependence structure dynamics between marine activities and commodities futures. The nature of the dependence is positives except for petroleum, all metals, livestock wheat, cattle and softs markets. Furthermore, we show the significant effects of global uncertainties, policy uncertainty and geopolitical risk on the dependence between sea transport and commodity assets. Finally, we note from the causality-in-mean and causality-in-variance analysis that the daily change of BDI is an efficient predictor (informational efficiency) of the daily changes of most of the commodity indices. Finally, we show that markets.

Investigating the Impact of Variable Dividends and Tail Dependence in a Compound Poisson Risk Model

Investigating the Impact of Variable Dividends and Tail Dependence in a Compound Poisson Risk Model

Does Green Energy Investment Effects on Islamic and Conventional Stock Markets? New Evidence from Advanced Economies

This study investigates the long-run and short-run nexus between renewable energy investment, and Islamic and conventional stock markets in advanced economies, namely the United States, United Kingdom, and European Union over the period from January 1, 2002 to August 1, 2023. The study uses the Nonlinear Autoregressive Distributed Lag (NARDL) model to examine the long-run and short-run asymmetric effects between selected variables under study. The results of empirical model estimation suggested that the green energy investment adjustment is running towards the long- and short-run steady increment regarding positive and negative shocks in conventional and Islamic stock markets. Indeed, the shocks of the green energy investment on conventional stock markets of the US and EU are positively asymmetric in the long run; the UK has an insignificant effect in the long run. Importantly, green energy investment positively has significant effects on conventional and Islamic stock markets in the US. UK and EU (except the conventional stock market of the US). The short-run coefficients of green energy investment have a significant positive effect on Islamic stock markets for the selected U.S, UK, and EU markets. Likewise, the EU conventional stock market has a significant positive effect in the short run. Change in green energy investment has a negatively insignificant impact on Islamic and conventional markets of the EU and the UK. Indeed, short-run coefficients of green energy investment negatively on the U.S. conventional stock market. The increase in green energy investment may result in an increase in the cost of energy generation, which reduces the renewable energy production firm’s profitability and finally decreases the stock market prices in the short run. These findings have important implications for the portfolio diversification and hedging decisions of environmentally concerned investors. In sum, investors should consider the presence of extreme tail dependence between green investments and stock markets an important factor in making more informed and rational choices.

Probability equivalent level for CoVaR and VaR

For a given risk, the well-known classical definition of Value-at-Risk (VaR) does not take into account possible interactions with other observable risks. For this reason, conditional VaRs that capture contagion effects and tail dependence among risks, such as the Co-Value-at-Risk (CoVaR), have been defined and studied in recent literature. In this paper we study conditions that guarantee, in the bivariate setting, the ordering between VaR and CoVaR, allowing to understand which, among the two measures, is more or less conservative than the other. By doing this, we introduce the notion of Probability Equivalent Level of CoVaR-VaR (PELCoV), which is the VaR value of the observable variable for which VaR and CoVaR coincide, and we study some of its properties such as uniqueness and boundedness. In particular, we show that its properties are entirely explained by the copula that describes the dependence between risks, and we provide a list of copulas for which PELCoV is explicitly available, and for which it is or not bounded. A practical applicative example is also presented.

Evaluating dependence between DeFi tokens and conventional cryptocurrencies

ABSTRACT This study analyzes the dependence structures of eight leading decentralized finance tokens using the GARCH-EVT-Copula models. The empirical results indicated that the dependencies between DeFi tokens and Bitcoin and Ethereum were positive and time-varying. DeFi tokens demonstrated a stronger association with Ethereum than with Bitcoin. DeFi tokens were found to exhibit weaker lower tail dependencies, revealing the unique feature of DeFi tokens in reducing extreme downside risks and enhancing portfolio diversification.

Exploring the joint probability of precipitation and soil moisture over Europe using copulas

Abstract. The joint probability of precipitation and soil moisture is here investigated over Europe with the goal to extrapolate meaningful insights into the potential joint use of these variables for the detection of agricultural droughts within a multivariate probabilistic modeling framework. The use of copulas is explored, being the framework often used in hydrological studies for the analysis of bivariate distributions. The analysis is performed for the period 1996–2020 on the empirical frequencies derived from ERA5 precipitation and LISFLOOD soil moisture datasets, both available as part of the Copernicus European Drought Observatory. The results show an overall good correlation between the two standardized series (Kendall's τ= 0.42±0.1) but also clear spatial patterns in the tail dependence derived with both non-parametric and parametric approaches. About half of the domain shows symmetric tail dependence, well reproduced by the Student's t copula, whereas the rest of the domain is almost equally split between low- and high-tail dependences (both modeled with the Gumbel family of copulas). These spatial patterns are reasonably reproduced by a random forest classifier, suggesting that this outcome is not driven by chance. This study stresses how a joint use of standardized precipitation and soil moisture for agriculture drought characterization may be beneficial in areas with strong low-tail dependence and how this behavior should be carefully considered in multivariate drought studies.

On a Compound Poisson Risk Model Perturbed by Brownian Motion with Variable Premium and Tail Dependence between Claims Amounts and Inter-Claim Time

On a Compound Poisson Risk Model Perturbed by Brownian Motion with Variable Premium and Tail Dependence between Claims Amounts and Inter-Claim Time

Breaking Boundaries: Lower Tail Dependence Can Triple the Economic Value of Index Insurance for Rural Households

Breaking Boundaries: Lower Tail Dependence Can Triple the Economic Value of Index Insurance for Rural Households

Can green investment funds hedge climate risk?

We analyze tail risk dependence between Green Investment Funds (GIFs) and climate risks, emphasizing diversification benefits, safe-haven, and hedge features from 2009 to 2022. Our methodology, using AGDCC-GARCH, time-varying optimal copula (TVOC), and conditional diversification benefits (CDB), reveal various tail dependence regimes, showcasing the role of GIFs in extreme climate events. Symmetrical co-movement, observed during COVID-19 and Russia-Ukraine war, indicates no asymmetric tail dependence. Notably, GIFs offer substantial diversification benefits against climate risks, providing insights for investors and policymakers to formulate proactive strategies in line with sustainability goals.

Efficiency gains in value-at-risk and expected shortfall estimation by using copulas and full maximum likelihood

We provide Monte Carlo evidence on the efficiency gains obtained in GARCH-based estimations of value-at-risk (VaR) and expected shortfall (ES) by incorporating dependence information through copulas and subsequently using full maximum likelihood (FML) estimates. First, we consider an individual returns series; in this case, the efficiency gain stems from exploiting the relationship with another returns series using a copula model. Second, we consider a portfolio returns series obtained as a linear combination of returns series related through a copula model; in this case, the efficiency gain stems from using FML estimates instead of two-stage ML estimates. We consider three copulas models in order to analyze the effect of the type and degree of tail dependence on the results. Our results show that, in these situations, using copula models and FML leads to a substantial reduction in the mean squared error of the VaR and ES estimates when there is a relatively high degree of dependence between returns (up to 70% in the presence of lower-tail dependence) and a notable improvement in the performance of backtesting procedures.

Vine Copula Model: Application to Chemical Elements in Water Samples

Copula can link the bivariate distribution function with marginal distribution functions without requiring specific information about the interdependence among random variables. There are several types of copulas, such as elliptical copulas, Archimedean copulas, and extreme value copulas. However, in multivariate modeling, each type of copula has limitations in modeling complex dependence structures in terms of symmetry and tail dependence properties. The class of vine copulas overcomes these limitations by constructing multivariate models using bivariate copulas in a tree-like structure. The bivariate copulas used in this study include the Clayton, Gumbel, Frank, Gaussian, and Student's t copula families. This study discusses the construction of vine copula models, parameter estimation, and their applications. The construction of vine copulas is done through the decomposition of conditional probability density functions and substituting bivariate copula density functions into the decomposition results. The data used in the study is the logarithm of the concentration of chemical elements in water samples in Colorado. The parameter estimation method used is pseudo-maximum likelihood with sequential estimation. Model selection is then performed using the Akaike information criterion (AIC) to determine the most suitable model. The results indicate that Caesium and Titanium have a dependency relationship with Scandium. Moreover, Scandium and Titanium exhibit the strongest dependence compared to other variable pairs.

A New Family of Archimedean Copulas: The Half-Logistic Family of Copulas

In this research, we introduce a truncation of the half-logistic distribution function as a multiplicative Archimedean generator. The corresponding Archimedean copula is obtained, namely the half-logistic family. The dependency structure of this copula is distinct from other well-known ones. Kendall’s tau correlation coefficient is obtained in exact form and found to cover the entire positive dependence range (i.e., [0, 1]). We have proven that this copula is positively ordered and has no tail dependencies. The density of this copula is shown to be totally positive of order two. An extension of this copula is also introduced by adding a second parameter. This extension allows for a negative correlation and connects the famous Frank copula to the half-logistic copula. Two datasets were used to compare the half-logistic copula with some other known copula models.

Basis risk and the demand for catastrophic rainfall insurance

Abstract Rainfall is an important source of covariate shock in developing countries. Insurance against a rainfall index has, therefore, held much promise as a formal insurance product to protect the livelihoods of poor farmers. But how good is rainfall as a measure of covariate shocks? The imperfect association between them has been flagged as a reason for low demand for index insurance. Using district crop yields and rainfall data for India, we find that tail dependence characterizes the association between aggregate crop yields and rainfall. Could this property be exploited to design catastrophic loss insurance programs that have low basis risk? Using simulations of the estimated copulas, we show that the value of index-based insurance relative to actuarial cost is higher for insurance against extreme losses (of the index) than for insurance against all losses. We conclude that index insurance could find greater acceptability if it only insured extreme losses.