Tail Copula Research Articles

Subsample analysis of stock market – cryptocurrency returns tail dependence: A copula approach for the tails

This paper describes the extremal and tail dependence between G7 stock market returns (USA, Canada, UK, Japan, Germany, France, Italy) and cryptocurrency returns (Bitcoin, Ethereum, Dash, Monero, Ripple) on the basis of the bivariate extremal dependence model (Padoan and Stupfler, 2022) and the bivariate test for equality of tail copulas (Can et al., 2023). Over a daily data period from January 1, 2016 through January 5, 2023, the findings reveal significant differences in tail risk between the categories of G7 stock market returns and cryptocurrency returns, highlighting the need to find an appropriate asset to hedge individual financial asset tail risks. The results also demonstrate a time-invariant tail dependence between G7 stock market returns and cryptocurrency returns, as the Wald test supports the null of an equal tail dependence structure between stock market and cryptocurrency returns during both calm and crises times. The finding offer important policy implications for portfolio managers and investors.

Unlocking the Impact of Climate Change Mitigation Policies: A Comprehensive Study of Clean and Dirty Innovation Dynamics

Achieving the Paris Climate Agreement’s goal of limiting global warming to 1.5 °C to 2 °C by the end of the century will require massive investments in environmental technologies and a drastic shift away from high-carbon technologies. This paper investigates the impact of climate change mitigation policies on clean energy innovation. A statistical evaluation of the impact of public policies on the rate and direction of innovation for a lowcarbon future is complicated by the nature of the data and the absence of benchmarks. In addition, the statistical analysis is further complicated by the spillover effects between clean and dirty innovation and by the lag effects. In this paper, the authors assess the effects of both public policies, such as carbon taxes and green subsidies, and economic and environmental conditions, such as oil prices, large recessions, climate-related disasters, etc., on clean innovation using a nonparametric method based on the copula distribution of clean innovation. The authors collect data from the European Patent Office (EPO) Worldwide Patent Statistical (PATSTAT) Database, both on clean and dirty patents. This database is managed by the EPO and compiles data from patent offices around the world. The emphasis is put on inventions for which a patent application has been submitted to the United States Patent and Trademark Office (USPTO). The inventions are dated based on the date of their first patent application. Clean innovation refers to patents in areas such as renewable energy generation and electric vehicles, while dirty innovation refers to fossil-based energy generation and internal combustion engines. The authors employ a novel nonparametric test against pairwise differences, especially in tail dependence structures, which we measure with tail copulas, thereby avoiding the possibility of parametric misspecification. This method also permits to examine the effects of various interventions and economic conditions on different portions of the distribution, with a particular emphasis on tail dependence. The authors identify nonlinear dependence structures between clean innovation, public policies, and economic determinants like the oil price and recession. By comparing the effects of clean and dirty innovation, we can determine whether the effect on clean innovation is distinct. The findings indicate that the tightening of environmental policies since the early 1990s has statistically and economically contributed to the increase in clean innovation. The findings can bolster public support for green R&D. In addition, they suggest that green policies may be able to increase the knowledge diffusion of clean innovation.

Measuring non-exchangeable tail dependence using tail copulas

AbstractQuantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpretations in that the MTCM evaluates the largest limiting probability among all comparable rectangles in the tail, and the ATCM is a normalized average of these limiting probabilities. In contrast to the TDC, the proposed measures can capture non-exchangeable tail dependence. Analytical forms of the proposed measures are also derived for various copulas. A real data analysis reveals striking tail dependence and tail non-exchangeability of the return series of stock indices, particularly in periods of financial distress.

Two-Sample Testing for Tail Copulas with an Application to Equity Indices

A novel, general two-sample hypothesis testing procedure is established for testing the equality of tail copulas associated with bivariate data. More precisely, using a martingale transformation of a natural two-sample tail copula process, a test process is constructed, which is shown to converge in distribution to a standard Wiener process. Hence, from this test process a myriad of asymptotically distribution-free two-sample tests can be obtained. The good finite-sample behavior of our procedure is demonstrated through Monte Carlo simulations. Using the new testing procedure, no evidence of a difference in the respective tail copulas is found for pairs of negative daily log-returns of equity indices during and after the global financial crisis.

Extremal Dependence-Based Specification Testing of Time Series

We propose a specification test for conditional location–scale models based on extremal dependence properties of the standardized residuals. We do so comparing the left-over serial extremal dependence—as measured by the pre-asymptotic tail copula—with that arising under serial independence at different lags. Our main theoretical results show that the proposed Portmanteau-type test statistics have nuisance parameter-free asymptotic limits. The test statistics are easy to compute, as they only depend on the standardized residuals, and critical values are likewise easily obtained from the limiting distributions. This contrasts with some extant tests (based, e.g., on autocorrelations of squared residuals), where test statistics depend on the parameter estimator of the model and critical values may need to be bootstrapped. We show that our tests perform well in simulations. An empirical application to S&P 500 constituents illustrates that our tests can uncover violations of residual serial independence that are not picked up by standard autocorrelation-based specification tests, yet are relevant when the model is used for, for example, risk forecasting.

Two-Sample Testing for Tail Copulas with an Application to Equity Indices

A novel, general two-sample hypothesis testing procedure is established for testing the equality of tail copulas associated with bivariate data. More precisely, using an ingenious transformation of a natural two-sample tail copula process, a test process is constructed, which is shown to converge in distribution to a standard Wiener process. Hence, from this test process a myriad of asymptotically distribution-free two-sample tests can be obtained. The good finite-sample behavior of our procedure is demonstrated through Monte Carlo simulations. Using the new testing procedure, no evidence of a difference in the respective tail copulas is found for pairs of negative daily log-returns of equity indices during and after the global financial crisis.

A measurement of affluence and poverty interdependence across countries: Evidence from the application of tail copula

AbstractThis paper examines affluence and poverty interdependence across 185 countries and its evolution over 1969–2014. To estimate affluence and poverty interdependence and derive tail interdependences the tail copulae are applied to multivariate density function. The tail coefficients are estimated in the non‐parametric way as in Schmidt and Stadtmüller (2006). The estimates show, that poverty is less interdependent and continue to decrease, while affluence has asymptotically high global dependence, meaning a higher global dependence on and sensitivity to the well‐being of the affluent countries.The received results derive the pattern of the extreme interdependence and can help to identify poverty and affluence spill‐over across countries and regions and calculate the average sensitivity of a country to these phenomena on the global level and can potentially help in poverty reduction strategies within the Sustainable Development Goal by the United Nations.

Empirical tail copulas for functional data

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula, uniformly over all finite subsets of at most D points (D fixed). An application for testing tail copula stationarity is presented. The main tool for deriving the result is the uniform asymptotic normality of all the D-variate tail empirical processes. The proof of the main result is nonstandard.

Empirical Tail Copulas for Functional Data

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula,uniformly over all finite subsets of at most D points (D fixed). As a special case we obtain the uniform asymptotic normality of all estimated uppertail dependence coeffcients. The main tool for deriving the result is the uniform asymptotic normality of all the D-variate tail empirical processes. The proof of the main result is non-standard.

Conditional excess risk measures and multivariate regular variation

Abstract Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable.

Detecting Structural Differences in Tail Dependence of Financial Time Series

ABSTRACTAn accurate assessment of tail inequalities and tail asymmetries of financial returns is key for risk management and portfolio allocation. We propose a new test procedure for detecting the full extent of such structural differences in the dependence of bivariate extreme returns. We decompose the testing problem into piecewise multiple comparisons of Cramér–von Mises distances of tail copulas. In this way, tail regions that cause differences in extreme dependence can be located and consequently be targeted by financial strategies. We derive the asymptotic properties of the test and provide a bootstrap approximation for finite samples. Moreover, we account for the multiplicity of the piecewise tail copula comparisons by adjusting individual p-values according to multiple testing techniques. Monte Carlo simulations demonstrate the test’s superior finite-sample properties for common financial tail risk models, both in the iid and the sequentially dependent case. During the last 90 years in U.S. stock markets, our test detects up to 20% more tail asymmetries than competing tests. This can be attributed to the presence of nonstandard tail dependence structures. We also find evidence for diminishing tail asymmetries during every major financial crisis—except for the 2007–2009 crisis—reflecting a risk-return trade-off for extreme returns. Supplementary materials for this article are available online.

Assessing the copula selection for bivariate frequency analysis based on the tail dependence test

The flood characteristics, namely, peak, duration and volume provide important information for the design of hydraulic structures, water resources planning, reservoir management and flood hazard mapping. Flood is a complex phenomenon defined by strongly correlated characteristics such as peak, duration and volume. Therefore, it is necessary to study the simultaneous, multivariate, probabilistic behaviour of flood characteristics. Traditional multivariate parametric distributions have widely been applied for hydrological applications. However, this approach has some drawbacks such as the dependence structure between the variables, which depends on the marginal distributions or the flood variables that have the same type of marginal distributions. Copulas are applied to overcome the restriction of traditional bivariate frequency analysis by choosing the marginals from different families of the probability distribution for flood variables. The most important step in the modelling process using copula is the selection of copula function which is the best fit for the data sample. The choice of copula may significantly impact the bivariate quantiles. Indeed, this study indicates that there is a huge difference in the joint return period estimation using the families of extreme value copulas and no upper tail copulas (Frank, Clayton and Gaussian) if there exists asymptotic dependence in the flood characteristics. This study suggests that the copula function should be selected based on the dependence structure of the variables. From the results, it is observed that the result from tail dependence test is very useful in selecting the appropriate copula for modelling the joint dependence structure of flood variables. The extreme value copulas with upper tail dependence have proved that they are appropriate models for the dependence structure of the flood characteristics and Frank, Clayton and Gaussian copulas are the appropriate copula models in case of variables which are diagnosed as asymptotic independence.

Nonparametric tests for constant tail dependence with an application to energy and finance

New tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas are presented. To obtain asymptotic properties, we derive a new limit result for the sequential empirical tail copula process. Moreover, consistency of both the tests and a break-point estimator are proven. We analyze the finite sample behavior of the tests by Monte Carlo simulations. Finally, and crucial from a risk management perspective, we apply the new findings to datasets from energy and financial markets.

Nonparametric Tests for Constant Tail Dependence with an Application to Energy and Finance

The present paper proposes new tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas. To obtain asymptotic properties, we derive a new limit result for the sequential empirical tail copula process. Moreover, consistency of both the tests and a break-point estimator are proven. We analyze the finite sample behavior of the tests by Monte Carlo simulations. Finally, and crucial from a risk management perspective, we apply the new findings to datasets from energy and financial markets.

Nonparametric estimation of the conditional tail copula

The tail copula is widely used to describe the dependence in the tail of multivariate distributions. In some situations such as risk management, the dependence structure may be linked with some covariate. The tail copula thus depends on this covariate and is referred to as the conditional tail copula. The aim of this paper is to propose a nonparametric estimator of the conditional tail copula and to establish its asymptotic normality. Some illustrations are presented both on simulated and real datasets.

Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas

Let (X1, Y1), … , (Xn, Yn) be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima max( Xi) and max(Yi) is then characterized by the marginal extreme value indices and the tail copula R. We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula R. The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric estimator of R. The transformed empirical process converges weakly to a standard Wiener process, paving the way for a multitude of asymptotically distribution-free goodness-of-fit tests. We also extend our results to the m-variate (m > 2) case. In a simulation study we show that the limit theorems provide good approximations for finite samples and that tests based on the transformed empirical process have high power.

Asymptotically distribution-free goodness-of-fit testing for tail copulas

Let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima $\bigvee_{i=1}^nX_i$ and $\bigvee_{i=1}^nY_i$ is then characterized by the marginal extreme value indices and the tail copula $R$. We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula $R$. The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric estimator of $R$. The transformed empirical process converges weakly to a standard Wiener process, paving the way for a multitude of asymptotically distribution-free goodness-of-fit tests. We also extend our results to the $m$-variate ($m>2$) case. In a simulation study we show that the limit theorems provide good approximations for finite samples and that tests based on the transformed empirical process have high power.

Estimation of risk measures in energy portfolios using modern copula techniques

The dependence structure between WTI and Brent crude oil spot log-returns is analysed using modern copula techniques. In a first step, to account for autocorrelation and volatility clustering in the marginals, several single equation models are applied. Second, to select both copulas and tail copulas characterising the joint dynamics between the time series, newly introduced bootstrap-based goodness-of-fit tests are implemented and evaluated. Based on each approach, a comprehensive backtesting is performed by simulating and comparing the risk measures Value-at-Risk and Expected Shortfall with observed values.

Multiplier bootstrap of tail copulas with applications

For the problem of estimating lower tail and upper tail copulas, we propose two bootstrap procedures for approximating the distribution of the corresponding empirical tail copulas. The first method uses a multiplier bootstrap of the empirical tail copula process and requires estimation of the partial derivatives of the tail copula. The second method avoids this estimation problem and uses multipliers in the two-dimensional empirical distribution function and in the estimates of the marginal distributions. For both multiplier bootstrap procedures, we prove consistency. For these investigations, we demonstrate that the common assumption of the existence of continuous partial derivatives in the the literature on tail copula estimation is so restrictive, such that the tail copula corresponding to tail independence is the only tail copula with this property. Moreover, we are able to solve this problem and prove weak convergence of the empirical tail copula process under nonrestrictive smoothness assumptions that are satisfied for many commonly used models. These results are applied in several statistical problems, including minimum distance estimation and goodness-of-fit testing.

Nonparametric Tail Copula Estimation: An Application to Stock and Volatility Index Returns

In this study, we measure asymmetric negative tail dependence and discuss their statistical properties. In a simulation study, we show the reliability of nonparametric estimators of tail copula to measure not only the common positive lower and upper tail dependence, but also the negative “lower–upper” and “upper–lower” tail dependence. The use of this new framework is illustrated in an application to financial data. We detect the existence of asymmetric negative tail dependence between stock and volatility indices. Many common parametric copula models used in finance fail to capture this characteristic.