Tail Dependence Research Articles

A non‐stationary factor copula model for non‐Gaussian spatial data

AbstractWe introduce a new copula model for non‐stationary replicated spatial data. It is based on the assumption that a common factor exists that controls the joint dependence of all the observations from the spatial process. As a result, our proposal can model tail dependence and tail asymmetry, unlike the Gaussian copula model. Moreover, we show that the new model can cover a full range of dependence between tail quadrant independence and tail dependence. Although the log‐likelihood of the model can be obtained in a simple form, we discuss its numerical computational issues and ways to approximate it for drawing inference. Using the estimated copula model, the spatial process can be interpolated at locations where it is not observed. We apply the proposed model to temperature data over the western part of Switzerland, and we compare its performance with that of its stationary version and with the Gaussian copula model.

Extreme Risk Spillovers From US Soybean Futures Market to China's Soybean‐Linked Futures Markets

ABSTRACTThis study investigates the cross‐border risk spillovers between the US soybean futures market and Chinese soybean‐related futures markets. We first confirm the existence of strong tail dependence between US soybean futures and four Chinese soybean‐related futures by conducting a novel quantile‐Granger causality test. Second, tests under MVMQ‐CAViaR further provide evidence of risk spillovers from CBOT soybean futures to the DCE No.1 soybean, No.2 soybean, soybean meal, and soybean oil futures in value‐at‐risk at different quantiles. Lastly, results from the quantile impulse‐response function reveal the time‐varying and asymmetric property of risk spillover effects. In addition, we compare the results from two subsample periods and identify different risk spillover effects across markets at different quantiles that may contribute to the investors' decision‐making under extreme market conditions.

Dynamic Dependence between Oil and Stock Markets: International Evidence with Stochastic Copula Approach

Modeling the dependency structure between variables has recently received increasing attention in many disciplines, especially finance and economics. In this study, the dependence between oil prices and stock markets is investigated through the stochastic copula approach, which is a class of time-varying copulas. This model enables to capture whole dependency between variables dynamically. Unlike time-varying copulas, it takes into account the latent process as well as observations in modeling dependency and thus evaluates the dependency structure in a more comprehensive framework. Empirical findings suggest that dependency between oil and stock markets evolve over time. There is a symmetric dependence between oil and the UK stock market, but the relationship between oil and the US stock market is measured by upper tail dependence. This indicates that oil and the US stock market are more likely to move together during periods of market uptrend.

Copula-based Bayesian estimation of probabilities for two dependent non homogenous compound Poisson processes

Bivariate copulas for two dependent non homogeneous compound Poisson processes (NCPP) are considered. In this scenario, the arrival times of events are assumed to follow two independent non homogeneous Poisson processes (NHPP), but the corresponding renewal variables are dependent through a bivariate copula. The article aims to provide a computational tool for the estimation of the probability that the random sum associated with the first NCPP is less than the other random sum by a truncated time. The proposed ML and Bayesian estimation methods in a general setting can be applied using bivariate copulas such as Gumbel (for upper tail dependency), Clayton (for lower tail dependency), and Frank (for symmetric dependency). Power Law (PL) intensity function for the NHPPs and Inverse-Gaussian as renewal distribution are considered as examples. Bayes estimates of NHPPs parameters use gamma priors, and for Bayes estimates of Inverse-Gaussian distribution, an MCMC algorithm is employed. Our method has a broad range of practical applications, especially in the insurance sector, where the renewal variable may represent the dollar amount associated with a claim. To illustrate our approach, we analyze injury claims in North Carolina and South Carolina over a specific period, using a dataset sourced from docs.databricks.com.

The Tail Dependence and Lead-Lag Relationship in Financial Markets

The Tail Dependence and Lead-Lag Relationship in Financial Markets

Likelihood Inference for Factor Copula Models with Asymmetric Tail Dependence.

For multivariate non-Gaussian involving copulas, likelihood inference is dominated by the data in the middle, and fitted models might not be very good for joint tail inference, such as assessing the strength of tail dependence. When preliminary data and likelihood analysis suggest asymmetric tail dependence, a method is proposed to improve extreme value inferences based on the joint lower and upper tails. A prior that uses previous information on tail dependence can be used in combination with the likelihood. With the combination of the prior and the likelihood (which in practice has some degree of misspecification) to obtain a tilted log-likelihood, inferences with suitably transformed parameters can be based on Bayesian computing methods or with numerical optimization of the tilted log-likelihood to obtain the posterior mode and Hessian at this mode.

Large Skew-t Copula Models and Asymmetric Dependence in Intraday Equity Returns

Skew-t copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A stochastic gradient ascent algorithm is used to solve the variational optimization. The methodology is used to estimate skew-t factor copula models with up to 15 factors for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. In a moving window study we show that the asymmetric dependencies also vary over time, and that intraday predictive densities from the skew-t copula are more accurate than those from benchmark copula models. Portfolio selection strategies based on the estimated pairwise asymmetric dependencies improve performance relative to the index.

Quantile connectedness and network among Green bonds, Renewable Energy, and G7 sustainability markets

This paper examines the quantile spillovers and interconnectedness between green bonds, renewable energy, and the G7 sustainability markets under extreme market conditions. It reaches beyond the widely employed studies based in the mean values of variables by resorting to the advanced quantile vector auto-regression methodology. The results show that the overall connectedness is stronger during extreme, either bearish or bullish market situations, than during normal market conditions. Furthermore, we show that all sustainability markets, except Japan, are the net transmitters of shocks, while all the green bonds are the net receivers. Moreover, we document that spillovers vary significantly over time and that extreme market conditions reduce the net spillover influences. The relative tail dependence analysis shows that the connectedness at the lower quantile relatively dominates the connectedness at the upper quantile. Finally, the total spillovers are higher at the extremes than at the median quantile, whereas the periods of major crises are associated with unusually high connectedness, as consistent with the literature on contagion effects.

The taxonomy of tail risk

AbstractWe use tail events at different levels of severity to define an asset's tail risk and to decompose the latter into a systematic and an idiosyncratic component. The systematic component captures an asset's tendency to experience joint tail losses with the market and generalizes a classic tail dependence coefficient. However, the idiosyncratic component consists of two parts: idiosyncratic tail risk that leads to asset‐specific tail losses and tail risk cushioning that dampens the tail losses emanating from the market. Tail risk cushioning is a novel concept that arises naturally in our framework, is consistent with the previous two and completes the taxonomy of tail risk. We examine the performance of our tail risk decomposition on a large dataset, confirming some previous results on tail risk and uncovering new theoretical and empirical findings.

Exploring the Relationship and Predictive Accuracy for the Tadawul All Share Index, Oil Prices, and Bitcoin Using Copulas and Machine Learning

Financial markets are increasingly interlinked. Therefore, this study explores the complex relationships between the Tadawul All Share Index (TASI), West Texas Intermediate (WTI) crude oil prices, and Bitcoin (BTC) returns, which are pivotal to informed investment and risk-management decisions. Using copula-based models, this study identified Student’s t copula as the most appropriate one for encapsulating the dependencies between TASI and BTC and between TASI and WTI prices, highlighting significant tail dependencies. For the BTC–WTI relationship, the Frank copula was found to have the best fit, indicating nonlinear correlation without tail dependence. The predictive power of the identified copulas were compared to that of Long Short-Term Memory (LSTM) networks. The LSTM models demonstrated markedly lower Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Scaled Error (MASE) across all assets, indicating higher predictive accuracy. The empirical findings of this research provide valuable insights for financial market participants and contribute to the literature on asset relationship modeling. By revealing the most effective copulas for different asset pairs and establishing the robust forecasting capabilities of LSTM networks, this paper sets the stage for future investigations of the predictive modeling of financial time-series data. The study highlights the potential of integrating machine-learning techniques with traditional econometric models to improve investment strategies and risk-management practices.

Comparing and quantifying tail dependence

We introduce a new stochastic order for the tail dependence between random variables. We then study different measures of tail dependence which are monotone in the proposed order, thereby extending various known tail dependence coefficients from the literature. We apply our concepts in an empirical study where we investigate the tail dependence for different pairs of S&P 500 stocks and indices, and illustrate the advantage of our measures of tail dependence over the classical tail dependence coefficient.

Traffic flows time series in a flood-prone area: modeling and clustering on extreme values with a spatial constraint

Time series of traffic flows, extracted from mobile phone origin–destination data, are employed for monitoring people crowding and mobility in areas subject to flooding risk. By applying a vector autoregressive model with exogenous covariates combined with dynamic harmonic regression to such time series, we detected the presence of many extreme events in the residuals, which exhibit heavy-tailed distribution. For this reason, we propose a time series clustering procedure based on tail dependence which is suitable for data characterized by a spatial dimension, since objects’ geographical proximity is taken into account. The final aim is to obtain clusters of areas characterized by the common tendency to the manifestation of extreme events, that in this case study are represented by extremely high incoming traffic flows. The proposed method is applied to the Mandolossa, a strongly urbanized area located on the western outskirts of Brescia (northern Italy) which is subject to frequent flooding.

Tail-dependence clustering of time series with spatial constraints

We introduce a clustering method for time series based on tail dependence. Such a method also considers spatial constraints by means of a suitable procedure merging temporal and spatial dependence via extreme-value copulas. The cluster composition depends on the choice of the hyper-parameter α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document} used to calibrate the contribution of the spatial dependence to the overall dissimilarity. A novel heuristic approach to select α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} based on a suitable connectedness index associated to each cluster of the partition is proposed.

Nonparametric estimator of the tail dependence coefficient: balancing bias and variance

Nonparametric estimator of the tail dependence coefficient: balancing bias and variance

Assessing the potential of compound extreme storm surge and precipitation along China's coastline

The occurrence of extreme storm surges and precipitation simultaneously or successively can lead to compound flooding. The interaction between extreme storm surges and precipitation holds significant implications for understanding the potential contributing to compound flood risk in coastal areas. This study examines the likelihood of joint occurrence for compound extreme storm surges and precipitation along the China's coastline using observations and model data sets based on tail dependence. We assess the complete characteristics of the tail dependence from observations at the spatio-temporal scale. Subsequently, we perform a principal component analysis to classify the compound flood into 6 synoptic patterns based on the mean sea level pressure data in two typical points (Xiamen and Shijiusuo). We analyze the structure dependence of both observed and simulated surge data and compare the dependence between the historical and the future tail dependence. The result shows that the Yellow Sea and East China Sea exhibit higher dependence compared to the Bohai Sea and South China Sea. The southeastern sea of China has more significant seasonal variation in dependence relative to the northern sea of China. The result indicates that the dominant weather type in Xiamen is associated with low sea pressure and high land pressure, while the type in Shijiusuo is located at the southern edge of a low-pressure center. Projected probabilities of future compound events (2015–2050) have shown substantial increases of 23.9%, 25.38%, 58.21%, and 119.47% over the current period (1979–2014), according to climate models CMCC-CM2-VHR4, GFDL-CMC192-SST, ECEarth3P-HR, and HadGEM3-GC31-HM, respectively. These findings emphasize the correlation between extreme precipitation and storm surges, contributing to a deeper understanding of the compound flood and promoting disaster prevention and control.

COVID-19 and Dependence Between Oil and Renewable Energy Stocks: C-Vine Copula Approach

This article analyses the dependence structure between traditional energy prices traditional oil indices and renewable energy stock indices employing Canonical Vine (C-Vine) copula based GARCH approach. The analysis is conducted before and during-COVID-pandemic period. Renewable energy stock is represented by SPGCE and CLEANTECH index, whereas traditional energy is represented by WTI crude oil and OVX. The empirical analysis indicated increasing asymmetric dependence between WTI crude and renewable energy indices during COVID-19. However, WTI crude oil is not found to be the primary determinant for influencing renewable energy stocks. In fact, we find a strong association between SPGCE and CLEANTECH and degree of dependence escalated during COVID-19 period. Besides this, the dependence structure between WTI crude and CLEANTECH changes from the upper tail dependence to lower tail dependence signifying the consequences of low economic activities on the technology sector. Furthermore, the dependence between OVX and SPGCE conditional on WTI reveal the scope of diversification as dependence parameters are insignificant. These findings are robust considering alternative GARCH specifications and alternative oil future benchmark and renewable energy stock index. The results provide useful implications for the energy policymakers and investors.

Max-convolution processes with random shape indicator kernels

In this paper, we introduce a new class of models for spatial data obtained from max-convolution processes based on indicator kernels with random shape. We show that these models have appealing dependence properties including tail dependence at short distances and independence at long distances. We further consider max-convolutions between such processes and processes with tail independence, in order to separately control the bulk and tail dependence behaviors, and to increase flexibility of the model at longer distances, in particular, to capture intermediate tail dependence. We show how parameters can be estimated using a weighted pairwise likelihood approach, and we conduct an extensive simulation study to show that the proposed inference approach is feasible in relatively high dimensions and it yields accurate parameter estimates in most cases. We apply the proposed methodology to analyze daily temperature maxima measured at 100 monitoring stations in the state of Oklahoma, US. Our results indicate that our proposed model provides a good fit to the data, and that it captures both the bulk and the tail dependence structures accurately.

On the Ratio-Type Family of Copulas

Investigating dependence structures across various fields holds paramount importance. Consequently, the creation of new copula families plays a crucial role in developing more flexible stochastic models that address the limitations of traditional and sometimes impractical assumptions. The present article derives some reasonable conditions for validating a copula of the ratio-type form uv/(1−θf(u)g(v)). It includes numerous examples and discusses the admissible range of parameter θ, showcasing the diversity of copulas generated through this framework, such as Archimedean, non-Archimedean, positive dependent, and negative dependent copulas. The exploration extends to the upper bound of a general family of copulas, uv/(1−θϕ(u,v)), and important properties of the copula are discussed, including singularity, measures of association, tail dependence, and monotonicity. Furthermore, an extensive simulation study is presented, comparing the performance of three different estimators based on maximum likelihood, ρ-inversion, and the moment copula method.

New Bivariate Copulas via Lomax Distribution Generated Distortions

We develop a framework for creating distortion functions that are used to construct new bivariate copulas. It is achieved by transforming non-negative random variables with Lomax-related distributions. In this paper, we apply the distortions to the base copulas of independence, Clayton, Frank, and Gumbel copulas. The properties of the tail dependence coefficient, tail order, and concordance ordering are explored for the new families of distorted copulas. We conducted an empirical study using the daily net returns of Amazon and Google stocks from January 2014 to December 2023. We compared the popular Clayton, Gumbel, Frank, and Gaussian copula models to their corresponding distorted copula models induced by the unit-Lomax and unit-inverse Pareto distortions. The new families of distortion copulas are equipped with additional parameters inherent in the distortion function, providing more flexibility, and are demonstrated to perform better than the base copulas. After analyzing the data, we have found that the joint extremes of Amazon and Google stocks are more likely for high daily net returns than for low daily net returns.

Extreme Value Statistics in Semi-Supervised Models

We consider extreme value analysis in a semi-supervised setting, where we observe, next to the n data on the target variable, n + m data on one or more covariates. This is called the semi-supervised model with n labeled and m unlabeled data. By exploiting the tail dependence between the target variable and the covariates, we derive estimators for the extreme value index and extreme quantiles of the target variable in this setting and establish their asymptotic behavior. Our estimators substantially improve the univariate estimators, based on only the n target variable data, in terms of asymptotic variances whereas the asymptotic biases remain unchanged. A simulation study confirms the substantially improved behavior of both estimators. Finally the estimation method is applied to rainfall data in France. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.