Elliptical Copulas Research Articles

Modeling Dependence Using Skew T Copulas: Bayesian Inference and Applications

We construct a copula from the skew t distribution of Sahu, Dey & Branco (2003). This copula can capture asymmetric and extreme dependence between variables, and is one of the few copulas that can do so and still be used in high dimensions effectively. However, it is difficult to estimate the copula model by maximum likelihood when the multivariate dimension is high, or when some or all of the marginal distributions are discrete-valued, or when the parameters in the marginal distributions and copula are estimated jointly. We therefore propose a Bayesian approach that overcomes all these problems. The computations are undertaken using a Markov chain Monte Carlo simulation method which exploits the conditionally Gaussian representation of the skew t distribution. We employ the approach in two contemporary econometric studies. The first is the modeling of regional spot prices in the Australian electricity market. Here, we observe complex non-Gaussian margins and nonlinear inter-regional dependence. Accurate characterization of this dependence is important for the study of market integration and risk management purposes. The second is the modeling of ordinal exposure measures for 15 major websites. Dependence between websites is important when measuring the impact of multi-site advertising campaigns. In both cases the skew t copula substantially out-performs symmetric elliptical copula alternatives, demonstrating that the skew t copula is a powerful modeling tool when coupled with Bayesian inference.

Detecting atypical observations in financial data: the forward search for elliptical copulas

In the last few years, copulas have been widely applied in many field of studies. Concentrating our attention on financial applications, we pursue the goal to detect multivariate atypical observations by extending to elliptical copulas the forward search originally introduced in linear and nonlinear regression by Atkinson and Riani (Robust diagnostic regression analysis. Springer, New York, 2000). Considering that, in the forward search, observations are ranked according to their closeness to the fitted data, we need to define a measure through which to initialize, progress and monitor the search. We achieve this goal building up the forward search for elliptical copulas relying on the squared Mahalanobis distance. Stressing the need to find theoretical boundaries for the inference on outliers, we introduce a procedure for computing envelopes as in Riani and Atkinson (Adv Data Anal Classif 1:123---141, 2007). Once defined our framework, we apply the forward search to a simulated environment where contaminations are exogenously introduced then, we carry out the analysis on n equity log-return real time series.

A comparison of three remotely sensed rainfall ensemble generators

In this paper three different stochastic ensemble generators are assessed based on their performance in ensemble generation of hourly and 15-min radar-based rainfall fields. The simulation methods include a Gaussian copula model, a t-copula-based approach and a non-Gaussian v-copula model. In the first and second models, two different elliptical copulas are employed to describe the dependence structures and to simulate multivariate rainfall error fields. In the third approach, a non-Gaussian v-transformed copula is applied for multivariate error simulations. Using this type of copula, one can describe asymmetrical dependencies through the copula parameters. In all three models, the outputs of the error simulators are used to perturb radar-based rainfall estimates to obtain an ensemble of precipitation estimates. In order to examine the reliability and performance of the presented models, several case studies are performed using radar reflectivity data (Level II) and Stage IV NEXRAD multi-sensor precipitation estimates as input to the models. Various statistical tests are employed to assess the models. The results indicate that the presented models are rather similar with respect to the dependence structure. However, using the t-copula model may have significant advantages over the other models, particularly, with respect to extremes. While all the presented models are able to simulate reasonable rainfall ensembles, the t-copula model seems to be more promising for practical applications.

Posouzení odhadu měnového rizika portfolia pomocí Lévyho modelů

Financial risk modeling, measuring, and managing are an inherent part of management in financial institutions. It is also an important step within the setting of optimal level of capital eligible to cover risk exposures. A significant portion of capital is usually assigned to cover the risk of unexpected changes in FX rates. FX rates (the returns) commonly exhibit significant skewness and relatively huge kurtosis. In this paper, we apply subordinated LĂŠvy models coupled together by ordinary elliptical copula functions in order to estimate the FX rate risk of normalized portfolio. Selected models are applied in order to estimate the risk ex-post, as well as ex-ante. The models are also compared to the more standard assumption of the joint normal distribution. Although the results for both types of modeling are quite different and LĂŠvy measure is ignored, suggested models deliver us improved risk estimation.

Projection Pursuit Through ϕ-Divergence Minimisation

In his 1985 article (“Projection pursuit”), Huber demonstrates the interest of his method to estimate a density from a data set in a simple given case. He considers the factorization of density through a Gaussian component and some residual density. Huber’s work is based on maximizing Kullback–Leibler divergence. Our proposal leads to a new algorithm. Furthermore, we will also consider the case when the density to be factorized is estimated from an i.i.d. sample. We will then propose a test for the factorization of the estimated density. Applications include a new test of fit pertaining to the elliptical copulas.

Copula‐based uncertainty modelling: application to multisensor precipitation estimates

AbstractThe multisensor precipitation estimates (MPE) data, available in hourly temporal and 4 km × 4 km spatial resolution, are produced by the National Weather Service and mosaicked as a national product known as Stage IV. The MPE products have a significant advantage over rain gauge measurements due to their ability to capture spatial variability of rainfall. However, the advantages are limited by complications related to the indirect nature of remotely sensed precipitation estimates. Previous studies confirm that efforts are required to determine the accuracy of MPE and their associated uncertainties for future use in hydrological and climate studies. So far, various approaches and extensive research have been undertaken to develop an uncertainty model. In this paper, an ensemble generator is presented for MPE products that can be used to evaluate the uncertainty of rainfall estimates. Two different elliptical copula families, namely, Gaussian and t‐copula are used for simulations. The results indicate that using t‐copula may have significant advantages over the well‐known Gaussian copula particularly with respect to extremes. Overall, the model in which t‐copula was used for simulation successfully generated rainfall ensembles with similar characteristics to those of the ground reference measurements. Copyright © 2010 John Wiley & Sons, Ltd.

An empirical analysis of multivariate copula models

Since the pioneering work of Embrechts and co-authors in 1999, copula models have enjoyed steadily increasing popularity in finance. Whereas copulas are well studied in the bivariate case, the higher-dimensional case still offers several open issues and it is far from clear how to construct copulas which sufficiently capture the characteristics of financial returns. For this reason, elliptical copulas (i.e. Gaussian and Student-t copula) still dominate both empirical and practical applications. On the other hand, several attractive construction schemes have appeared in the recent literature promising flexible but still manageable dependence models. The aim of this work is to empirically investigate whether these models are really capable of outperforming its benchmark, i.e. the Student-t copula and, in addition, to compare the fit of these different copula classes among themselves.

Elliptical families and copulas: tilting and premium; capital allocation

Elliptical copula measures with symmetrical marginals are proposed as a natural generalization of the elliptical family, which preserves the symmetrical character of marginals, but is more flexible in the choice of their shape parameters. The properties of these copulas are investigated and the elliptical copula tilting and corresponding premium are proposed as a natural tool for portfolio capital allocation. For the case of the multivariate normal family, such a tilting and premium coincide with the Esscher transform and premium.

Copula Structure Analysis

SummaryWe extend the standard approach of correlation structure analysis for dimension reduction of high dimensional statistical data. The classical assumption of a linear model for the distribution of a random vector is replaced by the weaker assumption of a model for the copula. For elliptical copulas a correlation-like structure remains, but different margins and non-existence of moments are possible. After introducing the new concept and deriving some theoretical results we observe in a simulation study the performance of the estimators: the theoretical asymptotic behaviour of the statistics can be observed even for small sample sizes. Finally, we show our method at work for a financial data set and explain differences between our copula-based approach and the classical approach. Our new method yielear models also.

A Method of Moments Estimator of Tail Dependence in Elliptical Copula Models

An elliptical copula model is a distribution function whose copula is that of an elliptical distribution. The tail dependence function in such a bivariate model has a parametric representation with two parameters: a tail parameter and a correlation parameter. The correlation parameter can be estimated by robust methods based on the whole sample. Using the estimated correlation parameter as plug-in estimator, we then estimate the tail parameter applying a modification of the method of moments approach proposed in the paper by J.H.J. Einmahl, A. Krajina and J. Segers [Bernoulli 14(4), 2008, 1003-1026]. We show that such an estimator is consistent and asymptotically normal. Also, we derive the joint limit distribution of the estimators of the two parameters. By a simulation study, we illustrate the small sample behavior of the estimator of the tail parameter and we compare its performance to that of the estimator proposed in the paper by C. Kluppelberg, G. Kuhn and L. Peng [Scandinavian Journal of Statistics 35(4), 2008, 701-718].

Optimal Dynamic Hedging via Copula-Threshold-GARCH Models

The contribution of this paper is twofold. First, we exploit copula methodology, with two threshold GARCH models as marginals, to construct a bivariate copula threshold GARCH model, simultaneously capturing asymmetric nonlinear behaviour in univariate stock returns of spot and futures markets and bivariate dependency, in a flexible manner. Two elliptical copulas (Gaussian and Student-t) and three Archimedean copulas (Clayton, Gumbel and the Mixture of Clayton and Gumbel) are utilized. Secondly, we employ the presenting models to investigate the hedging performance for five East Asian spot and futures stock markets: Hong Kong, Japan, Korea, Singapore and Taiwan. Compared with conventional hedging strategies, including Engle’s dynamic conditional correlation GARCH model, the results show that hedge ratios constructed by a Gaussian or Mixture copula are the best-performed in variance reduction for all markets except Japan and Singapore, and provide close to the best returns on a hedging portfolio over the sample period.

Do Rosenblatt and Nataf isoprobabilistic transformations really differ?

This article is the third in a series dedicated to the mathematical study of isoprobabilistic transformations and their relationship with stochastic dependence modelling, see [R. Lebrun, A. Dutfoy, An innovating analysis of the Nataf transformation from the viewpoint of copula, Probabilistic Engineering Mechanics (2008). doi: 10.1016/j.probengmech.2008.08.001] for an interpretation of the Nataf transformation in term of normal copula and [R. Lebrun, A. Dutfoy, A generalization of the Nataf transformation to distributions with elliptical copula, Probabilistic Engineering Mechanics (24) (2009), 172–178. doi:10.1016/j.probengmech.2008.05.001] for a generalisation of the Nataf transformation to any elliptical copula. In this article, we explore the relationship between two isoprobabilistic transformations widely used in the community of reliability analysts, namely the Generalised Nataf transformation and Rosenblatt transformation. First, we recall the elementary results of the copula theory that are needed in the remaining of the article, as a preliminary section to the presentation of both the Generalized Nataf transformation and the Rosenblatt transformation in the light of the copula theory. Then, we show that the Rosenblatt transformation using the canonical order of conditioning is identical to the Generalised Nataf transformation in the normal copula case, which is the most usual case in reliability analysis since it corresponds to the classical Nataf transformation. At this step, we also show that it is not possible to extend the Rosenblatt transformation to distributions with general elliptical copula the way the Nataf transformation has been generalised. Furthermore, we explore the effect of the conditioning order of the Rosenblatt transformation on the usual reliability indicators obtained from a FORM or SORM method. We show that in the normal copula case, all these reliability indicators, excepted the importance factors, are unchanged whatever the conditioning order one chooses. In the last section, we conclude the article with two numerical applications that illustrate the previous results: the equivalence between both transformations in the normal copula case, and the effect of the conditioning order in the normal and non-normal copula case.

Goodness-of-fit test for tail copulas modeled by elliptical copulas

Modeling and estimating a tail copula play an important role in forecasting rare events. Due to their easy simulation, elliptical copulas have been employed in risk management. Recently, Klüppelberg, [Klüppelber, C., Kuhn, G., Peng, L., 2007. Estimating the tail dependence function of an elliptical distribution. Bernoulli 13 (1), 229–251; Klüppelberg, C., Kuhn, G., Peng, L., 2008. Semi-parametric models for the multivariate tail dependence function—the asymptotically dependent case. Scandinavian Journal of Statistics 35, 701–718] proposed to model a tail copula by an elliptical copula, which results in an explicit parametric model for the tail copula. In this paper, we propose a goodness-of-fit test for such a parametric model and some real data analyses show that this fitting cannot be rejected. Therefore we demonstrate the practical applicability of this model.

Large Portfolio Risk Management with Dynamic Copulas

Modeling the dynamic high-dimensional multivariate distribution is very useful for active risk management and optimal portfolio allocation; however, available dynamic models are not easily applied for high-dimensional problems due to the curse of dimensionality. In the light of the recent development of multivariate GARCH techniques for a large number of underlying securities, I extend the framework of the Dynamic Conditional Correlation/Equicorrelation (DCC/DECO) (Engle, 2002 and Engle and Kelly, 2008) and an extreme value approach (McNeil and Frey, 2000) into a series of Dynamic Conditional Elliptical Copulas. By constructing portfolios of 89 stocks from CDX-listed …rms between 1995 and 2005, I examine Value at Risk (VaR) and Expected Shortfall (ES) by Monte Carlo simulation for passive portfolios and dynamic optimal portfolios through Mean-Variance and ES criteria. I …nd: (1) Modeling the marginal distribution is important for the dynamic high-dimensional multivariate models. (2) Neglecting the dynamic dependence in the copula causes over-aggressive risk management. (3) The DCC/DECO Gaussian copula and t-copula work very well for both VaR and ES. The DCC copula is necessary for value-weighted portfolios. For equally-weighted portfolios, the DECO copula performs about as well as the DCC copula. (4) Grouped t-copula and t-copula with dynamic degrees of freedom further match the fat tail. (5) Correctly modeling dependence structure makes an improvement in portfolio optimization. The optimal portfolio by ES does a good job against the tail risk in both DECO and DCC copulas. (6) Since portfolio optimization induces statistical error maximization, the assumption of multivariate t innovations with exogenously given degrees of freedom provides a ‡exible and applicable method for optimal portfolio risk management.

Optimal dynamic hedging via copula-threshold-GARCH models

The contribution of this paper is twofold. First, we exploit copula methodology, with two threshold GARCH models as marginals, to construct a bivariate copula-threshold-GARCH model, simultaneously capturing asymmetric nonlinear behaviour in univariate stock returns of spot and futures markets and bivariate dependency, in a flexible manner. Two elliptical copulas (Gaussian and Student's- t) and three Archimedean copulas (Clayton, Gumbel and the Mixture of Clayton and Gumbel) are utilized. Second, we employ the presenting models to investigate the hedging performance for five East Asian spot and futures stock markets: Hong Kong, Japan, Korea, Singapore and Taiwan. Compared with conventional hedging strategies, including Engle's dynamic conditional correlation GARCH model, the results show that hedge ratios constructed by a Gaussian or Mixture copula are the best-performed in variance reduction for all markets except Japan and Singapore, and provide close to the best returns on a hedging portfolio over the sample period.

Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case

Abstract. In general, the risk of joint extreme outcomes in financial markets can be expressed as a function of the tail dependence function of a high‐dimensional vector after standardizing marginals. Hence, it is of importance to model and estimate tail dependence functions. Even for moderate dimension, non‐parametrically estimating a tail dependence function is very inefficient and fitting a parametric model to tail dependence functions is not robust. In this paper, we propose a semi‐parametric model for (asymptotically dependent) tail dependence functions via an elliptical copula. Under this model assumption, we propose a novel estimator for the tail dependence function, which proves favourable compared to the empirical tail dependence function estimator, both theoretically and empirically.

A generalization of the Nataf transformation to distributions with elliptical copula

In the first article [Lebrun R, Dutfoy A. An innovating viewpoint of the isoprobabilistic Nataf transformation with the copula theory within exceedance threshold uncertainty analysis. Probabilistic Structure Engineering Structural Reliability. 2008 [in press]], we showed that the Nataf transformation is a way to model the dependence structure of a random vector by a normal copula, parameterized by its correlation matrix. Following this analysis, we propose an extension of this transformation to any elliptical copula, and give the equivalent of the First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM) for this generalization. In particular, we derive the Breitung asymptotic approximation in this new context.

Estimating the Probability of a Rare Event via Elliptical Copulas

A rare event happens with an extremely small probability but may cost billions of dollars. How to model and estimate the small probability of such an event is of importance to the insurance industry. Based on multivariate extreme value theory, methods have been proposed to extrapolate data into a far tail region. However, questions still remain open, such as the direction of extrapolation for a multivariate distribution and threshold selection for both marginals and the tail dependence function. In this paper we provide a way to estimate the probability of a rare event via modeling marginals and dependence by heavy tailed distributions and elliptical copulas, respectively. Hence, the direction of extrapolation becomes irrelevant. Moreover we employ recent threshold selection procedures to choose tuning parameters automatically.

The effects of misspecified marginals and copulas on computing the value at risk: A Monte Carlo study

The effect on the estimation of the Value at Risk when dealing with multivariate portfolios when there is a misspecification both in the marginals and in the copulas is investigated. It is first shown that, when there is skewness in the data and symmetric marginals are used, the estimated elliptical (normal or t) copula correlations are negatively biased, reaching values as high as 70% of the true values. Besides, the bias almost doubles if negative correlations are considered, compared to positive correlations. As for the t copula degrees of freedom parameter, the use of wrong marginals delivers large positive biases, instead. If the dependence structure is represented by a copula which is not elliptical, e.g. the Clayton copula, the effects of marginal misspecifications on the copula parameter estimation can be rather different, depending on the sign of marginal skewness. Extensive Monte Carlo studies then show that the misspecifications in the marginal volatility equation more than offset the biases in copula parameters when VaR forecasting is of concern, small samples are considered and the data are leptokurtic. The biases in the volatility parameters are much smaller, whereas those ones in the copula parameters remain almost unchanged or even increase when the sample dimension increases. In this case, copula misspecifications do play a role for VaR estimation. However, these effects depend heavily on the sign of the dependence.

Heavy-tailed longitudinal data modeling using copulas

In this paper, we consider “heavy-tailed” data, that is, data where extreme values are likely to occur. Heavy-tailed data have been analyzed using flexible distributions such as the generalized beta of the second kind, the generalized gamma and the Burr. These distributions allow us to handle data with either positive or negative skewness, as well as heavy tails. Moreover, it has been shown that they can also accommodate cross-sectional regression models by allowing functions of explanatory variables to serve as distribution parameters. The objective of this paper is to extend this literature to accommodate longitudinal data, where one observes repeated observations of cross-sectional data. Specifically, we use copulas to model the dependencies over time, and heavy-tailed regression models to represent the marginal distributions. We also introduce model exploration techniques to help us with the initial choice of the copula and a goodness-of-fit test of elliptical copulas for model validation. In a longitudinal data context, we argue that elliptical copulas will be typically preferred to the Archimedean copulas. To illustrate our methods, Wisconsin nursing homes utilization data from 1995 to 2001 are analyzed. These data exhibit long tails and negative skewness and so help us to motivate the need for our new techniques. We find that time and the nursing home facility size as measured through the number of beds and square footage are important predictors of future utilization. Moreover, using our parametric model, we provide not only point predictions but also an entire predictive distribution.