Class Of Copulas Research Articles

Efficient Bayesian inference for stochastic time-varying copula models

There is strong empirical evidence that dependence in multivariate financial time series varies over time. To model this effect, a time varying copula class is developed, which is called the stochastic copula autoregressive (SCAR) model. Dependence at time t is modeled by a real-valued latent variable, which corresponds to the Fisher Z transformation of Kendall’s τ for the chosen copula family. This allows for a common scale so that a general range of copula families including the Gaussian, Clayton and Gumbel copulas can be used and compared in our modeling framework. The inclusion of latent variables makes maximum likelihood estimation computationally difficult, therefore a Bayesian approach is followed. This approach allows the computation of credibility intervals in addition to point estimates. Two Markov Chain Monte Carlo (MCMC) sampling algorithms are proposed. The first one is a naïve approach using Metropolis–Hastings within Gibbs, while the second is a more efficient coarse grid sampler. The performance of these samplers are investigated in a simulation study and are applied to data involving financial stock indices. It is shown that time varying dependence is present for this data and can be quantified by estimating the underlying time varying Kendall’s τ with point-wise credible intervals.

Sklar’s theorem obtained via regularization techniques

Sklar’s theorem establishes the connection between a joint d -dimensional distribution function and its univariate marginals. Its proof is straightforward when all the marginals are continuous. The hard part is the extension to the case where at least one of the marginals has a discrete component. We present a new proof of this extension based on some analytical regularization techniques (i.e., mollifiers) and on the compactness (with respect to the L ∞ norm) of the class of copulas.

Application of copulas for derivation of drought severity–duration–frequency curves

AbstractThis study presents copula‐based multivariate probabilistic approach to model severity–duration–frequency (S‐D‐F) relationship of drought events in western Rajasthan, India. Drought occurrences are analysed using standardized precipitation index computed on monthly mean areal precipitation, aggregated at a time scale of 6 months. After testing with a series of probability density functions, the drought variable severity is found to be better represented with log‐normal distribution, whereas duration is well fitted with exponential distribution. Four different classes of bivariate copulas – Archimedean, extreme value, Plackett, and elliptical families are evaluated for modelling joint distribution of drought characteristics. It is observed that the extreme value copula – Gumbel–Hougaard copula – performed better as compared with other classes of copulas, based on results of various statistical tests and upper tail dependence coefficient. The joint distribution obtained from best performing copula is then employed to determine conditional return period and to derive drought severity‐duration‐frequency (S‐D‐F) curves for the study region. The results of the study suggests that the copula method can be used effectively to derive the drought S‐D‐F curves, which can be helpful in planning and adopting suitable drought mitigation strategies in drought‐prone areas. Copyright © 2011 John Wiley & Sons, Ltd.

On a strong metric on the space of copulas and its induced dependence measure

Using the one-to-one correspondence between copulas and Markov operators on L 1 ( [ 0 , 1 ] ) and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a metric D 1 on the space of copulas C that is a metrization of the strong operator topology of the corresponding Markov operators. It is shown that the resulting metric space ( C , D 1 ) is complete and separable and that the induced dependence measure ζ 1 , defined as a scalar times the D 1 -distance to the product copula Π, has various good properties. In particular the class of copulas that have maximum D 1 -distance to the product copula is exactly the class of completely dependent copulas, i.e. copulas induced by Lebesgue-measure preserving transformations on [ 0 , 1 ] . Hence, in contrast to the uniform distance d ∞ , Π cannot be approximated arbitrarily well by completely dependent copulas with respect to D 1 . The interrelation between D 1 and the so-called ∂-convergence by Mikusinski and Taylor as well as the interrelation between ζ 1 and the mutual dependence measure ω by Siburg and Stoimenov is analyzed. ζ 1 is calculated for some well-known parametric families of copulas and an application to singular copulas induced by certain Iterated Functions Systems is given.

A generalized beta copula with applications in modeling multivariate long-tailed data

This work proposes a new copula class that we call the MGB2 copula. The new copula originates from extracting the dependence function of the multivariate GB2 distribution (MGB2) whose marginals follow the univariate generalized beta distribution of the second kind (GB2). The MGB2 copula can capture non-elliptical and asymmetric dependencies among marginal coordinates and provides a simple formulation for multi-dimensional applications. This new class features positive tail dependence in the upper tail and tail independence in the lower tail. Furthermore, it includes some well-known copula classes, such as the Gaussian copula, as special or limiting cases. To illustrate the usefulness of the MGB2 copula, we build a trivariate MGB2 copula model of bodily injury liability closed claims. Extended GB2 distributions are chosen to accommodate the right-skewness and the long-tailedness of the outcome variables. For the regression component, location parameters with continuous predictors are introduced using a nonlinear additive function. For comparison purposes, we also consider the Gumbel and t copulas, alternatives that capture the upper tail dependence. The paper introduces a conditional plot graphical tool for assessing the validation of the MGB2 copula. Quantitative and graphical assessment of the goodness of fit demonstrate the advantages of the MGB2 copula over the other copulas.

Recovering copulas from limited information and an application to asset allocation

This paper proposes an entropy-based method to construct a new class of copulas – the most entropic canonical copulas (MECC). Our empirical study focuses on an investment problem for an investor with a constant relative risk aversion (CRRA) utility function allocating wealth between the Dow Jones Large-Cap and Small-Cap indices, of which the contemporaneous dependence can be modeled by the MECC or other commonly-used copulas. Both the theoretical analysis of the method and the empirical study indicate the potential for enormous statistical and economic gains as a result of using the MECC.

Adaptive Importance Sampling for simulating copula-based distributions

In this paper, we propose a generalization of importance sampling, called Adaptive Importance Sampling, to approximate simulation of copula-based distributions. Unlike existing methods for copula simulation that have appeared in the literature, this algorithm is broad enough to be used for any absolutely continuous copula. We provide details of the algorithm including rules for stopping the iterative process and consequently assess its performance using extensive Monte Carlo experiments. To assist in its extension to several dimensions, we discuss procedures for identifying the crucial parameters in order to achieve desirable results especially as the size of the dimension increases. Finally, for practical illustration, we demonstrate the use of the algorithm to price First-to-Default credit swap, an important credit derivative instrument in the financial market. The method works exquisitely well even for large dimensions making it a valuable tool for simulating from many different classes of copulas including those which have been difficult to sample from using traditional techniques.

Monotonicity properties of multivariate distribution and survival functions — With an application to Lévy-frailty copulas

The monotonicity properties of multivariate distribution functions are definitely more complicated than in the univariate case. We show that they fit perfectly well into the general theory of completely monotone and alternating functions on abelian semigroups. This allows us to prove a correspondence theorem which generalizes the classical version in two respects: the function in question may be defined on rather arbitrary product sets in R ¯ n , and it need not be grounded, i.e. disappear at the lower-left boundary. In 2009 a greatly interesting class of copulas was discovered by Mai and Scherer (cf. Mai and Scherer (2009) [4]), connecting in a very surprising way complete monotonicity with respect to the maximum operation on R + n and with respect to ordinary addition on N 0 . Based on the preceding results, we give another proof of this result.

Bivariate income distributions for assessing inequality and poverty under dependent samples

As indicators of social welfare, the incidence of inequality and poverty is of ongoing concern to policy makers and researchers alike. Of particular interest are the changes in inequality and poverty over time, which are typically assessed through the estimation of income distributions. From this, income inequality and poverty measures, along with their differences and standard errors, can be derived and compared. With panel data becoming more frequently used to make such comparisons, traditional methods which treat income distributions from different years independently and estimate them on a univariate basis, fail to capture the dependence inherent in a sample taken from a panel study. Consequently, parameter estimates are likely to be less efficient, and the standard errors for between-year differences in various inequality and poverty measures will be incorrect. This paper addresses the issue of sample dependence by suggesting a number of bivariate distributions, with Singh–Maddala or Dagum marginals, for a partially dependent sample of household income for two years. Specifically, the distributions considered are the bivariate Singh–Maddala distribution, proposed by Takahasi (1965), and bivariate distributions belonging to the copula class of multivariate distributions, which are an increasingly popular approach to modelling joint distributions. Each bivariate income distribution is estimated via full information maximum likelihood using data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey for 2001 and 2005. Parameter estimates for each bivariate income distribution are used to obtain values for mean income and modal income, the Gini inequality coefficient and the headcount ratio poverty measure, along with their differences, enabling the assessment of changes in such measures over time. In addition, the standard errors of each summary measure and their differences, which are of particular interest in this analysis, are calculated using the delta method.

Lipschitz continuity of copulas w.r.t. [formula omitted]-norms

In the framework of the stability analysis of real functions, the Lipschitz continuity of copulas w.r.t. L p -norms is investigated. Emphasis is put on 1-Lipschitz continuity, as it is the strongest type possible for copulas. After illustrating how to identify 1-Lipschitz copulas w.r.t. some L p -norm, the preservation of this 1-Lipschitz continuity by certain construction methods, such as ordinal sums and convex sums, is demonstrated. Special attention is given to three important classes of copulas, namely Archimedean copulas, extreme value copulas and Archimax copulas.

De copulis non est disputandum

Normal distribution of residuals is a traditional assumption in multivariate models. It is, however, not very often consistent with real data. Copulae allow for an extension of dependency models to nonellipticity and for separation of margins from the dependency. This paper provides a survey of copulae where different copula classes, estimation and simulation techniques and goodness-of-fit tests are considered. In the empirical section we apply different copulae to the static and dynamic Value-at-Risk of portfolio returns and Profit-and-Loss function.

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.

COPULA-BASED CHARACTERIZATIONS FOR HIGHER ORDER MARKOV PROCESSES

In this paper, we obtain characterizations of higher order Markov processes in terms of copulas corresponding to their finite-dimensional distributions. The results are applied to establish necessary and sufficient conditions for Markov processes of a given order to exhibitm-dependence,r-independence, or conditional symmetry. The paper also presents a study of applicability and limitations of different copula families in constructing higher order Markov processes with the preceding dependence properties. We further introduce new classes of copulas that allow one to combine Markovness withm-dependence orr-independence in time series.

A class of copulas with piecewise linear horizontal sections

We introduce the class of bivariate copulas with piecewise linear horizontal sections whose graph is composed, at most, of two segments. It is a wide class of copulas which contains some known copulas. We study several properties of the copulas in the new class concerning absolute continuity, singular components, measures of association, concordance ordering, dependence concepts and symmetry. Finally, we provide several examples.

Three-stage semi-parametric estimation of [formula omitted]-copulas: Asymptotics, finite-sample properties and computational aspects

A two-stage semi-parametric estimation procedure for a broad class of copulas satisfying minimal regularity conditions has been recently proposed. In addition, a three-stage semi-parametric estimation method based on Kendall’s tau in order to estimate the Student’s t copula has also been designed. Its major advantage is to allow for greater computational tractability when dealing with high dimensional issues, where two-stage procedures are no more a viable choice. The asymptotic properties of this methodology are developed and its finite-sample behavior are examined via simulations. The advantages and disadvantages of this methodology are analyzed in terms of numerical convergence and positive definiteness of the estimated T -copula correlation matrix.

Investing Dynamic Dependence Using Copulae

A general methodology for time series modelling is developed which works down from distributional properties to implied structural models including the standard regression relationship. This general to specific approach is important since it can avoid spurious assumptions such as linearity in the form of the dynamic relationship between variables. It is based on splitting the multivariate distribution of a time series into two parts: (i) the marginal unconditional distribution, (ii) the serial dependence encompassed in a general function , the copula. General properties of the class of copula functions that fulfill the necessary requirements for Markov chain construction are exposed. Special cases for the gaussian copula with AR(p) dependence structure and for archimedean copulae are presented. We also develop copula based dynamic dependency measures — auto-concordance in place of autocorrelation. Finally, we provide empirical applications using financial returns and transactions based forex data. Our model encompasses the AR(p) model and allows non-linearity. Moreover, we introduce non-linear time dependence functions that generalize the autocorrelation function.

Bounds for Value at Risk for Multiasset Portfolios

The theory of copulas provides a useful tool for modeling dependence in risk management. In insurance and flnance, as well as in other applications, dependence of extreme events is particularly important, hence there is a need for the detailed study of the tail behaviour of the multivariate copulas. In this paper we investigate the class of copulas being the weighted means of copulas having homogeneous lower tails. We show that having only such information on the structure of dependence of returns from assets is enough to get estimates on value at risk of the multiasset portfolio in terms of value at risk of one-asset portfolios. PACS numbers: 89.65.Gh

Linear B-spline copulas with applications to nonparametric estimation of copulas

In this paper, we propose a method for constructing a new class of copulas. They are called linear B-spline copulas which are a good approximation of a given complicated copula by using finite numbers of values of this copula without the loss of some essential properties. Moreover, rigorous analysis shows that the empirical linear B-spline copulas are more effective than empirical copulas to estimate perfectly dependent copulas. For the cases of nonperfectly dependent copulas, simulations show that the empirical linear B-spline copulas also improve the empirical copulas to estimate the underlying copula structure. Furthermore, we compare the performance of parametric estimation of copulas based on the empirical copulas with that based on the empirical linear B-spline copulas by simulations. In most of the cases, the latter are better than the former.

Simulating from Exchangeable Archimedean Copulas

Multivariate exchangeable Archimedean copulas are one of the most popular classes of copulas that are used in actuarial science and finance for modeling risk dependencies and for using them to quantify the magnitude of tail dependence. Owing to the increase in popularity of copulas to measure dependent risks, generating from multivariate copulas has become a very crucial exercise. Current methods for generating from multivariate Archimedean copulas could become a very difficult task as the number of dimension increases. The resulting analytical procedures suggested in the existing literature do not offer much guidance for practical implementation. This article presents an algorithm for generating values from the multivariate exchangeable Archimedean copulas based on a multivariate extension of a bivariate result. A procedure for generating values from bivariate Archimedean copulas has been originally proposed in Genest and Rivest (1993) and again later described in Nelsen (1999) and Embrechts et al. (2001ab). Using a proof that is simply based on fundamental Jacobian techniques for deriving distributions of transformed random variables, we are able to extend the bivariate result into the multivariate case allowing us to develop an interesting algorithm to generate values from exchangeable Archimedean copulas. As auxiliary results, we are able to derive the distribution function of an n-dimensional Archimedean copula, a result already known in Genest and Rivest (2001), but our approach of proving this result is based on a different perspective. This article focuses on this class of copulas that has one generating function and one parameter that characterizes the dependence structure of the joint distribution function.

A weakening of Schur-concavity for copulas

The class of copulas satisfying a new property called weak Schur-concavity is introduced and studied and the relationships of this property with other known notions of concavity are investigated.