Semiparametric bivariate Archimedean copulas

While parametric copulas often lack expressive capacity to capture the complex dependencies that are usually found in empirical data, non-parametric copulas can have poor generalization performance because of overfitting. A semiparametric copula method based on the family of bivariate Archimedean copulas is introduced as an intermediate approach that aims to provide both accurate and robust fits. The Archimedean copula is expressed in terms of a latent function that can be readily represented using a basis of natural cubic splines. The model parameters are determined by maximizing the sum of the log-likelihood and a term that penalizes non-smooth solutions. The performance of the semiparametric estimator is analyzed in experiments with simulated and real-world data, and compared to other methods for copula estimation: three parametric copula models, two semiparametric estimators of Archimedean copulas previously introduced in the literature, two flexible copula methods based on Gaussian kernels and mixtures of Gaussians and finally, standard parametric Archimedean copulas. The good overall performance of the proposed semiparametric Archimedean approach confirms the capacity of this method to capture complex dependencies in the data while avoiding overfitting.     

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.     

Smooth semiparametric and nonparametric Bayesian estimation of bivariate densities from bivariate histogram data

Penalized B-splines combined with the composite link model are used to estimate a bivariate density from a histogram with wide bins. The goals are multiple: they include the visualization of the dependence between the two variates, but also the estimation of derived quantities like Kendall’s tau, conditional moments and quantiles. Two strategies are proposed: the first one is semiparametric with flexible margins modeled using B-splines and a parametric copula for the dependence structure; the second one is nonparametric and is based on Kronecker products of the marginal B-spline bases. Frequentist and Bayesian estimations are described. A large simulation study quantifies the performances of the two methods under different dependence structures and for varying strengths of dependence, sample sizes and amounts of grouping. It suggests that Schwarz’s BIC is a good tool for classifying the competing models. The density estimates are used to evaluate conditional quantiles in two applications in social and in medical sciences.     

Conditional copulas, association measures and their applications

One way to model a dependence structure is through the copula function which is a mean to capture the dependence structure in the joint distribution of variables. Association measures such as Kendall’s tau or Spearman’s rho can be expressed as functionals of the copula. The dependence structure between two variables can be highly influenced by a covariate, and it is of real interest to know how this dependence structure changes with the value taken by the covariate. This motivates the need for introducing conditional copulas, and the associated conditional Kendall’s tau and Spearman’s rho association measures. After the introduction and motivation of these concepts, two nonparametric estimators for a conditional copula are proposed and discussed. Then nonparametric estimates for the conditional association measures are derived. A key issue is that these measures are now looked at as functions in the covariate. The performances of all estimators are investigated via a simulation study which also includes a data-driven algorithm for choosing the smoothing parameters. The usefulness of the methods is illustrated on two real data examples.     

Semiparametric analysis of randomized response data with missing covariates in logistic regression

In this article, two semiparametric approaches are developed for analyzing randomized response data with missing covariates in logistic regression model. One of the two proposed estimators is an extension of the validation likelihood estimator of Breslow and Cain [Breslow, N.E., and Cain, K.C. 1988. Logistic regression for two-stage case-control data. Biometrika. 75, 11-20]. The other is a joint conditional likelihood estimator based on both validation and non-validation data sets. We present a large sample theory for the proposed estimators. Simulation results show that the joint conditional likelihood estimator is more efficient than the validation likelihood estimator, weighted estimator, complete-case estimator and partial likelihood estimator. We also illustrate the methods using data from a cable TV study.     

Archimedean copula estimation using Bayesian splines smoothing techniques

Copulas enable to specify multivariate distributions with given marginals. Various parametric proposals were made in the literature for these quantities, mainly in the bivariate case. They can be systematically derived from multivariate distributions with known marginals, yielding e.g. the normal and the Student copulas. Alternatively, one can restrict his/her interest to a sub-family of copulas named Archimedean. They are characterized by a strictly decreasing convex function on (0,1) which tends to +∞ at 0 (when strict) and which is 0 at 1. A ratio approximation of the generator and of its first derivative using B-splines is proposed and the associated parameters estimated using Markov chains Monte Carlo methods. The estimation is reasonably quick. The fitted generator is smooth and parametric. The generated chain(s) can be used to build “credible envelopes” for the above ratio function and derived quantities such as Kendall's tau, posterior predictive probabilities, etc. Parameters associated to parametric models for the marginals can be estimated jointly with the copula parameters. This is an interesting alternative to the popular two-step procedure which assumes that the regression parameters are fixed known quantities when it comes to copula parameter(s) estimation. A simulation study is performed to evaluate the approach. The practical utility of the method is illustrated by a basic analysis of the dependence structure underlying the diastolic and the systolic blood pressures in male subjects.     

Semiparametric multivariate density estimation for positive data using copulas

The estimation of density functions for positive multivariate data is discussed. The proposed approach is semiparametric. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with parametric copulas to model the dependence. This semiparametric approach is robust both to the well-known boundary bias problem and the curse of dimensionality problem. Mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality are derived. A simulation study investigates the finite sample performance of the estimator. The proposed estimator performs very well, also for data without boundary bias problems. For bandwidths choice in practice, the univariate least squares cross validation method for the bandwidth of the marginal density estimators is investigated. Applications in the field of finance are provided.     

Multivariate distributions with proportional reversed hazard marginals

Several univariate proportional reversed hazard models have been proposed in the literature. Recently, Kundu and Gupta (2010) proposed a class of bivariate models with proportional reversed hazard marginals. It is observed that the proposed bivariate proportional reversed hazard models have a singular component. In this paper we introduce the multivariate proportional reversed hazard models along the same manner. Moreover, it is observed that the proposed multivariate proportional reversed hazard model can be obtained from the Marshall–Olkin copula. The multivariate proportional reversed hazard models also have a singular component, and their marginals have proportional reversed hazard distributions. The multivariate ageing and the dependence properties are discussed in details. We further provide some dependence measure specifically for the bivariate case. The maximum likelihood estimators of the unknown parameters cannot be expressed in explicit forms. We propose to use the EM algorithm to compute the maximum likelihood estimators. One trivariate data set has been analysed for illustrative purposes.     

Nonparametric regression models for right-censored data using Bernstein polynomials

In some applications of survival analysis with covariates, the commonly used semiparametric assumptions (e.g., proportional hazards) may turn out to be stringent and unrealistic, particularly when there is scientific background to believe that survival curves under different covariate combinations will cross during the study period. We present a new nonparametric regression model for the conditional hazard rate using a suitable sieve of Bernstein polynomials. The proposed nonparametric methodology has three key features: (i) the smooth estimator of the conditional hazard rate is shown to be a unique solution of a strictly convex optimization problem for a wide range of applications; making it computationally attractive, (ii) the model is shown to encompass a proportional hazards structure, and (iii) large sample properties including consistency and convergence rates are established under a set of mild regularity conditions. Empirical results based on several simulated data scenarios indicate that the proposed model has reasonably robust performance compared to other semiparametric models particularly when such semiparametric modeling assumptions are violated. The proposed method is further illustrated on the gastric cancer data and the Veterans Administration lung cancer data.     

Copula density estimation by total variation penalized likelihood with linear equality constraints

A copula density is the joint probability density function (PDF) of a random vector with uniform marginals. An approach to bivariate copula density estimation is introduced that is based on maximum penalized likelihood estimation (MPLE) with a total variation (TV) penalty term. The marginal unity and symmetry constraints for copula density are enforced by linear equality constraints. The TV-MPLE subject to linear equality constraints is solved by an augmented Lagrangian and operator-splitting algorithm. It offers an order of magnitude improvement in computational efficiency over another TV-MPLE method without constraints solved by the log-barrier method for the second order cone program. A data-driven selection of the regularization parameter is through K-fold cross-validation (CV). Simulation and real data application show the effectiveness of the proposed approach. The MATLAB code implementing the methodology is available online.     

Absolute penalty and shrinkage estimation in partially linear models

In the context of a partially linear regression model, shrinkage semiparametric estimation is considered based on the Stein-rule. In this framework, the coefficient vector is partitioned into two sub-vectors: the first sub-vector gives the coefficients of interest, i.e., main effects (for example, treatment effects), and the second sub-vector is for variables that may or may not need to be controlled. When estimating the first sub-vector, the best estimate may be obtained using either the full model that includes both sub-vectors, or the reduced model which leaves out the second sub-vector. It is demonstrated that shrinkage estimators which combine two semiparametric estimators computed for the full model and the reduced model outperform the semiparametric estimator for the full model. Using the semiparametric estimate for the reduced model is best when the second sub-vector is the null vector, but this estimator suffers seriously from bias otherwise. The relative dominance picture of suggested estimators is investigated. In particular, suitability of estimating the nonparametric component based on the B-spline basis function is explored. Further, the performance of the proposed estimators is compared with an absolute penalty estimator through Monte Carlo simulation. Lasso and adaptive lasso were implemented for simultaneous model selection and parameter estimation. A real data example is given to compare the proposed estimators with lasso and adaptive lasso estimators.     

Computing the volume of a high-dimensional semi-unsupervised hierarchical copula

We propose an algorithm for the computation of the volume of a multivariate copula function (and the probability distribution of the counting variable linked to this multidimensional copula function), which is very complex for large dimensions. As is common practice for large dimensional problem, we restrict ourselves to positive orthant dependence and we construct a Hierarchical copula which describes the joint distribution of random variables accounting for dependence among them. This approach approximates a multivariate distribution function of heterogenous variables with a distribution of a fixed number of homogenous clusters, organized through a semi-unsupervised clustering method. These clusters, representing the second-level sectors of hierarchical copula function, are characterized by an into-sector dependence parameter determined by a method which is very similar to the Diversity Score method. The algorithm, implemented in MatLab^? code, is particularly efficient allowing us to treat cases with a large number of variables, as can be seen in our scalability analysis. As an application, we study the problem of valuing the risk exposure of an insurance company, given the marginals i.e. the risks of each policy.     

Sampling algorithms for generating joint uniform distributions using the vine-copula method

An n-dimensional joint uniform distribution is defined as a distribution whose one-dimensional marginals are uniform on some interval I. This interval is taken to be [0,1] or, when more convenient . The specification of joint uniform distributions in a way which captures intuitive dependence structures and also enables sampling routines is considered. The question whether every n-dimensional correlation matrix can be realized by a joint uniform distribution remains open. It is known, however, that the rank correlation matrices realized by the joint normal family are sparse in the set of correlation matrices. A joint uniform distribution is obtained by specifying conditional rank correlations on a regular vine and a copula is chosen to realize the conditional bivariate distributions corresponding to the edges of the vine. In this way a distribution is sampled which corresponds exactly to the specification. The relation between conditional rank correlations on a vine and correlation matrix of corresponding distribution is complex, and depends on the copula used. Some results for the elliptical copulae are given.     

Aggregation of Dependent Risks Using the Koehler–Symanowski Copula Function

This study examines the Koehler and Symanovski copula function with specific marginals, such as the skew Student-t, the skew generalized secant hyperbolic, and the skew generalized exponential power distributions, in modelling financial returns and measuring dependent risks. The copula function can be specified by adding interaction terms to the cumulative distribution function for the case of independence. It can also be derived using a particular transformation of independent gamma functions. The advantage of using this distribution relative to others lies in its ability to model complex dependence structures among subsets of marginals, as we show for aggregate dependent risks of some market indices.     

Modeling spot price dependence in Australian electricity markets with applications to risk management

We examine the dependence structure of electricity spot prices across regional markets in Australia. One of the major objectives in establishing a national electricity market was to provide a nationally integrated and efficient electricity market, limiting market power of generators in the separate regional markets. Our analysis is based on a GARCH approach to model the marginal price series in the considered regions in combination with copulae to capture the dependence structure between the marginals. We apply different copula models including Archimedean, elliptical and copula mixture models. We find a positive dependence structure between the prices for all considered markets, while the strongest dependence is exhibited between markets that are connected via interconnector transmission lines. Regarding the nature of dependence, the Student-t copula provides a good fit to the data, while the overall best results are obtained using copula mixture models due to their ability to also capture asymmetric dependence in the tails of the distribution. Interestingly, our results also suggest that for the four major markets, NSW, QLD, SA and VIC, the degree of dependence has decreased starting from the year 2008 towards the end of the sample period in 2010. Examining the Value-at-Risk of stylized portfolios constructed from electricity spot contracts in different markets, we find that the Student-t and mixture copula models outperform the Gaussian copula in a backtesting study. Our results are important for risk management and hedging decisions of market participants, in particular for those operating in several regional markets simultaneously.     

Comparison of semiparametric and parametric methods for estimating copulas

Copulas have attracted significant attention in the recent literature for modeling multivariate observations. An important feature of copulas is that they enable us to specify the univariate marginal distributions and their joint behavior separately. The copula parameter captures the intrinsic dependence between the marginal variables and it can be estimated by parametric or semiparametric methods. For practical applications, the so called inference function for margins (IFM) method has emerged as the preferred fully parametric method because it is close to maximum likelihood (ML) in approach and is easier to implement. The purpose of this paper is to compare the ML and IFM methods with a semiparametric (SP) method that treats the univariate marginal distributions as unknown functions. In this paper, we consider the SP method proposed by Genest et al. [1995. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), 543-552], which has attracted considerable interest in the literature. The results of an extensive simulation study reported here show that the ML/IFM methods are nonrobust against misspecification of the marginal distributions, and that the SP method performs better than the ML and IFM methods, overall. A data example on household expenditure is used to illustrate the application of various data analytic methods for applying the SP method, and to compare and contrast the ML, IFM and SP methods. The main conclusion is that, in terms of statistical computations and data analysis, the SP method is better than ML and IFM methods when the marginal distributions are unknown which is almost always the case in practice.     

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.     

A new algorithm based on copulas for VaR valuation with empirical calculations

This paper concerns the application of copula functions in VaR valuation. The copula function is used to model the dependence structure of multivariate assets. After the introduction of the traditional Monte Carlo simulation method and the pure copula method we present a new algorithm based on mixture copula functions and the dependence measure, Spearman’s rho. This new method is used to simulate daily returns of two stock market indices in China, Shanghai Stock Composite Index and Shenzhen Stock Composite Index, and then empirically calculate six risk measures including VaR and conditional VaR. The results are compared with those derived from the traditional Monte Carlo method and the pure copula method. From the comparison we show that the dependence structure between asset returns plays a more important role in valuating risk measures comparing with the form of marginal distributions.     

Copula model evaluation based on parametric bootstrap

Copulas are used to model multivariate data as they account for the dependence structure and provide a flexible representation of the multivariate distribution. A great number of copulas has been proposed with various dependence aspects. One important issue is the choice of an appropriate copula from a large set of candidate families to model the data at hand. A large number of copulas are compared via likelihood principle, showing that it is hard to recognize the true underlying copula from real data since copulas with similar dependence properties are very close together. A goodness of fit test based on Mahalanobis squared distance between original and simulated log-likelihoods through parametric bootstrap techniques is also proposed. The advantage of this approach is that it is applicable to all families of copulas.