Selection of Vine Copulas
Vine copula models have proven themselves as a very flexible class of multivariate copula models with regard to symmetry and tail dependence for pairs of variables. The full specification of a vine model requires the choice of a vine tree structure, the c
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Selection of Vine Copulas Claudia Czado, Eike Christian Brechmann, and Lutz Gruber
Abstract Vine copula models have proven themselves as a very flexible class of multivariate copula models with regard to symmetry and tail dependence for pairs of variables. The full specification of a vine model requires the choice of a vine tree structure, the copula families for each pair copula term and their corresponding parameters. In this survey we discuss the different approaches, both frequentist and Bayesian, for these model choices so far and point to open problems.
2.1 Introduction The analysis of high-dimensional data sets requires flexible multivariate stochastic models that can capture the inherent dependency patterns. The copula approach, which separates the modeling of the marginal distributions from modeling the dependence characteristics, is a natural one to follow in this context. This development has spawned a tremendous increase in copula-based applications in the last 10 years, especially in the areas of finance, economics, and hydrology. Considerable efforts have been undertaken to increase the flexibility of multivariate copula models beyond the scope of elliptical and Archimedean copulas. Vine copulas are among the best-received of such efforts. Vine copulas use (conditional) bivariate copulas as the so-called pair copula building blocks to describe a multivariate distribution (see [37]). A set of linked trees—the “vine”— describes a vine copula’s factorization of the multivariate copula density function into the density functions of its pair copulas (see [8,9]). The article by [1] illustrates a first application of the vine copula concept using non-Gaussian pair copulas to financial data. The first comprehensive account of vine copulas is found in [45], a
C. Czado () E.C. Brechmann L. Gruber Zentrum Mathematik, Technische Universit¨at M¨unchen, Munich, Germany e-mail: [email protected]; [email protected]; [email protected] P. Jaworski et al. (eds.), Copulae in Mathematical and Quantitative Finance, Lecture Notes in Statistics 213, DOI 10.1007/978-3-642-35407-6 2, © Springer-Verlag Berlin Heidelberg 2013
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recent survey in [20], and the current developments of this active research area in [46]. Elliptical copulas as well as Archimedean copulas have been shown to be inadequate models to describe the dependence characteristics of real data applications (see, for example, [1, 23, 24]). As a pair copula construction, vine copulas allow different structural behaviors of pairs of variables to be modeled suitably, in particular so with regard to their symmetry, or lack thereof, strength of dependence, and tail dependencies. Such flexibility requires well-designed model selection procedures to realize the full potential of vine copulas as dependence models. Successful applications of vines can be found, amongst others, in [10, 12, 17, 22, 24, 32, 48, 51, 54, 61]. A parametric vine copula consists of three components: a set of linked trees identifying the pairs of variables and
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