Dependence Modeling

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction

• Investment Example

• Vines
  • Graphical representation
  • Vine density
  • Estimation
  • Properties and applications

• Outline

• Glossary and Notation

• References

Though dating back to 1959 when the term “copulae” was coined, copula models only started their triumphal procession in the mid-1990s. Application of copulae was primarily restricted to the world of finance and insurance but now the copula concept has found its way into nearly all relevant statistical and mathematical literature where multivariate dependence structures are involved. Whereas the bivariate case was central in most of the publications and seems to be well-explored at present, there is still an ongoing and active debate on the construction of multivariate copula models. Apart from pair-copula constructions, which are the focus of this book and intensively discussed in the following chapters, this chapter briefly reviews both different copula classes and construction schemes of multivariate models.

An introduction to the main idea of vines as graphical models is presented, along with various notation and graphs for representing vines. The early history of vines is summarized, together with the motivation for their construction. The relation to compatibility of subsets of marginal distributions is given to provide some intuition. Important properties and applications of vines are included.

Erhardt and Czado¹¹ suggest an approximative method for sampling high-dimensional count random variables with a specified Pearson correlation. They utilize Gaussian copulae for the construction of multivariate discrete distributions. A major task is to determine the appropriate copula parameters for the achievement of a specified target correlation. Erhardt and Czado¹¹ develop an optimization routine to determine these copula parameters sequentially. Thereby, they use pair-copula decompositions of n-dimensional distributions, i.e., a decomposition consisting only of bivariate copula with one parameter each. C-vines, a graphical tool to organize such pair-copula decompositions, are used to select a possible decomposition. In the paper mentioned, the approach was compared to the NORTA method for discrete margins described in Ref. 2. Here, we will compare it to a widely used naive sampling approach for an even larger variety of marginal distributions such as the Poisson, generalized Poisson, negative binomial and zero-inflated generalized Poisson distributions.

An elementary though seemingly underappreciated finding shows that small global correlations are amplified by aggregation. We observe this behavior in flood damage claims in the US. We also observe that upper tail dependence seems to be amplified by aggregation in these data. We seek to understand this behavior. For sums of exponential variables which are conditionally independent given a gamma-distributed rate, we derive explicit expressions for upper tail dependence and prove that it goes to one as the number of summands goes to infinity, and that the lower tail dependence is zero. We also study sums of events under a latent variable model, where each event occurs if a uniform variable exceeds a threshold, and all uniform variables are conditionally independent given a “latent variable”. We obtain a necessary and sufficient condition for strong asymptotic upper tail dependence as the number of summands goes to infinity. Curiously, the normal copula satisfies this condition, although it is not tail dependent via the usual definition. Thus, sums of events under the normal copula latent variable model have upper tail dependence increasing to 1. We also identify tail dependent-like behavior in finite sums of events with the latent variable model.

This chapter surveys the asymptotic theory of estimation of a copula from a frequentistic perspective and presents the problems involved in frequentistic model selection among several candidate copulae when using the maximum pseudo-likelihood estimator (MPLE). Frequentistic copula model selection has recently been addressed through the development of the copula information criterion (CIC) — a model selection formula which extends the maximum likelihood-based Akaike information criterion (AIC) to the MPLE. We present the developments leading to the CIC with a focus on its implications, while deferring proofs of underlying limit theorems to the original CIC paper.

The CIC is in fact two different formulae, one for misspecified copula models and another for correctly specified copula models, paralleling the Takeuchi information criterion and the Akaike information criterion respectively.

These formulae show that there does not exist (in a certain technical sense) an AIC formula for MPL estimation when the parametric copula has extreme behavior near the edge of the unit cube. This means that one cannot make first-order bias-correction terms of a desired part of the attained Kullback-Leibler divergence between the MPL-estimated copula and the data-generating copula in a class of copulae which has received much attention in econometrics. This may be seen as a demarcation of which types of copulae that should be estimated with the MPLE. Interestingly, the main motivating factor for using the MPLE is also the reason for the non-existence of a general MPLE-based AIC formula. A further conclusion is that the CIC provides a counterexample to the often acclaimed intrinsic connection between the AIC and Occam's Razor.

It is known that there is one equivalence class of regular vines for n = 3 and two equivalence classes for n = 4. Through an enumeration, we show that there are six equivalence classes for n = 5 with boundary classes of C-vines and D-vines, so that the dimension of n = 5 is manageable in terms of studying the intermediate regular vines. For vine copulae, which obtain from associating a bivariate copula with each pair in the vine, we develop an approach that leads to algorithms for C-vines, D-vines and intermediate vines. Some similarities between different vine copulae can be seen from how they are simulated. For Gaussian and non-Gaussian vines, we compare the marginal bivariate dependence of different vine copulae when the level ℓ bivariate linking copulae are all set at C^ℓ, ℓ = 1,…, n − 1.

Definitions and properties of conditional and multivariate tail dependence functions are given and applied to vine copulae. We show that vine copulae can have flexible dependence with asymmetry in the joint upper and lower tails, by using appropriate choices of bivariate linking copulae that are reflection asymmetric and have upper/lower tail dependence parameters λ_L, λ_U that independently take values in (0, 1).

In this chapter, three algorithms for producing and enumerating regular vines are presented. The first one produces all possible vines on n nodes and regular vines are found by inspection. The second one uses the concept of line graphs to produce only regular vines. The third algorithm produces regular vines by extending a regular vine on three nodes to a regular vine on n nodes. The first and second algorithms presented have been used for the construction of a catalogue of labeled regular vines on at most nine nodes and tree-equivalent regular vines on at most seven. This catalogue is presented as an appendix to the chapter.

A natural order for a regular vine on n variables is an assignment of indices to the variables such that variables indexed with j and j + 1 occur as conditioned variables in a node of tree j, j = 1,…,n−1. Regular vines V and U on n variables are equivalent if there is a permutation π ∈ n! such that π(V) = U. U and V are equivalent if and only if the regular vines in natural order corresponding to U and V are equivalent. The number of equivalence classes for regular vines is obtained by counting the number of equivalence classes for regular vines in natural order.

The vine copulae representation is a very flexible model of n-dimensional random vectors. However when n becomes too large, some simplifying assumptions have to be made as fitting this model becomes too cumbersome. In this chapter truncations of vines are discussed. We could reduce a vine to a Markov tree structure but Markov trees allow n −1 copulae to be specified out of possible for vines. Hence trees may be too restrictive for a given set of data. Another possibility would be to model subsets of variables with vines and connect these smaller vines in a tree structure. We suggest one more strategy of choosing the “most suitable” vine for the correlation matrix. The “best vine” is the one whose nodes of top trees (tree with the most conditioning) correspond to the smallest absolute values of partial correlations. To search for the “best vine” we developed a new algorithm of generating a regular vine. We start building the vine from the top node (node in tree n−1) and progress to the lower trees, ensuring that the regularity condition is satisfied and that the partial correlations corresponding to these nodes are the smallest. If we assume that we can assign the independent copula to nodes of the vine with small absolute values of partial correlations, then this algorithm can be used to find an optimal truncation of a vine structure. We advocate using it as a preprocessing step of fitting a vine to the data.

In the last two decades the advent of fast computers has made Bayesian inference, based on Markov chain Monte Carlo (MCMC) methods, very popular in many fields of science. These Bayesian methods, are good alternatives to traditional maximum likelihood (ML) methods since they can often estimate complicated statistical models for which an ML approach fails. In this chapter we review available MCMC estimation and model selection algorithms as well as their possible extensions for D-vine pair-copula constructions (PCC) based on bivariate t-copulae. However the discussed methods can easily be extended for an arbitrary regular vine PCC based on any bivariate copulae. A Bayesian inference for Australian electricity loads demonstrates the addressed algorithms at work.

Sklar's theorem allows the construction of models for dependent components using a multivariate copula together with marginal distributions. For estimation of the copula and marginal parameters, a two-step procedure is often used to avoid high-dimensional optimization. Here, marginal parameters are estimated first, then used to transform to uniform margins and in a second step, the copula parameters are estimated. This procedure is not efficient. Therefore, we follow a joint estimation approach in a Bayesian framework using Markov chain Monte Carlo (MCMC) methods. This allows also for the assessment of parameter uncertainty using credible intervals. D-vine copulae are utilized and as marginal models we allow for autoregressive models of first order. Finally, we apply these methods to Australian electricity loads, demonstrating the usefulness of this approach. Bayesian model selection is also discussed and applied using a method suggested by Congdon.

This chapter reviews aspects of non-parametric Bayesian belief nets (NPBBN). The theory behind NPBBNs is closely related to that of regular vines and it has benefited from developments in the latter. It also offers an alternative to undirected graphical models in general, and to regular vines in particular. The differences and similarities in modeling using directed versus undirected graphs are discussed in this chapter from the perspective of NPBBNs and vines. Until recently, Bayesian belief nets (BBNs) were either discrete or discrete-normal. Despite their popularity, both suffer from severe limitations. Discrete BBNs are limited by size and complexity, discrete-normal BBNs are limited by the assumption of joint normality. NPBBNs were introduced to overcome these limitations. Algorithms for specifying, sampling and analyzing high-dimensional distributions using NPBBNs have been developed and successfully applied in decision support systems.

In this chapter, we compare three constructions for modeling higher-dimensional dependence: the Student copula, the partially nested Archimedean construction (PNAC) and the pair-copula construction. For the latter two, a multivariate data set is modeled using a cascade of lower-dimensional copulae. They differ, however, in their construction of the dependence structure. The PNAC is more restrictive than the PCC in two respects. There are strong limitations on the degree of dependence in each level of the PNAC, and all the bivariate copulae in this construction has to be Archimedean. The PCC, on the other hand, can be built using copulae from any class and there are no constraints on the parameters. We show through two applications that the PCC provides a better fit to financial data than the two other structures.

We model the dependence structure of multivariate financial returns with a time-varying D-vine copula. Vine copulae are flexible multivariate copulae that are obtained by a hierarchical construction, with bivariate copulae as building blocks. We focus on D-vines, which are a subclass of vine copulae. In order to take into account the fact that the dependence structure between financial returns is not constant over time, we allow each of the possible bivariate copulae to be time-varying. We use two different data sets, six exchange rates and five Asian equity indices. We find that most of the time variation is found in the first tree of the D-vine. Moreover, while currencies can be adequately modeled with symmetric copulae, Asian equity indices require asymmetric copulae.

The following sections are included:

• Summary

• Future Research Directions

The following sections are included:

• Index