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Keywords:

  • Multivariate copula;
  • regular vines;
  • simplified vines;
  • truncated canonical vines;
  • MSC 2010: Primary 62H15;
  • secondary 62H12

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

Using only bivariate copulas as building blocks, regular vine copulas constitute a flexible class of high-dimensional dependency models. However, the flexibility comes along with an exponentially increasing complexity in larger dimensions. In order to counteract this problem, we propose using statistical model selection techniques to either truncate or simplify a regular vine copula. As a special case, we consider the simplification of a canonical vine copula using a multivariate copula as previously treated by Heinen & Valdesogo (2009) and Valdesogo (2009). We validate the proposed approaches by extensive simulation studies and use them to investigate a 19-dimensional financial data set of Norwegian and international market variables. The Canadian Journal of Statistics 40: 68–85; 2012 © 2012 Statistical Society of Canada

En utilisant uniquement des copules bidimensionnelles comme unités de base, les copules en arborescence régulière constituent une classe flexible pour modéliser la dépendance pour les grandes dimensions. Toutefois, en grandes dimensions, la flexibilité s'accompagne d'une croissance exponentielle de la complexité. Pour contrecarrer ce problème, nous proposons l'utilisation des techniques de sélection de modèles statistiques afin de tronquer ou encore de simplifier la copule en arborescence régulière. Comme cas particulier, nous considérons la simplification de la copule en arborescence canonique par l'utilisation d'une copule multidimensionnelle telle que présentée dans Heinen et Valdesogo (2009) et Valdesogo (2009). Nous validons les approches proposées par de vastes études de simulation et nous les utilisons pour analyser un jeu de données financières de dimension 19 sur des variables des marchés norvégien et internationaux. La revue canadienne de statistique 40: 68–85; 2012 © 2012 Société statistique du Canada


1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

A copula is a multivariate distribution with standard uniform marginal distributions. While the literature on copulas is substantial, most of the research is still limited to the bivariate case. However, recently hierarchical copula-based structures have been proposed as an alternative to the standard copula methodology. One of the most promising of these structures is the pair-copula construction (PCC). The PCC was originally proposed by Joe (1996) and has further been explored by Bedford & Cooke (2001, 2002) and Kurowicka & Cooke (2006). After being set in an inferential context by Aas et al. (2009), the PCC has been used in various applications, see, for example, Schirmacher & Schirmacher (2008), Chollete, Heinen, & Valdesogo (2009), Heinen & Valdesogo (2009), Berg & Aas (2009), Min & Czado (2010, 2011), Czado, Schepsmeier, & Min (2011), and Smith et al. (2010).

Pair-copula constructions are also called regular vine (R-vine) copulas. They are hierarchical in nature, the various levels (also called trees) corresponding to the incorporation of more variables in the conditioning sets, using bivariate copulas as simple building blocks, the so-called pair-copulas. Until now, the concentration has been on two special cases of R-vine copulas; drawable vine (D-vine) and canonical vine (C-vine) copulas. However, very recently, there has been considerable progress in constructing R-vine copulas even in general, using graph theoretic algorithms (Dißmann et al., 2011).

The growing interest for pair-copula constructions/R-vine copulas is probably due to their high flexibility, which makes them able to model a wide range of complex dependencies. Nevertheless, these structures have some shortcomings, the most important being that the computational effort required to estimate all parameters grows exponentially with the dimension. For the R-vine copulas to be really useful in practice, one needs to be able to fit such structures to data with more than 20 dimensions. Hence, in this paper we treat the problem of determining whether an R-vine copula can be either truncated or simplified. By a pairwisely truncated R-vine copula at level K, we mean an R-vine copula where all pair-copulas with conditioning set equal to or larger than K are replaced by independence copulas. The subject of optimal truncation of vine copulas has previously been treated by Kurowicka (2011), who constructs R-vine copulas bottom-up beginning with the highest level and iteratively moving to the first level. Her approach is however based on Pearson product–moment correlations and therefore does not reflect nonelliptical dependence adequately. The approach suggested here is very different. It sequentially proceeds top-down and does not rely on the assumption of elliptical dependence. Additionally, our approach allows to identify independence of variables based on a statistical test in contrast to the ad-hoc procedure of Kurowicka (2011).

An R-vine copula is defined to be pairwisely simplified at level K if all pair-copulas with conditioning set equal to or larger than K instead are replaced by Gaussian copulas. We advocate using Gaussian copulas for the following reasons. They mean a simplification since they are easy to specify and faster to estimate than, for example, t copulas. Moreover, they are easy to interpret in terms of the correlation parameter. Most common Archimedean copulas such as the Clayton or the Gumbel, on the other hand, have asymmetric tail dependence and are therefore not suitable for simplification, since such asymmetry for a large number of pair-copulas is a very strict assumption. Thus the choice of the Gaussian copula as “neutral” copula is reasonable. It will be shown in our 19-dimensional application how specification and simulation times can be significantly improved using simplification with Gaussian copulas.

To identify the most appropriate truncation/simplification level, we use a heuristic procedure based on statistical model selection methods; more specifically, AIC, BIC and the likelihood-ratio based test proposed by Vuong (1989). We first evaluate the performance of the different methods in a simulation study, and then we investigate whether it is possible to simplify or truncate the R-vine copula specification corresponding to a 19-dimensional data set consisting of Norwegian and international market variables.

For the special case of a C-vine copula, the product of all pair-copulas with conditioning set equal to or larger than K (i.e., the pair-copulas involved in trees higher than K) gives a (equation image)-variate copula, where d is the total number of variables. Hence, in this case one may in addition to the above-mentioned model selection methods, use copula goodness-of-fit tests to determine the truncation/simplification level. The first kind of methods are hereafter referred to as pairwise truncation or simplification and the latter as joint truncation or simplification. It should be noted that joint simplification of C-vine copulas previously has been treated by Heinen & Valdesogo (2009) and Valdesogo (2009). There, it is referred to as “truncation”, while using our notation it would be called “simplification” (truncation in our meaning of the word is not explicitly discussed in Valdesogo, 2009). Pairwisely simplified R-vine copulas and the identification of truncation/simplification levels are not considered in these papers.

The rest of this paper is organized as follows. In Section 2 we provide necessary background on R-vine copulas and their likelihood. In Section 3 we introduce the pairwise simplification and truncation of R-vine copulas in general, while Section 4 treats the joint simplification and truncation of the special case of C-vine copulas. The heuristic selection of an appropriate truncation and simplification level in the general case and in the special case of C-vine copulas is discussed in Section 5. The performance of the different selection methods is studied in Section 6, while in Section 7 we apply the methodology in the context of a financial data set. Finally, Section 8 contains some concluding remarks.

2. Multivariate copulas and regular vines

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

Consider a vector equation image of random variables with a joint density function f. Sklar's theorem (Sklar, 1959) states that every multivariate distribution F with marginals equation image can be written as

  • equation image(1)

for some appropriate d-dimensional copula C. Using the chain rule, we further have for an absolutely continuous F with strictly increasing continuous marginals equation image that

  • equation image

Here equation image denotes the copula density. More details on copula theory can be found in the books by Joe (1997) and Nelsen (2006).

In higher-dimensional practical applications the choice of adequate copulas is limited. Multivariate copulas such as elliptical or exchangeable Archimedean are rather restricted and often not appropriate for dependence modeling. Hence, there is a growing need for more flexible copulas.

The notion of a regular vine distribution was introduced by Bedford & Cooke (2001, 2002) and described in more detail in Kurowicka & Cooke (2006). It involves the specification of a sequence of trees where each edge corresponds to a bivariate copula, a so-called pair-copula. These pair-copulas then constitute the building blocks of the joint regular vine distribution. According to Definition 4.4 of Kurowicka & Cooke (2006) a regular vine (R-vine) equation image on d variables consists of trees equation image with nodes equation image and edges equation image for equation image, which satisfy the following:

  • 1.
    equation image has nodes equation image and edges equation image.
  • 2.
    For equation image the tree equation image has nodes equation image.
  • 3.
    (proximity condition) If two edges in tree equation image are to be joined by an edge in tree equation image they must share a common node.

To build up a statistical model on R-vine trees with the node set equation image and the edge set equation image, one associates each edge equation image in equation image with a bivariate copula density equation image. The nodes equation image and equation image are called the conditioned nodes, while equation image is the conditioning set. An R-vine distribution is defined as the distribution of the random vector equation image with conditional copula density of equation image given the variables equation image specified as equation image for the R-vine trees with node set equation image and edge set equation image. equation image denotes the subvector of equation image determined by the indices in equation image. As in Aas et al. (2009) it is assumed here that equation image is independent of the conditioning variable equation image. Hobæk Haff, Aas, & Frigessi (2010) call this the simplified PCC which must not be confused with simplification discussed in the following.

In Theorem 4.2 of Kurowicka & Cooke (2006) it is proven that the joint density of equation image is uniquely determined and given by

  • equation image(2)

where equation image denotes the subvector of equation image determined by the indices in equation image. The rightmost part of Equation (2), which involves equation image bivariate copula densities, is called an R-vine copula.

An example of a seven-dimensional R-vine tree specification together with its edge indices is given in the left panel of Figure 1. This tree specification was found for a subset of the financial data in Section 7 (the considered variables and their corresponding numbers in the left panel of Figure 1 are: V17 (1), V20 (2), V18 (3), V1 (4), V10 (5), V14 (6) and V15 (7); see Table 4) according to the selection criteria discussed at the end of this section.

thumbnail image

Figure 1. An R-vine tree specification on seven variables (left panel) and a C-vine tree specification on five variables (right panel) with edge indices.

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Table 4. Variables of the Norwegian financial data set.
IDDescriptionIDDescription
V1Norwegian Financial IndexV125-year US Government Rate
V2USD-NOK exchange rateV13Norwegian bond index (BRIX)
V3 V4EURO-NOK exchange rate YEN-NOK exchange rateV14Citigroup World Government Bond Index (WGBI)
V5GBP-NOK exchange rateV15Norwegian 6-year Swap Rate
V73-month Norwegian Inter Bank Offered RateV16ST2X—Government Bond Index (fix modified duration of 0.5 years)
V8Norwegian 5-year Swap RateV17Morgan Stanley World Index (MSCI)
V93-month Euro Interbank Offered RateV18OSEBX—Oslo Stock Exchange main index
V105-year German Government RateV19Oslo Stock Exchange Real Estate Index
V113-month US Libor RateV20S&P Hedge Fund Index

Until now, the concentration has been on two special cases of regular vines; drawable vines (D-vines) and canonical vines (C-vines). In particular an R-vine is called

  • a D-vine if each node in equation image has a degree of at most 2, where the degree of a node denotes the number of connections or edges the node has to other nodes, and

  • a C-vine if each tree equation image has a unique node with degree equation image, the root node.

The corresponding R-vine distribution is called a D-vine or a C-vine distribution, respectively. For distinct indices equation image with equation image and equation image we use the abbreviation

  • equation image

Using this notation the D-vine density is given by

  • equation image(3)

and the C-vine density by

  • equation image(4)

See Aas et al. (2009) for more on simulation, inference and applications of C- and D-vine copulas. A five-dimensional C-vine tree specification is shown in the right panel of Figure 1.

For R-vines in general, there are no expressions like (3) and (4). Hence, an efficient way of storing the indices of the pair-copulas required in the joint density expression (2) is needed. One such approach was recently proposed by Morales-Napoles (2011) and explored in more detail in Dißmann et al. (2011). It involves the specification of a lower triangular matrix equation image with equation image. That is, the diagonal entries of M are the numbers equation image in decreasing order. In this matrix, according to a rather technical condition, each row from the bottom up represents a tree, where the conditioned set is identified by a diagonal entry and by the corresponding column entry of the row under consideration, while the conditioning set is given by the column entries below this row. Corresponding copula types and parameters can conveniently be stored in matrices related to M.

The R-vine matrix corresponding to the R-vine in Figure 1 is

  • equation image(5)

where all other entries are zero. The bottom row of M corresponds to equation image, the second row from the bottom to equation image, and so on. To determine the edges in equation image, we combine the numbers in the bottom row with the diagonal elements in the corresponding columns, that is, the edges are (7,5), (6,5), (5,1) and so on. To determine the edges in equation image, we combine the numbers in the second row from the bottom with the diagonal elements in the corresponding columns, and condition on the elements in the bottom row, giving the edges (7,1equation image5), (6,1equation image5), etc. Proceeding like this, the only edge in equation image is found by combining the two upper elements in the leftmost column of the matrix and condition on the remaining 5 entries in the same column, that is, (7,4equation image62315).

This matrix specification of R-vines at the same time directly allows for the derivation of the pair-copula decomposition (see Aas et al., 2009) of the corresponding R-vine distribution. Let equation image be an R-vine matrix corresponding to the R-vine equation image. Then, according to Dißmann et al. (2011), the R-vine density is:

  • equation image(6)

where the pair-copulas have arguments equation image and equation image.

The number of different possible R-vines in d dimensions is very large (Morales-Napoles, 2011). Hence, we need a way of selecting reasonable R-vine trees. Here, we will heuristically proceed as follows. We want to model the most important dependencies in the first trees. We therefore construct a graph on d nodes corresponding to the d variables, where all nodes are connected by a common edge, that is, have equation image neighbours. These edges have a weight according to a measure of pairwise dependence between the respective two variables, for example, empirical Kendall's τ or tail dependence. For this graph, we then find a maximum spanning tree (using the well-known algorithm of Prim, 1957), which is a tree on all nodes that maximizes the pairwise dependencies. Given this tree, we can now select pair-copulas, estimate parameters and compute transformed observations equation image for the next level, which in general are given by

  • equation image(7)

Here equation image is a bivariate copula, equation image is an arbitrary component of equation image and equation image denotes the vector equation image excluding equation image.

At the second level, we repeat the first level procedure, and iterate until all trees are constructed and their pair-copulas sequentially estimated. See Dißmann et al. (2011) for more details, and Brechmann (2010, Section 3.2) for construction methods for the special cases of C-and D-vines. In the latter case, the root node in each tree is found by choosing the variable with maximum sum of column entries in the matrix of pairwise dependencies.

Unfortunately vine copulas estimated in this way are not robust against misspecification of the pair-copulas, as also noted by Hobæk Haff (2010). To the best of our knowledge reliable alternatives are however not yet available. Goodness-of-fit tests for each pair-copula term may help reduce this uncertainty. These tests are however computationally not feasible in higher dimensions, since the number of pair-copulas grows quadratically with the dimension. Moreover, a large-scale simulation study in Brechmann (2010, Section 5.4) showed that copula selection using the AIC is more reliable than using goodness-of-fit tests.

3. Pairwise simplification and truncation

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

For R-vine copulas to be useful for risk analysis of market portfolios, one needs to be able to fit the parameters of such models in the high-dimensional case, for example, for 20–100 stocks. The computational effort needed to estimate all required parameters of an R-vine copula increases with the dimension. Hence, we take a pragmatic approach. We do not attempt to find the best fitting R-vine copula, but try to find the best fitting one under limited time and computational resources. To fix ideas, we want to allow for best possible specification of the first K trees in the R-vine copula, while higher order trees should only involve simple pair-copula terms, according to the idea that the most important dependencies are captured in the first trees.

Specifically, we denote an R-vine copula a pairwisely simplified K level one, if we replace all pair-copula terms which involve a conditioning set of size larger or equal to K by bivariate Gaussian copulas. Furthermore, we speak of a pairwisely truncated R-vine copula at level K, if all pair-copulas with conditioning set equal to or larger than K are set to bivariate independence copulas. If equation image, the truncated R-vine copula becomes a Markov tree distribution, where all conditional relationships are modeled as independent. Truncation may also be regarded as a special case of simplification, using Gaussian pair-copulas with correlation parameter equal to zero. Hence, it constitutes the greatest possible simplification.

In order to discuss the selection of simplification and truncation levels and propose appropriate procedures, we introduce some notation. First, we denote a pairwisely truncated R-vine copula at level K by tRV(K) and pairwisely simplified K level ones by sRV(K). Further, let equation image be the pair-copula parameters of the truncated R-vine copula, that is, equation image, equation image, where equation image denotes the parameter(s) of the copula density equation image. Then, the density of a truncated R-vine copula at level K is given by

  • equation image(8)

where equation image.

The density in (8) may be interpreted as a composite likelihood (see Lindsay (1988) and Varin, Reid, & Firth (2011)). This can be shown as follows. The joint density in (6) may also be decomposed as

  • equation image(9)

where the order of the conditioning variables is fixed, and given by the column entries of the R-vine matrix M. If the R-vine copula is truncated at level K, the density in (9) reduces to

  • equation image(10)

since the pair-copulas in trees equation image are set to independence copulas. Such an approximate likelihood, where the conditioning set is a subset of the variables equation image, has been proposed by Vecchia (1988) in the context of spatial models. For example, for equation image we obtain a Markov structure of order 1. If we set all margins and all pair-copulas to be Gaussian, (10) corresponds to the model of Vecchia (1988). In contrast to general composite likelihoods, the expression in (10) however does not require the choice of any weights and is in fact a valid probability density. The increasing conditional dependence in higher order trees directly leads to compatibility with composite likelihood methods. From their theory, we hence directly obtain consistency of the composite maximum likelihood estimates equation image based on the density in (8).

The density of a simplified K level R-vine copula is given by

  • equation image(11)

where, equation image denote Gaussian pair-copula densities with correlation parameter equation image, and the arguments of the copula densities have been omitted for simplicity. Further, equation image is the parameter set of equation image, that is,

  • equation image(12)

with equation image denoting the parameter(s) of the copula equation image.

In Sections 5.1 and 5.2 we will develop heuristic procedures for the selection of truncation and simplification levels, respectively, but first, in Section 4 we will describe the special case of a C-vine copula, for which joint simplification of the remaining equation image trees is possible.

4. Joint simplification and truncation

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

If we consider the special case of a C-vine copula, all pair-copulas with a conditioning set larger than or equal to K dimensions can be modeled jointly by a equation image-dimensional copula as shown in Valdesogo (2009). Typically we will choose a simple shape for this equation image-dimensional copula, such as the independence copula or multivariate Gaussian copula. In the case of an independence copula we speak of a jointly truncated C-vine copula, while in the Gaussian case, the resulting C-vine copula is denoted as jointly simplified. In the following, jointly simplified K level C-vine copulas will be denoted by jsCV(K). Simplification of C-vine copulas has previously been treated by Heinen & Valdesogo (2009) and Valdesogo (2009) who refer to it as “truncation”. Truncation in our meaning of the word is however not discussed in these papers.

For C-vine copulas the second component of the product in (11) reduces to a (equation image)-dimensional Gaussian copula. Hence, we obtain the density of a jointly simplified K level C-vine copula by rewriting (11) to

  • equation image

where equation image is a (equation image)-dimensional Gaussian copula density with arguments equation image. The parameter set equation image is defined similarly to (12) as

  • equation image

where equation image are the parameters of the pair-copulas equation image, while equation image denote the entries of the correlation matrix of the multivariate Gaussian copula equation image.

Finally, note that for D-vine copulas, joint simplification as described above is not possible. The reason is that while C-vine copulas have a common conditioning set in each tree as shown in (4), this is not the case for the D-vine copula (see (3)). For instance, in a five-dimensional D-vine copula, the arguments to the pair-copula densities in tree equation image are equation image, equation image, equation image, equation image, equation image and equation image. Crosswise relationships such as equation image and equation image complicate the situation.

5. Selection of truncation and simplification levels

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

5.1. Selection of Truncation Level in the General Case

We will now consider the selection of the truncation level in the general case. Note that the vine copula tRV(K) is nested in tRV(equation image), since equation image. The log likelihood for tRV(K) is given by

  • equation image(13)

where n is the number of data points equation image, equation image. From now on we assume that data has been transformed to the unit hypercube using the respective marginal distribution functions of the variables. In practice this is either done parametrically by selecting (and estimating) appropriate marginal distributions, or nonparametrically by using the empirical distribution functions. Here, we assume that the latter is the case, because it eliminates the risk of misspecification (and possible influences on the selection of the truncation or simplification level). For more details on these issues see Genest, Ghoudi, & Rivest (1995), Joe (1997) and Kim, Silvapulle, & Silvapulle (2007). However estimating the true margins by empirical ones alters the asymptotic distribution of the maximum likelihood (ML) estimates of the dependence parameters and hence influences the estimation of standard errors and critical values for hypothesis tests.

If there are sufficient computing resources, we can maximize the full log likelihood (13). Alternatively, one may use the stepwise/sequential ML-estimator originally proposed by Aas et al. (2009) and further explored by Hobæk Haff (2010, 2011). Based on an extensive simulation study, the latter paper shows that the performance of the stepwise estimator is satisfactory compared to the full log likelihood method. To determine the sequential parameter estimates of equation image, it is enough to only use transformed variables (7) up to tree K. These sequential estimates can then be used as starting values for maximizing equation image. For K small, the number of parameters to be maximized over is considerably reduced compared to a full R-vine copula specification.

We will start with equation image and fit a truncated R-vine copula (for equation image a pre-test of joint independence can be performed). We thereafter increase K by one and assess how much gain we get by fitting the extra tree. If the gain is negligible we stop and use the resulting specification. If the gain is large enough, we increase K by one again, and proceed in this way until we have reached a truncation level equation image, which either gives a sufficient fit, or we have reached the computational time frame we allowed for the estimation process.

To assess whether there is gain to move from model tRV(K) to tRV(equation image), we now consider two kinds of statistical model selection techniques; AIC/BIC and the likelihood-ratio based test proposed by Vuong (1989). First, since tRV(K) is nested within tRV(equation image), we can compare the AIC or BIC values of the two models to quantify the marginal gain of an additionally fitted tree. In particular, these quantities are given by

  • equation image

where equation image denotes the dimension of equation image. We choose the one of tRV(K) and tRV(equation image) with the smaller AIC or BIC value. If for some equation image the smaller model is chosen, we stop, and declare tRV(equation image) as the best fitting model among the model sequence equation image, equation image.

Alternatively, we can use the likelihood-ratio based test proposed by Vuong (1989). In order to compare two competing nonnested models equation image and equation image with estimated parameters equation image and equation image, respectively, we compute the standardized sum, ν, of the log differences of their pointwise likelihoods equation image for observations equation image, equation image. Under fairly general regularity conditions ν is shown to be asymptotically standard normal, leading to the following test. We prefer model 1 to model 2 at level α if

  • equation image(14)

where equation image denotes the inverse of the standard normal distribution function. If equation image we choose model 2. If, however, equation image, no decision among the models is possible. Like AIC and BIC, the Vuong test statistic may be corrected for the number of parameters used in the models. There are two possible corrections; the Akaike and the Schwarz corrections, which correspond to the penalty terms in the AIC and the BIC, respectively.

When dealing with truncated R-vines, models are nested and a classical likelihood-ratio test could be used to compare tRV(K) and tRV(equation image). As we want to allow for misspecification of the models, the asymptotic distribution of the test is however hardly tractable (see Vuong (1989)). Moreover, a likelihood-ratio test cannot be corrected for the number of model parameters as conveniently as the Vuong test.

We therefore heuristically apply the Vuong test to compare tRV(K) (model equation image) and tRV(equation image) (model equation image). If equation image, we stop with tRV(K), since tRV(K) is preferred to, or indistinguishable from tRV(equation image), at level α. It thus determines the truncation level as the level equation image for which tRV(equation image) does not provide a significant gain in the model fit. In the light of the regularity conditions of Vuong (1989), it is important to note that the log likelihood (13) is in fact a valid log likelihood under certain conditional independence conditions and that the sequential estimates are consistent and asymptotically normal as shown by Hobæk Haff (2010).

Table 1 describes the truncation procedure based on the Vuong test. Truncation using information criteria proceeds in the same way, where we need to compute only the contribution from tree equation image to the AIC of tRV(equation image). This is due to the fact that the AICs of tRV(K) and tRV(equation image) are equal with the exception of the contribution from tree equation image. Since all copulas in tree equation image of tRV(K) are independence copulas, the contribution from tree equation image to the AIC of tRV(K) is zero. Hence, if the contribution from tree equation image to the AIC of tRV(equation image) is greater than zero, we truncate at level j.

Table 1. Algorithm for truncation of R-vine copulas based on the Vuong test.
Input: Observations of d variables, significance level α.
forequation imagedo
Specify tRV(equation image) by additionally constructing tree equation image with appropriate pair-copulas.
Perform a Vuong test for tRV(j) (model equation image) and tRV(equation image) (model equation image), that is, determine test
statistic ν as in (14), possibly with Akaike or Schwarz correction.
ifequation imagethen
Truncate the R-vine copula at level equation image, that is, exit the loop with tRV(j).
end if
end for
Output: Pairwisely truncated K level R-vine copula, or fully specified R-vine copula, if no
truncation is possible.

Before we move on to selection of the simplification level in the general case, we turn to an illustrative example.

Example 1 (Pairwise truncation of R-vine copulas.) We consider a five-dimensional D-vine copula. Assume that we have already appropriately specified the pair-copulas of the first two trees equation image and equation image. We now want to determine whether the D-vine copula can be truncated or simplified at level 2. This is done by measuring the marginal gain of a third tree equation image. If the marginal gain is too small either in terms of AIC/BIC or as determined by a Vuong test, we truncate at level equation image. Hence, we simply have to compare the smaller model (equation image) to the larger model (equation image) as illustrated in Figure 2. Note that this is not an exact model comparison between a truncated D-vine copula and a fully specified one (equation image), but only an approximation to the truth, since possible dependencies in the fourth tree equation image are ignored in the comparison. However, under the assumption that most dependencies are captured in the first trees, this should be a reasonable approximation.

thumbnail image

Figure 2. Pair-copula density terms of five-dimensional D-vine copulas truncated after the second tree equation image (smaller model) and after the third tree equation image (larger model), respectively, where equation image denote densities of independence copulas.

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5.2. Selection of Simplification Level in the General Case

Selection of simplification levels, that is, model selection between sRV(K) and sRV(equation image), proceeds in essentially the same way as for truncation. However, on the contrary to tRV(K) and tRV(equation image), the models sRV(K) and sRV(equation image) are not nested, in general equation image. Nonnested models may be compared using the Vuong test under the assumption that models are not equal (A pre-test for overlapping (partially nested) models as outlined in Vuong, 1989 is not performed, since it is numerically not feasible and the assumption of unequal models is sensible here.). If we use AIC or BIC, however, we have to deal with an increased variability (Ripley, 2008, pp. 34–35). Since we build models according to the paradigm that the most important dependencies are captured in the first trees, we assume that the specifications of trees equation image to equation image are equal in models sRV(K) and sRV(equation image) if we work with AIC and BIC. Moreover, we know that trees equation image to equation image are the same. Hence, we have achieved “as much nestedness as possible”, since only tree equation image is different in both models.

The simplification procedure based on the Vuong test is outlined in Table 2. Simplification using information criteria proceeds similarly but under the assumption that trees equation image are specified with bivariate Gaussian copulas according to the discussion of nestedness above. In Example 1 this means that we assume that the Gaussian pair-copulas equation image and equation image in equation image of the smaller and the larger model, respectively, are equal if we work with AIC and BIC.

Table 2. Algorithm for simplification of R-vine copulas based on the Vuong test.
Input: Observations of d variables, significance level α.
forequation imagedo
Specify sRV(j) by constructing higher order trees equation image with bivariate Gaussian
copulas.
Specify sRV(equation image) by additionally constructing tree equation image with appropriate pair-copulas,
and by constructing higher order trees equation image with bivariate Gaussian copulas.
Perform a Vuong test for sRV(j) (model equation image) and sRV(equation image) (model equation image), that is, determine test
statistic ν as in (14), possibly with Akaike or Schwarz correction.
ifequation imagethen
Simplify the R-vine copula at level equation image, that is, exit the loop with sRV(j).
end if
end for
Output: Pairwisely simplified K level R-vine copula, or fully specified R-vine copula, if no
simplification is possible.

5.3. Selection of Simplification and Truncation Levels for C-vine Copulas

The procedures described in Sections 5.1 and 5.2 may of course be used also for the special case of a C-vine copula. However, in this case, we may alternatively use multivariate independence tests or copula goodness-of-fit tests to determine whether we can truncate or simplify the model at level K, respectively. If the P-value of an independence test is larger than a preliminarily chosen level, we truncate the structure at level K. Also for the purpose of simplification, if the P-value of an appropriate copula goodness-of-fit test for the multivariate Gaussian copula is larger than a preliminarily chosen level, we simplify the structure at level K. As previously described, this way of truncation/simplification is denoted joint truncation/simplification. Table 3 outlines the joint simplification procedure, while joint truncation proceeds in exactly the same way but with the use of an independence test in the second line.

Table 3. Algorithm for joint simplification of C-vine copulas.
Input: Observations of d variables, significance level α.
forequation imagedo
Perform a copula goodness-of-fit test for jsCV(j) to test if the transformed observations from
tree equation image can be appropriately modeled with a equation image-dimensional Gaussian copula.
ifP-value equation imagethen
Simplify the C-vine copula at level equation image, that is, exit the loop with jsCV(j).
end if
Specify tree equation image with appropriate pair-copulas.
end for
Output: Jointly simplified K level C-vine copula, or fully specified C-vine copula, if no
simplification is possible.

6. Simulation studies

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

We have evaluated the performance of the heuristic simplification and truncation procedures presented in Section 5 in extensive simulation studies. A thorough description of the results can be found in the supplementary material. To summarize the main findings, the procedures based on the Vuong test with or without correction for the number of parameters should be used in most cases, even for joint simplification/truncation where tailor-made procedures are available. The procedures based on AIC/BIC can be regarded as “quick and dirty” alternatives. These criteria tend to identify truncation and especially simplification too late, but they are very fast compared to the Vuong test (in a 52-dimensional example, AIC identified the truncation/simplification level 44%/80% faster than the Vuong test). Parsimonious models can be obtained by using the Vuong test with Schwarz correction.

7. Application

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

In this section, we analyze a 19-dimensional data set consisting of Norwegian and international financial variables. See Table 4 for a description. The variables constitute the market portfolio of a large Norwegian financial institution and hence, it is very important to correctly model the dependencies between them. The observed time period is from 3/25/2003 to 3/26/2008, resulting in 1,107 daily observations. As previously stated, the computational effort needed to estimate all required parameters of an R-vine copula increases with the dimension. Hence, the aim of the work presented here was to investigate whether simplification or truncation of the R-vine copula specification corresponding to this 19-dimensional data set is possible.

Before analyzing the dependence in the data set, we selected appropriate ARMA-GARCH time series models for the univariate margins (see the supplementary material for more details). After filtering the original returns with the chosen univariate models, the standardized residual vectors are converted to uniform pseudo-observations using their empirical distribution functions. In the light of results due to Chen & Fan (2006), the method of maximum pseudo likelihood is consistent even when time series are fitted to the margins.

For the sake of reference, we first fit a full R-vine copula to this data set, using the approach described in Section 2. We use Kendall's τ's as edge weights, and pair-copulas are selected from a range of 11 bivariate families using AIC: independence copula, Gaussian, t, Clayton, rotated Clayton (90equation image), Gumbel, rotated Gumbel (90equation image), Frank, Joe, Clayton-Gumbel (BB1), Joe-Clayton (BB7). For more information on copula types, see, for example, Nelsen (2006) or Joe (1997). The independence copula is chosen according to the bivariate independence test based on Kendall's τ as described in Genest & Favre (2007). If the P-value is larger than 5%, the independence copula is chosen to obtain more parsimonious models and therefore results in an additional inherent truncation.

Figure 3 shows the first tree of the fitted R-vine copula. See Table 4 for the correspondence between the IDs and the variable descriptions. The edge labels represent the empirical Kendall's τ's between the respective variables. The corresponding R-vine matrix specifications with copula types and parameters can be found in the supplementary material. In particular, the first tree pair-copula terms identify strong to medium tail dependence and some asymmetries. Dependencies modeled in higher order trees are much weaker.

thumbnail image

Figure 3. First tree of the full R-vine copula model for the Norwegian financial data set. The edge labels indicate empirical Kendall's τ's between the respective variables.

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In economical terms, the tree in Figure 3 has an evident interpretation. It identifies three clusters of economically similar variables. The first cluster consists of the stock indices, the hedge fond index and the real estate index (variables V1, V17, V18, V19 and V20). The second cluster consists of the interest rates and the bond indices (V7, V8, V9, V10, V11, V12, V13, V14, V15 and V16). Finally, the exchange rates (V2, V3, V4, and V5) constitute the third cluster. The stock and interest rate clusters are linked through the variables V10 and V17 (the 5-year German Government Rate and the MSCI World index), while the interest rate and exchange rate clusters are connected by V14 and V2 (the WGBI bond index and the USD-NOK exchange rate).

In addition to the R-vine copula with different copulas for different pairs, we also fitted an R-vine copula with t copulas for all pairs. The BIC-values in the two upper rows of Table 5 show however that the R-vine copula with mixed copulas is superior to the one with only t copulas.

Table 5. R-vine copula specifications of the Norwegian financial data set obtained by different procedures (full maximum likelihood estimation). Vuong test statistics (V/1, V/2, V/3, V/4) with a “*” imply that the considered model is indistinguishable from or superior to the full R-vine copula model (columns 7 and 8) or the multivariate t copula (columns 9 and 10), respectively, at the equation image level; test statistics V/2 and V/4 are adjusted according to the Schwarz correction. Models considered are the truncated/simplified R-vine copulas as well as the R-vine copula with only pair t copulas.
ModelLvl.Log lik.#Par.BICV/1V/2V/3V/4
Full model6390.7592−12130.22−1.93*−10.22*
Pair t copulas6378.33104−12020.420.82*3.61−1.71*−9.44*
Multivar. t copula6324.98172−11432.341.93*10.22
Trunc.Vuong66274.4777−12003.837.253.941.34*−7.59*
 V.Schw.46234.0568−11986.727.973.652.39−7.28*
 AIC/BIC66274.4777−12003.837.253.941.34*−7.59*
Simpl.Vuong26350.0984−12105.523.190.97*−0.75*−10.07*
 V.Schw.26350.0984−12105.523.190.97*−0.75*−10.07*
 AIC/BIC66373.8088−12124.632.460.41*−1.41*−10.02*

Having fitted the full R-vine copula, we apply the different statistical model selection criteria from Sections 5.1 and 5.2 to investigate whether truncation and/or simplification of this R-vine is possible. Table 5 shows the results. We report log likelihood values, the number of parameters and BIC for the truncated/simplified models obtained using the different criteria (truncation/simplification based on AIC and BIC turned out to give the same results for this data set). In addition, the table shows the test statistics of Vuong tests (with and without Schwarz correction) with respect to the null hypothesis that the fully specified model and simplified/truncated model are equivalent. Test statistics indicated by “*” imply that the null hypothesis cannot be rejected at the equation image level or that the simplified/truncated model is even superior. Note that the number of parameters of a full R-vine copula with d variables is equation image if all pair-copulas have one parameter each. The reason why the number of parameters shown in Table 5 is much smaller than this, is that many of the pair-copulas in the full R-vine copula (both the one with mixed copulas and the one with only t copulas) are estimated to be independence copulas. The R-vine copula with only t copulas would otherwise have 342 parameters rather than 104!

If we first turn to the truncation results, they show that truncation at level 6 seems to give a slightly better model than truncation at level 4. The hypothesis that the fully specified model and the truncated model are equivalent is however rejected for both tRV(4) and tRV(6), meaning that there still seems to be significant dependencies after tree equation image. In Brechmann (2010, Section 11.2.2) we have studied the model tRV(4) in more detail, by considering, among others, joint tail behaviour, copula Q-Q plots and Kendall's τ's of simulated observations. The results showed that although this model did not fully reproduce the observed data characteristics, it may be viewed as an adequate specification for the data. Hence, we conclude that the most important dependencies in this data set are actually captured in the first four to six trees, meaning that the corresponding R-vine copula may be truncated at level 6, or even at level 4, depending on the desired level of parsimonity (and of course at the expense of accuracy).

As far as simplification is concerned, sRV(6) seems to be slightly better than sRV(2) in terms of BIC. However, the hypothesis that the fully specified model and the simplified model are equivalent is not rejected even for sRV(2). Based on this, and also on a more thorough study of sRV(2) in Brechmann (2010, Section 11.2.2) we conclude that all important (asymmetric) tail dependencies seem to be captured in the first two trees. Hence, simplification at level 2 seems appropriate.

The d-dimensional t copula with one common degrees of freedom parameter is currently the state-of-the-art approach for modeling financial return data. A number of papers, such as Mashal & Zeevi (2002), have shown that the fit of this copula is generally superior to that of other d-dimensional copulas for such data. Hence, we wanted also to compare the truncated and simplified R-vine copulas to this structure. The parameters of the t copula were estimated in two steps. First, the correlation matrix of the t copula was determined by inversion of bivariate Kendall's τ-values, and then the degrees of freedom parameter was found using maximum likelihood estimation. As indicated by the results in Table 5, the t copula is statistically equivalent or even inferior to all truncated and simplified R-vine copula models, in particular when taking the number of parameters into account. The latter is due to the fact that the t copula requires the specification of the full correlation matrix, while even the full R-vine copula might be reduced with bivariate independence tests, and hence leads to more parsimonious copula models.

Finally, computing times relative to the full R-vine model are shown in Table 6. If we first turn to the sequential estimation, we see that the AIC/BIC based procedures confirm their naming as “quick and dirty”. The procedures based on the Vuong test require more time, since the comparisons are more complex and require the estimation of additional trees in the case of simplification. Using the full maximum likelihood estimation, however, the truncated and simplified R-vines can be fitted much faster than a full R-vine. Moreover, simulation from the truncated/simplified models is of course more computationally efficient than from the full R-vine model.

Table 6. Computing times relative to the full R-vine copula model for the models identified in Table 5.
ModelEstimationEstimationEstimationSimulation
  (Part I: sequential)(Part II: full MLE)(Part I & II) 
Pair t copulas0.741.271.251.04
Trunc.Vuong1.180.470.500.89
 V.Schwarz1.030.340.370.81
 AIC/BIC0.630.470.480.90
Simpl.Vuong2.160.470.530.73
 V.Schwarz2.120.460.520.73
 AIC/BIC0.720.580.590.92

8. Conclusions

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

In this paper we considered the problem of determining whether R-vine copulas can be pairwisely truncated or alternatively, simplified with Gaussian pair-copulas, after a certain tree. In extensive simulations different procedures for truncation and simplification were proposed and evaluated. The results showed that Vuong test based procedures performed particularly well.

We also considered truncating or simplifying the special case of a C-vine copula. In this case, the remaining dependencies may be captured by a multivariate copula; the independence copula for the truncation alternative and the Gaussian copula for the simplification one. Hence, simplification/truncation levels may be determined using a multivariate copula goodness-of-fit-test. However, simulations showed that our procedures developed for the general R-vine copula overall seemed to detect the simplification/truncation levels more accurately than the multivariate goodness-of-fit-tests.

Finally, we have investigated whether it is possible to simplify or truncate the R-vine copula specification corresponding to a 19-dimensional data set consisting of Norwegian and international market variables. This study showed that the most important dependencies in the Norwegian data set are captured in the first 4–6 trees, meaning that the corresponding R-vine copula may be truncated at level 6, or even at level 4. Moreover, simplification at level 2 seemed to be appropriate, indicating that all important (asymmetric) tail dependencies are captured in the first two trees.

To summarize, the methods discussed in this paper allow to efficiently construct R-vine copula models even in higher dimensions and under time or resource restrictions. As such, R-vine copula models constitute a flexible and powerful class of high-dimensional dependency models, available for a wide range of applications.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY

We like to thank Arnoldo Frigessi and Ingrid Hobæk Haff for contributing valuable ideas and information. We acknowledge the helpful comments of the referees, which further improved the manuscript. The work conducted by Kjersti Aas is sponsored by Statistics for Innovation, (sfi)2. Claudia Czado is supported by the DFG (German Research Foundation) grant CZ 86/1-3. The numerical computations were performed on a Linux cluster supported by DFG grant INST 95/919-1 FUGG.

BIBLIOGRAPHY

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. Multivariate copulas and regular vines
  5. 3. Pairwise simplification and truncation
  6. 4. Joint simplification and truncation
  7. 5. Selection of truncation and simplification levels
  8. 6. Simulation studies
  9. 7. Application
  10. 8. Conclusions
  11. Acknowledgements
  12. BIBLIOGRAPHY
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