### Abstract

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

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. Multivariate copulas and regular vines
- 3. Pairwise simplification and truncation
- 4. Joint simplification and truncation
- 5. Selection of truncation and simplification levels
- 6. Simulation studies
- 7. Application
- 8. Conclusions
- Acknowledgements
- 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 ()-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

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

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.

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.

Table 4. Variables of the Norwegian financial data set.ID | Description | ID | Description |
---|

V1 | Norwegian Financial Index | V12 | 5-year US Government Rate |

V2 | USD-NOK exchange rate | V13 | Norwegian bond index (BRIX) |

V3 V4 | EURO-NOK exchange rate YEN-NOK exchange rate | V14 | Citigroup World Government Bond Index (WGBI) |

V5 | GBP-NOK exchange rate | V15 | Norwegian 6-year Swap Rate |

V7 | 3-month Norwegian Inter Bank Offered Rate | V16 | ST2X—Government Bond Index (fix modified duration of 0.5 years) |

V8 | Norwegian 5-year Swap Rate | V17 | Morgan Stanley World Index (MSCI) |

V9 | 3-month Euro Interbank Offered Rate | V18 | OSEBX—Oslo Stock Exchange main index |

V10 | 5-year German Government Rate | V19 | Oslo Stock Exchange Real Estate Index |

V11 | 3-month US Libor Rate | V20 | S&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 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 has a unique node with degree , the *root node*.

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

Using this notation the D-vine density is given by

- (3)

and the C-vine density by

- (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.

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

- (5)

where all other entries are zero. The bottom row of *M* corresponds to , the second row from the bottom to , and so on. To determine the edges in , 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 , 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 (7,15), (6,15), etc. Proceeding like this, the only edge in 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, (7,462315).

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 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 for the next level, which in general are given by

- (7)

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

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

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

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

- (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

- (10)

since the pair-copulas in trees are set to independence copulas. Such an approximate likelihood, where the conditioning set is a subset of the variables , has been proposed by Vecchia (1988) in the context of spatial models. For example, for 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 based on the density in (8).

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 trees is possible.

### 7. Application

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. Multivariate copulas and regular vines
- 3. Pairwise simplification and truncation
- 4. Joint simplification and truncation
- 5. Selection of truncation and simplification levels
- 6. Simulation studies
- 7. Application
- 8. Conclusions
- Acknowledgements
- 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 (90), Gumbel, rotated Gumbel (90), 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.

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 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.Model | Lvl. | Log lik. | #Par. | BIC | V/1 | V/2 | V/3 | V/4 |
---|

Full model | — | 6390.75 | 92 | −12130.22 | — | — | *−1.93** | *−10.22** |

Pair t copulas | — | 6378.33 | 104 | −12020.42 | *0.82** | 3.61 | *−1.71** | *−9.44** |

Multivar. t copula | — | 6324.98 | 172 | −11432.34 | *1.93** | 10.22 | — | — |

Trunc. | Vuong | 6 | 6274.47 | 77 | −12003.83 | 7.25 | 3.94 | *1.34** | *−7.59** |

| V.Schw. | 4 | 6234.05 | 68 | −11986.72 | 7.97 | 3.65 | 2.39 | *−7.28** |

| AIC/BIC | 6 | 6274.47 | 77 | −12003.83 | 7.25 | 3.94 | *1.34** | *−7.59** |

Simpl. | Vuong | 2 | 6350.09 | 84 | −12105.52 | 3.19 | *0.97** | *−0.75** | *−10.07** |

| V.Schw. | 2 | 6350.09 | 84 | −12105.52 | 3.19 | *0.97** | *−0.75** | *−10.07** |

| AIC/BIC | 6 | 6373.80 | 88 | −12124.63 | 2.46 | *0.41** | *−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 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 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 . 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.Model | Estimation | Estimation | Estimation | Simulation |
---|

| | (Part I: sequential) | (Part II: full MLE) | (Part I & II) | |
---|

Pair t copulas | 0.74 | 1.27 | 1.25 | 1.04 |

Trunc. | Vuong | 1.18 | 0.47 | 0.50 | 0.89 |

| V.Schwarz | 1.03 | 0.34 | 0.37 | 0.81 |

| AIC/BIC | 0.63 | 0.47 | 0.48 | 0.90 |

Simpl. | Vuong | 2.16 | 0.47 | 0.53 | 0.73 |

| V.Schwarz | 2.12 | 0.46 | 0.52 | 0.73 |

| AIC/BIC | 0.72 | 0.58 | 0.59 | 0.92 |

### 8. Conclusions

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