The partial copula: Properties and associated dependence measures

Spanhel, Fabian; Kurz, Malte S.

doi:10.1016/j.spl.2016.07.014

Cited by 13 publications

(14 citation statements)

References 15 publications

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“…Recently, there has been a renewed interest in the partial copula. Spanhel and Kurz [18] investigate properties of the partial copula and mention some explicit examples whereas Gijbels et al [16,17] and Portier and Segers [19] focus on the non-parametric estimation of the partial copula.…”

Section: Definition 24 (Conditional Probability Integral Transform (mentioning

confidence: 99%

“…a regular vine structure and that they coincide with their conditional analogues if the simplifying assumption holds. A partial correlation coefficient of zero is commonly interpreted as an indication of conditional independence, although this can be quite misleading if the underlying distribution is not close to a Normal distribution (Spanhel and Kurz [18]). Therefore, one might wonder to what extent higher-order partial copulas can be used to check for conditional independencies.…”

Section: Properties Of the Partial Vine Copula And Examplesmentioning

confidence: 99%

“…Let C Fr (θ) denote the bivariate Frank copula with dependence parameter θ and C P-Fr (θ) be the partial Frank copula [18] with dependence parameter θ. Let C 1:3 be the true copula with (C 12 , C 23 , C 13; 2 ) = (C Fr (5.74), C Fr (5.74), C P-Fr (5.74)), i.e., C 1:3 = C PVC 1:3 , and C SVC 1:3 (θ) = (C Fr (θ 12 ), C Fr (θ 23 ), C P-Fr (θ 13;2 )) be the parametric SVC that is fitted to data generated from C 1:3 .…”

Section: Example 61 (Pvc Of the Frank Copula)mentioning

confidence: 99%

“…As a result, researchers have recently started to investigate new dependence concepts that are related to the simplifying assumption and arise if it does not hold. In particular, studies on the bivariate partial copula, a generalization of the partial correlation coefficient, have (re-)emerged lately [15,16,17,18,19].…”

Section: Introductionmentioning

confidence: 99%

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Simplified vine copula models: Approximations based on the simplifying assumption

Spanhel¹,

Kurz²

2019

Electron. J. Statist.

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Simplified vine copulas (SVCs), or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. So far, specification and estimation of SVCs has been conducted under the simplifying assumption, i.e., all bivariate conditional copulas of the vine are assumed to be bivariate unconditional copulas. We introduce the partial vine copula (PVC) which provides a new multivariate dependence measure and which plays a major role in the approximation of multivariate distributions by SVCs. The PVC is a particular SVC where to any edge a j-th order partial copula is assigned and constitutes a multivariate analogue of the bivariate partial copula.We investigate to what extent the PVC describes the dependence structure of the underlying copula. We show that the PVC does not minimize the Kullback-Leibler divergence from the true copula and that the best approximation satisfying the simplifying assumption is given by a vine pseudo-copula. However, under regularity conditions, stepwise estimators of pair-copula constructions converge to the PVC irrespective of whether the simplifying assumption holds or not. Moreover, we elucidate why the PVC is the best feasible SVC approximation in practice.

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Section: Definition 24 (Conditional Probability Integral Transform (mentioning

confidence: 99%

Section: Properties Of the Partial Vine Copula And Examplesmentioning

confidence: 99%

Section: Example 61 (Pvc Of the Frank Copula)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simplified vine copula models: Approximations based on the simplifying assumption

Spanhel¹,

Kurz²

2019

Electron. J. Statist.

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“…In this case the pair copula density c je,ke;De G je|De (u je |u De ), G ke|De (u ke |u De ) encodes the partial dependence between U je and U ke conditional on U De [19,33]. This simplifying assumption is often valid for financial data and even if it is violated, it may serve as a useful approximation to the truth.…”

Section: Vine Copulasmentioning

confidence: 99%

Model selection in sparse high-dimensional vine copula models with an application to portfolio risk

Nagler

Bumann

Czado

2019

Journal of Multivariate Analysis

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Vine copulas allow to build flexible dependence models for an arbitrary number of variables using only bivariate building blocks. The number of parameters in a vine copula model increases quadratically with the dimension, which poses new challenges in high-dimensional applications. To alleviate the computational burden and risk of overfitting, we propose a modified Bayesian information criterion (BIC) tailored to sparse vine copula models. We show that the criterion can consistently distinguish between the true and alternative models under less stringent conditions than the classical BIC. The new criterion can be used to select the hyper-parameters of sparse model classes, such as truncated and thresholded vine copulas. We propose a computationally efficient implementation and illustrate the benefits of the new concepts with a case study where we model the dependence in a large stock stock portfolio.

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Estimating non-simplified vine copulas using penalized splines

Schellhase

Spanhel

2017

Stat Comput

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Vine copulas (or pair-copula constructions) have become an important tool for highdimensional dependence modeling. Typically, so called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first non-parametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse of dimensionality, we approximate conditional copulas with more than one conditioning argument by a conditional copula with the first principal component as conditioning argument. An extensive simulation study is conducted, showing a substantial improvement in the out-of-sample Kullback-Leibler divergence if the null hypothesis of a simplified vine copula can be rejected. We apply our method to the famous uranium data and present a classification of an eye state data set, demonstrating the potential benefit that can be achieved when conditional copulas are modeled.

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The partial copula: Properties and associated dependence measures

Cited by 13 publications

References 15 publications

Simplified vine copula models: Approximations based on the simplifying assumption

Simplified vine copula models: Approximations based on the simplifying assumption

Model selection in sparse high-dimensional vine copula models with an application to portfolio risk

Estimating non-simplified vine copulas using penalized splines

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