Archimedean Copulas and Temporal Dependence

Beare, Brendan K.

doi:10.1017/s0266466612000126

Cited by 33 publications

(33 citation statements)

References 33 publications

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“…Finally, the obtained results could be used to develop statistical inference procedures for Markovian copula time series models as introduced in Darsow, Nguyen and Olsen [18]. Based on recent results from Beare [5] on the mixing properties of these time series, one could, for instance, apply the proposed multiplier bootstrap to derive uniform confidence bands for the empirical copula or to develop tests for simple goodness-of-fit hypotheses on the copula in theses models.…”

Section: Introductionmentioning

confidence: 99%

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

Bücher¹,

Kojadinovic²

2016

Bernoulli

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Two key ingredients to carry out inference on the copula of multivariate observations are the empirical copula process and an appropriate resampling scheme for the latter. Among the existing techniques used for i.i.d. observations, the multiplier bootstrap of Rémillard and Scaillet (J. Multivariate Anal. 100 (2009) 377-386) frequently appears to lead to inference procedures with the best finite-sample properties. Bücher and Ruppert (J. Multivariate Anal. 116 (2013) 208-229) recently proposed an extension of this technique to strictly stationary strongly mixing observations by adapting the dependent multiplier bootstrap of Bühlmann (The blockwise bootstrap in time series and empirical processes (1993) ETH Zürich, Section 3.3) to the empirical copula process. The main contribution of this work is a generalization of the multiplier resampling scheme proposed by Bücher and Ruppert along two directions. First, the resampling scheme is now genuinely sequential, thereby allowing to transpose to the strongly mixing setting many of the existing multiplier tests on the unknown copula, including nonparametric tests for change-point detection. Second, the resampling scheme is now fully automatic as a data-adaptive procedure is proposed which can be used to estimate the bandwidth parameter. A simulation study is used to investigate the finite-sample performance of the resampling scheme and provides suggestions on how to choose several additional parameters. As by-products of this work, the validity of a sequential version of the dependent multiplier bootstrap for empirical processes of Bühlmann is obtained under weaker conditions on the strong mixing coefficients and the multipliers, and the weak convergence of the sequential empirical copula process is established under many serial dependence conditions. Keywords: lag window estimator; multiplier central limit theorem; multivariate observations; partial-sum process; ranks; serial dependence This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2016, Vol. 22, No. 2, 927-968. This reprint differs from the original in pagination and typographic detail.

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Section: Introductionmentioning

confidence: 99%

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

Bücher¹,

Kojadinovic²

2016

Bernoulli

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“…It follows from the results in Beare (2010) that (Y i ) i∈Z is alpha-mixing for many commonly used copula families, and it is easy to see that this property transfers to (X i ) i∈Z .…”

Section: The General Frameworkmentioning

confidence: 98%

“…In these time series models, copulas are used to model the serial dependence at lag one. We refer to Beare (2010), who derived the corresponding mixing properties, for a number of references to applications of these models. Extensions to the multivariate case which also cover a joint modeling of the contemporary and the serial dependence are given in Rémillard et al (2012).…”

Section: Introductionmentioning

confidence: 99%

Goodness-of-Fit Tests for Multivariate Copula-Based Time Series Models

Berghaus

Bücher

2016

Econom. Theory

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In recent years, stationary time series models based on copula functions became increasingly popular in econometrics to model nonlinear temporal and cross-sectional dependencies. Within these models, we consider the problem of testing the goodness-of-fit of the parametric form of the underlying copula. Our approach is based on a dependent multiplier bootstrap and it can be applied to any stationary, strongly mixing time series. The method extends recent i.i.d. results by Kojadinovic, Yan and Holmes [I. Kojadinovic, Y. Yan and M. Holmes, Fast large sample goodness-of-fit tests for copulas, Statistica Sinica 21 (2011), 841-871] and shares the same computational benefits compared to methods based on a parametric bootstrap. The finite-sample performance of our approach is investigated by Monte Carlo experiments for the case of copula-based Markovian time series models.

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“…Theorem 5.1 (Theorem 3.1. in Beare (2010)) Suppose {U t : t ∈ Z} is a stationary Markov chain whose invariant distribution is uniform on (0, 1). Let C denote the joint distribution function of (U 0 , U 1 ).…”

Section: Transformed Geometrically Ergodic Markov Chainmentioning

confidence: 99%

“…Remark that Clayton, Ali-Mikhail-Haq, Gumbel, Frank and Joe copulas among others, with suitable parameter ranges, satisfy assumptions of Theorem 5.1 (see Examples 3.1-3.11 in Beare (2010)). Conversely, generator (20) in Table 4.1 of Nelsen (1999) does not satisfy Assumptions of Theorem 5.1.…”

Section: Transformed Geometrically Ergodic Markov Chainmentioning

confidence: 99%

On tail dependence coefficients of transformed multivariate Archimedean copulas

Bernardino¹,

Rullière²

2016

Fuzzy Sets and Systems

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This paper presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part of this paper, we calculate multivariate tail dependence coefficients when the generator of the considered copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. This first part exploits previous works of Charpentier and Segers (2009) and extends some results of Juri and Wüthrich (2003) and De Luca and Rivieccio (2012). In the second part of the paper we analyse the impact in the upper and lower multivariate tail dependence coefficients of a large class of transformations of dependence structures. These results are based on the transformations exploited by Di Bernardino and Rullière (2013a) and Di Bernardino and Rullière (2013b). We extend some bivariate results of Durante et al. (2010) in a multivariate setting by calculating multivariate tail dependence coefficients for transformed copulas. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. In the third part, we show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients. Furthermore, using results in Larsson and Nešlehová (2011), we detail the extreme behaviour of the transformed radial part of Archimedean copulas. At last, we explain possible applications with Markov chains with specific dependence structure.

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Archimedean Copulas and Temporal Dependence

Cited by 33 publications

References 33 publications

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing

Goodness-of-Fit Tests for Multivariate Copula-Based Time Series Models

On tail dependence coefficients of transformed multivariate Archimedean copulas

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