Vine Copula Specifications for Stationary Multivariate Markov Chains

Abstract

Vine copulae provide a graphical framework in which multiple bivariate copulae may be combined in a consistent fashion to yield a more complex multivariate copula. In this article, we discuss the use of vine copulae to build flexible semiparametric models for stationary multivariate higher‐order Markov chains. We propose a new vine structure, the M‐vine, that is particularly well suited to this purpose. Stationarity may be imposed by requiring the equality of certain copulae in the M‐vine, while the Markov property may be imposed by requiring certain copulae to be independence copulae.

Copula-Based Factor Models for Multivariate Asset Returns

“…Thus, one departs from the COPAR model and generalizes it as well as the copula based multivariate time series models of Beare and Seo (2015) and Smith (2015). In general, one can completely revise our approach and alternatively develop dynamic versions of copula factor models as proposed by Krupskii and Joe (2013) and Oh and Patton (2017) for iid data.…”

“…Moreover, Brechmann and Czado (2015), Beare and Seo (2015), and Smith (2015) have applied vine copulas to model temporal dependence of multivariate time series as well as the cross-sectional dependence between univariate time series. In this section, we review the COPAR model of Brechmann and Czado (2015), which is used to describe the stochastic dynamics and the dependence structure of the estimated factors from Section 2.…”

“…Recently, Brechmann and Czado (2015), Beare and Seo (2015), and Smith (2015) simultaneously developed copula-based models for stationary multivariate time series. Although these models differ from each other, their generality consists of an underlying R-vine pair-copula construction (see Aas et al 2009) to describe the cross-sectional and temporal dependence jointly.…”

“…Although these models differ from each other, their generality consists of an underlying R-vine pair-copula construction (see Aas et al 2009) to describe the cross-sectional and temporal dependence jointly. To capture the cross-sectional dependence, Brechmann and Czado (2015) employ C-Vine, while Beare and Seo (2015) and Smith (2015) consider D-vine. Further, Brechmann and Czado (2015) and Beare and Seo (2015) assume the existence of a key variable, whose temporal dependence was explicitly modeled, and this assumption combined with C-or D-vine for the cross-sectional dependence results in a corresponding R-vine for a multivariate time series.…”

“…To capture the cross-sectional dependence, Brechmann and Czado (2015) employ C-Vine, while Beare and Seo (2015) and Smith (2015) consider D-vine. Further, Brechmann and Czado (2015) and Beare and Seo (2015) assume the existence of a key variable, whose temporal dependence was explicitly modeled, and this assumption combined with C-or D-vine for the cross-sectional dependence results in a corresponding R-vine for a multivariate time series. In contrast, Smith (2015) explicitly modeled the temporal dependence between multivariate observations and constructed a general D-vine for them consisting of D-vines for the cross-sectional dependence.…”