Mixed data kernel copulas

Abstract

A number of approaches towards the kernel estimation of copula have appeared in the literature. Most existing approaches use a manifestation of the copula that requires kernel density estimation of bounded variates lying on a d-dimensional unit hypercube. This gives rise to a number of issues as it requires special treatment of the boundary and possible modifications to bandwidth selection routines, among others. Furthermore, existing kernel-based approaches are restricted to continuous date types only, though there is a growing interest in copula estimation with discrete marginals (see e.g. Smith & Khaled (2012) for a Bayesian approach). We demonstrate that using a simple inversion method (cf Nelsen (2006), Fermanian & Scaillet (2003)) can sidestep boundary issues while admitting mixed data types directly thereby extending the reach of kernel copula estimators. Bandwidth selection proceeds by the recently proposed method of Li & Racine (2013). Furthermore, there is no curse-of-dimensionality for the kernel-based copula estimator (though there is for the copula density estimator, as is the case for existing kernel copula density methods).

Price linkages in the international skim milk powder market: empirical evidence from nonparametric and time-varying copulas

“…Price transmission between Oceania and the EU has been further enhanced by the gradual reduction in the EU intervention price for SMP since the early 2000s (Thiele et al 2013). The kernel-based nonparametric estimation of the copula functions has been implemented here using the 'inversion approach' (Fermanian and Scaillet 2003;Balakrishnan and Lai 2009;Racine 2015). In particular, solving Equation (1) obtains…”

“…We are not aware of any earlier empirical works on price interrelationships in the international markets of dairy commodities, in general. Second, for the investigation of price co-movement, it relies on the kernelbased nonparametric approach (Fermanian and Scaillet 2003;Racine 2015) which provides a smooth (differentiable) reconstruction of the copula function dispensing with the need to specify a mathematical form. To the best of our knowledge, there have been no earlier published empirical studies on spatial market integration with kernel-based copulas.…”

“…We have carried out the nonparametric kernel-based estimation of the copula functions using the np package in R (Version 0.60-2; R Development Core Team, 2014), multivariate least-squares cross-validation and Gaussian kernel functions(Li and Racine 2013;Racine 2015).© 2016 Australian Agricultural and Resource Economics Society Inc.Price linkages in skim milk market…”

Extreme Dependence under Uncertainty: an application to Stock, Currency and Oil Markets

“…According to Sklar's theorem where F represents the joint cdf of ( x l,t , x j,t ), and C the copula function (Zimmer ). For each state‐pair combination, we proceed to implementing Racine's () version of Fermanian and Scaillet's () nonparametric estimator of F , that is, where , and the integral corresponds to a standard kernel density estimator that utilizes an adaptive nearest‐neighbor bandwidth with a second‐order Gaussian kernel. In terms of empirical implementation, the interesting feature in this type of modeling is that the bandwidth changes with each sample realization in the dataset when estimating the cdfs at any point of the multivariate (copula) distribution and it demonstrates superior numerical performance when estimating extreme tails (Li and Racine ).…”

A survey of kernel-type estimators for copula and their applications

“…Another approach of estimating copula is proposed by [10] by inversion of marginal cumulative distribution functions and probability density functions.…”