Recent Developments in Copula Models

Copula models have become very popular and well studied among the scientific community. Now, most academic researchers, engineers, modelers, etc, own at least some basic copula toolkit and are able to apply it in real situations. Based on the famous Sklar’s theorem (Sklar 1959), copulas allow to put in place the fruitful idea of splitting the specification of a multivariate model into two parts: the marginal distributions on one side, the dependence structure (copula) on the other part. This simple way of thinking has induced an impressive number of theoretical and applied papers during the last two decades and this trend is exponentially increasing. Indeed, all families of multivariate models and their associated statistical techniques (inference, testing, simulation, etc) potentially have to be revisited under a copula point of view for theoretical and practical reasons. This huge and necessary task has just started. On the theoretical side, a renewed focus on semiparametric techniques has been fuelled because the underlying marginal distributions are often replaced by their empirical counterparts in copula models. Moreover, twoor three-stage estimators are common under a copula point of view, inducing particular asymptotics and finite distance performances. The earliest applications of copulas have been proposed in survival analysis (biostatistics, reliability, actuarial science), but the scientific community rapidly understood the far wider scope of dependence modelling. All applied fields are now affected by copulas. In economics and finance, this path is now clearly engaged. There, a particular difficulty is due to time-dependencies that cannot be straightforwardly managed by (sequences of) copulas: see Darsow et al. (1992) for markovian features with copulas, Cherubini et al. (2011) for a recent survey of results, etc. Nonetheless, significant advances have been observed in terms of copula modelling for univariate or even multivariate time series in the last years. The objective of this special issue is to provide new contributions into the field of copula models in general, with applications in financial econometrics. Standard families of multivariate models for financial time series are of GARCH-type and/or stochastic volatility-type: see the surveys Asai et al. (2006) and Bauwens et al. (2006), for instance. Once the first two conditional moments of asset returns are controlled, we are faced with dependencies across the estimated (vectors of) residuals. It has been found in Chen and Fan (2006) that the maximum-likelihood estimator of the copula parameter that is associated to residuals does not depend on the first-level estimates, under some conditions of regularity. Here, Bruno Rémillard provides the limiting distribution of the sequential empirical process and the sequential empirical copula process that are associated to estimated residuals in a general multivariate GARCH framework. He proves that the limiting behavior of the latter process does not depend on the conditional mean and conditional variance estimated parameters, when the conditional correlations are constant in time. As a by-product, the limiting distribution of rank-based dependence measures computed with the residuals are the same as if the dependence measures were computed with the innovations. He applies such results to tests of structural change and specification tests of copulas. Beside multivariate extensions of standard “historical” models in econometrics, copulas have induced a remarkable and fruitful new approach called “pair-copula constructions”: combinations of bivariate copulas allow to build very flexible models through so-called “vines”, that are connected trees with some particular features (Bedford and Cooke 2002). Kjersti Aas explains the main intuition and techniques behind such vine approaches: specification, inference, testing, parsimony, the “simplifying