Copula-Based Dependence Characterizations and Modeling for Time Series

This paper develops a new unified approach to copula-based modeling and characterizations for time series and stochastic processes. We obtain complete characterizations of many time series dependence structures in terms of copulas corresponding to their finite-dimensional distributions. In particular, we focus on copula- based representations for Markov chains of arbitrary order, m-dependent and r-independent time series as well as martingales and conditionally symmetric processes. Our results provide new methods for modeling time series that have prescribed dependence structures such as, for instance, higher order Markov processes as well as non-Markovian processes that nevertheless satisfy Chapman-Kolmogorov stochastic equations. We also focus on the construction and analysis of new classes of copulas that have flexibility to combine many different dependence properties for time series. Among other results, we present a study of new classes of cop- ulas based on expansions by linear functions (Eyraud-Farlie-Gumbel-Mongenstern copulas), power functions (power copulas) and Fourier polynomials (Fourier copulas) and introduce methods for modeling time series using these classes of dependence functions. We also focus on the study of weak convergence of empirical copula processes in the time series context and obtain new results on asymptotic gaussianity of such processes for a wide class of beta mixing sequences.

Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C ψ: [0, 1]2 → [0, 1], C ψ(x, y) = ψ{C[ψ−1(x), ψ−1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails

Simplified Pair Copula Constructions --- Limits and Extensions

So called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on is popular. In this paper, we show for which classes of distributions such a simplification is applicable, significantly extending the discussion of Hob{\ae}k Haff et al. (2010). In particular, we show that the only Archimedean copula in dimension d \geq 4 which is of the simplified type is that based on the gamma Laplace transform or its extension, while the Student-t copula is the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on

An overview of the goodness-of-fit test problem for copulas

We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc, are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized

Multiplier bootstrap of tail copulas with applications

For the problem of estimating lower tail and upper tail copulas, we propose two bootstrap procedures for approximating the distribution of the corresponding empirical tail copulas. The first method uses a multiplier bootstrap of the empirical tail copula process and requires estimation of the partial derivatives of the tail copula. The second method avoids this estimation problem and uses multipliers in the two-dimensional empirical distribution function and in the estimates of the marginal distributions. For both multiplier bootstrap procedures, we prove consistency. For these investigations, we demonstrate that the common assumption of the existence of continuous partial derivatives in the the literature on tail copula estimation is so restrictive, such that the tail copula corresponding to tail independence is the only tail copula with this property. Moreover, we are able to solve this problem and prove weak convergence of the empirical tail copula process under nonrestrictive smoothness assumptions that are satisfied for many commonly used models. These results are applied in several statistical problems, including minimum distance estimation and goodness-of-fit testing.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ425 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The joint distribution of stock returns is not elliptical

Using a large set of daily US and Japanese stock returns, we test in detail the relevance of Student models, and of more general elliptical models, for describing the joint distribution of returns. We find that while Student copulas provide a good approximation for strongly correlated pairs of stocks, systematic discrepancies appear as the linear correlation between stocks decreases, that rule out all elliptical models. Intuitively, the failure of elliptical models can be traced to the inadequacy of the assumption of a single volatility mode for all stocks. We suggest several ideas of methodological interest to efficiently visualise and compare different copulas. We identify the rescaled difference with the Gaussian copula and the central value of the copula as strongly discriminating observables. We insist on the need to shun away from formal choices of copulas with no financial interpretation.Comment: 12 figure