[Paper Review] Empirical and sequential empirical copula processes under serial dependence

Axel Bücher, Stanislav Volgushev|arXiv (Cornell University)|Nov 11, 2011

Financial Risk and Volatility Modeling19 references7 citations

TL;DR

This paper extends the asymptotic theory of empirical and sequential empirical copula processes to strictly stationary, weakly dependent multivariate time series under minimal smoothness assumptions. Using the functional delta method, it establishes weak convergence of the empirical copula process and its sequential variants without requiring continuous partial derivatives of the copula, enabling robust inference and bootstrap consistency under serial dependence.

ABSTRACT

The empirical copula process plays a central role for statistical inference on copulas. Recently, Segers (2011) investigated the asymptotic behavior of this process under non-restrictive smoothness assumptions for the case of i.i.d. random variables. In the present paper we extend his main result to the case of serial dependent random variables by means of the powerful and elegant functional delta method. Moreover, we utilize the functional delta method in order to obtain conditional consistency of certain bootstrap procedures. Finally, we extend the results to the more general sequential empirical copula process under serial dependence.

Motivation & Objective

• To extend Segers' (2011) weak convergence results for the empirical copula process to strictly stationary, weakly dependent sequences.
• To establish conditional consistency of bootstrap procedures for copula inference under serial dependence using the functional delta method.
• To generalize R"uschendorf's (1976) sequential empirical process results to the sequential empirical copula process under weak dependence.
• To provide minimal smoothness conditions—only continuity of the copula—under which weak convergence holds, improving applicability to real-world copula models.
• To unify and strengthen existing asymptotic theory for copula-based inference in time series by leveraging functional delta method techniques.

Proposed method

• Applies the functional delta method to link weak convergence of the sequential empirical process to that of the empirical copula process.
• Uses the functional delta method to derive weak convergence of the sequential empirical copula processes C#_n and C+_n under minimal smoothness.
• Imposes Condition 2.1 (weak convergence of the empirical process on U_i) and Condition 3.1 (weak convergence of the sequential empirical process) as minimal assumptions.
• Employs Hadamard differentiability of the mappings Ψ and Γ to transfer weak convergence from the empirical process to the copula processes.
• Derives limiting distributions as Gaussian fields: G+_C(s,u) = B#_C(s,u) - s B#_C(1,u) and G#_C(s,u) = B#_C(s,u) - s Σ ∂_p C(u) B#_C(1,u(p)).
• Utilizes the generalized inverse of empirical distribution functions and probability integral transforms to define the empirical copula.

Experimental results

Research questions

• RQ1 Under what minimal smoothness conditions does the empirical copula process converge weakly under serial dependence?
• RQ2 Can the functional delta method be used to derive weak convergence of the sequential empirical copula processes without assuming differentiability of the copula?
• RQ3 How can bootstrap procedures for copula inference be shown to be conditionally consistent under weak dependence?
• RQ4 What are the limiting distributions of the sequential empirical copula processes C#_n and C+_n under weak dependence and minimal smoothness?
• RQ5 Can the results be generalized to arbitrary copulas without requiring continuous partial derivatives?

Key findings

• The empirical copula process √n(C_n - C) converges weakly to a Gaussian process G+_C(s,u) = B#_C(s,u) - s B#_C(1,u) under Condition 3.1, without requiring smoothness of the copula C.
• The sequential empirical copula process C+_n(s,u) converges weakly to G+_C(s,u) = B#_C(s,u) - s B#_C(1,u), with the limiting process depending on the sequential empirical process B#_C.
• For the process C#_n(s,u), weak convergence holds to G#_C(s,u) = B#_C(s,u) - s Σ_p ∂_p C(u) B#_C(1,u_p), provided C satisfies Condition 2.3 (partial differentiability).
• The functional delta method allows for bootstrap consistency without smoothness assumptions, improving on prior methods requiring continuous partial derivatives.
• The results generalize prior work by Segers (2011) and Doukhan et al. (2009) by removing the need for smoothness and extending to weakly dependent data.
• The limiting distributions are Gaussian fields on ℓ∞([0,1]^{d+1}), with explicit covariance structures derived from the dependence structure of the underlying sequence.