Of Copulas, Quantiles, Ranks and Spectra - An L1-Approach to Spectral Analysis

By Holger Dette, Marc Hallin, Tobias Kley, Stanislav Volgushev. 2011

75 citations recorded by the source corpus.

In this paper we present an alternative method for the spectral analysis of a strictly stationary time series {Yt}t2Z. We define a “new” spectrum as the Fourier transform of the differences between copulas of the pairs (Yt, Yt−k) and the independence copula. This object is called copula spectral density kernel and allows to separate marginal and serial aspects of a time series. We show that it is intrinsically related to the concept of quantile regression. Like in quantile regression, which provides more information about the conditional distribution than the classical location-scale model, the copula spectral density kernel is more informative than the spectral density obtained from the autocovariances. In particular the approach provides a complete description of the distributions of all pairs (Yt, Yt−k). Moreover, it inherits the robustness properties of classical quantile regression, because it does not require any distributional assumptions such as the existence of finite moments. In order to estimate the copula spectral density kernel we introduce rank-based Laplace periodograms which are calculated as bilinear forms of weighted L1-projections of the ranks of the observed time series onto a harmonic regression model. We establish the asymptotic distribution of those periodograms, and the consistency of adequately smoothed versions. The finite-sample properties of the new methodology, and its potential for applications are briefly investigated by simulations and a short empirical example.

Subfields: Analytical Chemistry, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c4d80296f0f807fd7c9d9cfb5e84699231beef1e

Citations in the corpus

• Quantile coherency: A general measure for dependence between cyclical economic variables (2015, unknown): “As noted by Hagemann (2013) other approaches aim at discovering “the presence of any type of dependence structure in time series data”, referring to work of Dette et al. (2015) and Lee and Rao (2012).”
• Quantile Spectral Processes: Asymptotic Analysis and Inference (2014, unknown): “[12], and generalize the τ th quantile spectral densities of Hagemann [19], with which they coincide for τ1 = τ2 = τ ; an integrated version actually was first considered by Hong [27].”