TY - JOUR
T1 - Of copulas, quantiles, ranks and spectra: an L1-approach to spectral analysis
AU - Dette, Holger
AU - Hallin, Marc
AU - Kley, Tobias
AU - Volgushev, Stanislav
PY - 2015
Y1 - 2015
N2 - In this paper, we present an alternative method for the spectral analysis of a univariate, strictly stationary time series {Yt}t∈Z. 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 a copula spectral density kernel and allows to separate the marginal and serial aspects of a time series. We show that this spectrum is closely related to the concept of quantile regression. Like quantile regression, which provides much more information about conditional distributions than classical location-scale regression models, copula spectral density kernels are more informative than traditional spectral densities obtained from classical autocovariances. In particular, copula spectral density kernels, in their population versions, provide (asymptotically provide, in their sample versions) a complete description of the copulas of all pairs (Yt,Yt−k). Moreover, they inherit the robustness properties of classical quantile regression, and do 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.
AB - In this paper, we present an alternative method for the spectral analysis of a univariate, strictly stationary time series {Yt}t∈Z. 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 a copula spectral density kernel and allows to separate the marginal and serial aspects of a time series. We show that this spectrum is closely related to the concept of quantile regression. Like quantile regression, which provides much more information about conditional distributions than classical location-scale regression models, copula spectral density kernels are more informative than traditional spectral densities obtained from classical autocovariances. In particular, copula spectral density kernels, in their population versions, provide (asymptotically provide, in their sample versions) a complete description of the copulas of all pairs (Yt,Yt−k). Moreover, they inherit the robustness properties of classical quantile regression, and do 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.
KW - copulas
KW - periodogram
KW - quantile regression
KW - ranks
KW - spectral analysis
KW - time reversibility
KW - time series
UR - https://arxiv.org/abs/1111.7205
U2 - 10.3150/13-BEJ587
DO - 10.3150/13-BEJ587
M3 - Article (Academic Journal)
VL - 21
SP - 781
EP - 831
JO - Bernoulli
JF - Bernoulli
SN - 1350-7265
IS - 2
ER -