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Session 22. Multivariate stochastic modelling in finance, insurance and risk management
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Of copulas, quantiles, ranks, and spectra: an \(L_1\)-approach to spectral analysis |
Holger Dette, Ruhr-Universität, Germany
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The talk is based on the joint work with Marc Hallin (ECARES, Universit Libre de Bruxelles, and ORFE, Princeton University and Tobias Kley, Stanislav Volgushev, Stefan Skowronek, Ruhr-Universität, Bochum)
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In this paper, we present an alternative method for the spectral analysis of
{a univariate, strictly stationary} time series
\(\{Y_t\}_{t\in \mathbb{Z} }\).
We define a ``new'' spectrum as the Fourier transform of the
differences between copulas
of the pairs \((Y_t,Y_{t-k})\) 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 \((Y_t, \, Y_{t-k})\). 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 \(L_1\)-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.
We also discuss the advantages of a localized version of the method.
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