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2021 | OriginalPaper | Chapter

Quantile and Copula Spectrum: A New Approach to Investigate Cyclical Dependence in Economic Time Series

Authors : Gilles Dufrénot, Takashi Matsuki, Kimiko Sugimoto

Published in: Recent Econometric Techniques for Macroeconomic and Financial Data

Publisher: Springer International Publishing

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Abstract

This chapter presents a survey of some recent methods used in economics and finance to account for cyclical dependence and account for their multifaced dynamics: nonlinearities, extreme events, asymmetries, non-stationarity, time-varying moments. To circumvent the caveats of the standard spectral analysis, new tools are now used based on copula spectrum, quantile spectrum and Laplace periodogram in both non-parametric and parametric contexts. The chapter presents a comprehensive overview of both theoretical and empirical issues as well as a computational approach to explain how the methods can be implemented using the R Package.

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When \(\left\{ {\varepsilon_{t} } \right\}\) is a white noise process with a finite variance \(\sigma^{2}\), \(L(\omega ) = \eta^{2}\) \(( = 1/\left( {4f^{2} (0)} \right)\) and \(G(\omega ) = \sigma^{2}\). Obviously, to obtain the spectrum as the mean of the asymptotic distribution, the ordinary periodogram needs the existence of a finite variance, while the Laplace periodogram needs only the condition of \(f(0) > 0\).

\(I_{n,R}^{{\tau_{1} ,\tau_{2} }} (\omega )\) is one-variate case of \(I_{n,R}^{{l_{1} l_{2} }} \left( {\omega ; \tau_{1} , \tau_{2} } \right)\) for \(\varvec{X}_{t} = X_{t,1}\) in Eq. (41), which is called the CR periodogram. Its smoothed version \(\widehat{G}_{n,R} \left( {\omega ;\tau_{1} ,\tau_{2} } \right)\) is also a special case of \(\widehat{G}_{n, R}^{{l_{1} l_{2} }} \left( {\omega ; \tau_{1} , \tau_{2} } \right)\) in Eq. (43).

The smoothed rank-based Laplace periodogram is defined as \(\hat{f}_{n,R} \left( {\omega ;\tau_{1} ,\tau_{2} } \right) : = \frac{2\pi }{n}\sum\nolimits_{s = 1}^{n - 1} W_{n} \left( {\omega - 2\pi s/n} \right)L_{{n,\tau_{1} ,\tau_{2} }}^{R} \left( {\frac{2\pi s}{n}} \right)\), where \(W_{n}\) denotes a sequence of weight functions.

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Title: Quantile and Copula Spectrum: A New Approach to Investigate Cyclical Dependence in Economic Time Series
Authors: Gilles Dufrénot
Takashi Matsuki
Kimiko Sugimoto
Publisher: Springer International Publishing
Book: Recent Econometric Techniques for Macroeconomic and Financial Data
Print ISBN: 978-3-030-54251-1

Electronic ISBN: 978-3-030-54252-8

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-54252-8_1