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The Exponential Model for the Spectrum of a Time Series: Extensions and Applications

Author

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  • Tommaso Proietti

    () (University of Rome “Tor Vergata” and Creates)

  • Alessandra Luati

    () (University of Bologna)

Abstract

The exponential model for the spectrum of a time series and its fractional extensions are based on the Fourier series expansion of the logarithm of the spectral density. The coefficients of the expansion form the cepstrum of the time series. After deriving the cepstrum of important classes of time series processes, also featuring long memory, we discuss likelihood inferences based on the periodogram, for which the estimation of the cepstrum yields a generalized linear model for exponential data with logarithmic link, focusing on the issue of separating the contribution of the long memory component to the log-spectrum. We then propose two extensions. The first deals with replacing the logarithmic link with a more general Box-Cox link, which encompasses also the identity and the inverse links: this enables nesting alternative spectral estimation methods (autoregressive, exponential, etc.) under the same likelihood-based framework. Secondly, we propose a gradient boosting algorithm for the estimation of the log-spectrum and illustrate its potential for distilling the long memory component of the log-spectrum.

Suggested Citation

  • Tommaso Proietti & Alessandra Luati, 2013. "The Exponential Model for the Spectrum of a Time Series: Extensions and Applications," CREATES Research Papers 2013-34, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2013-34
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    File URL: ftp://ftp.econ.au.dk/creates/rp/13/rp13_34.pdf
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    References listed on IDEAS

    as
    1. Donald W. K. Andrews & Patrik Guggenberger, 2003. "A Bias--Reduced Log--Periodogram Regression Estimator for the Long--Memory Parameter," Econometrica, Econometric Society, vol. 71(2), pages 675-712, March.
    2. Rosen, Ori & Stoffer, David S. & Wood, Sally, 2009. "Local Spectral Analysis via a Bayesian Mixture of Smoothing Splines," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 249-262.
    3. Alessandra Luati & Tommaso Proietti & Marco Reale, 2012. "The Variance Profile," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 607-621, June.
    4. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.
    5. Alessandra Luati & Tommaso Proietti, 2010. "Hyper-spherical and elliptical stochastic cycles," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(3), pages 169-181, May.
    6. Koenker, Roger & Yoon, Jungmo, 2009. "Parametric links for binary choice models: A Fisherian-Bayesian colloquy," Journal of Econometrics, Elsevier, vol. 152(2), pages 120-130, October.
    7. Masaki Narukawa & Yasumasa Matsuda, 2011. "Broadband semi‐parametric estimation of long‐memory time series by fractional exponential models," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(2), pages 175-193, March.
    8. Ori Rosen & Sally Wood & David S. Stoffer, 2012. "AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1575-1589, December.
    9. Barndorff-Nielsen, O. & Schou, G., 1973. "On the parametrization of autoregressive models by partial autocorrelations," Journal of Multivariate Analysis, Elsevier, vol. 3(4), pages 408-419, December.
    10. Hurvich, Clifford M., 2002. "Multistep forecasting of long memory series using fractional exponential models," International Journal of Forecasting, Elsevier, vol. 18(2), pages 167-179.
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    More about this item

    Keywords

    Frequency Domain Methods; Generalized linear models; Long Memory; Boosting.;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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