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Efficient Nonparametric Estimation of Generalized Autocovariances

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This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalized autocovariance function of a stationary random process. The generalized autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. The general result on the asymptotic efficiency is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule-Walker system of equations in the generalized autocovariance estimator.

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  • Alessandra Luati & Francesca Papagni & Tommaso Proietti, 2021. "Efficient Nonparametric Estimation of Generalized Autocovariances," CEIS Research Paper 515, Tor Vergata University, CEIS, revised 14 Oct 2021.
    • Handle: RePEc:rtv:ceisrp:515
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    References listed on IDEAS

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    1. A. M. Walker, 1995. "On Results Of Porat Concerning Asymptotic Efficiency Of Sample Covariances Of Gaussian Arma Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(2), pages 237-248, March.
    2. Francesco Battaglia, 1983. "Inverse Autocovariances And A Measure Of Linear Determinism For A Stationary Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(2), pages 79-87, March.
    3. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.
    4. 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.
    5. Yoshihide Kakizawa, 1999. "Note on the Asymptotic Efficiency of Sample Covariances in Gaussian Vector Stationary Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 20(5), pages 551-558, September.
    6. Yoshihide Kakizawa & Masanobu Taniguchi, 1994. "Asymptotic Efficiency Of The Sample Covariances In A Gaussian Stationary Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 15(3), pages 303-311, May.
    7. Boshnakov, Georgi N., 2005. "On the asymptotic properties of multivariate sample autocovariances," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 42-52, January.
    8. Francesco Battaglia, 1988. "On The Estimation Of The Inverse Correlation Function," Journal of Time Series Analysis, Wiley Blackwell, vol. 9(1), pages 1-10, January.
    9. Boaz Porat, 1987. "Some Asymptotic Properties Of The Sample Covariances Of Gaussian Autoregressive Moving‐Average Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(2), pages 205-220, March.
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    Keywords

    Cramér-Rao lower bound; Frequency Domain; Minimum Contrast Estimation; Periodogram;
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