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The Variance Profile

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  • Alessandra Luati
  • Tommaso Proietti
  • Marco Reale

Abstract

The variance profile is defined as the power mean of the spectral density function of a stationary stochastic process. It is a continuous and nondecreasing function of the power parameter, p , which returns the minimum of the spectrum ( p →−∞), the interpolation error variance (harmonic mean, p =−1), the prediction error variance (geometric mean, p =0), the unconditional variance (arithmetic mean, p =1), and the maximum of the spectrum ( p →∞). The variance profile provides a useful characterization of a stochastic process; we focus in particular on the class of fractionally integrated processes. Moreover, it enables a direct and immediate derivation of the Szegö-Kolmogorov formula and the interpolation error variance formula. The article proposes a nonparametric estimator of the variance profile based on the power mean of the smoothed sample spectrum, and proves its consistency and its asymptotic normality. From the empirical standpoint, we propose and illustrate the use of the variance profile for estimating the long memory parameter in climatological and financial time series and for assessing structural change.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:498:p:607-621
    DOI: 10.1080/01621459.2012.682832
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    References listed on IDEAS

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    8. Nidhan Choudhuri & Subhashis Ghosal & Anindya Roy, 2004. "Bayesian Estimation of the Spectral Density of a Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 1050-1059, December.
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    Cited by:

    1. Proietti, Tommaso & Luati, Alessandra, 2013. "The Exponential Model for the Spectrum of a Time Series: Extensions and Applications," MPRA Paper 45280, University Library of Munich, Germany.
    2. 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.
    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, 2015. "Generalised partial autocorrelations and the mutual information between past and future," CEIS Research Paper 344, Tor Vergata University, CEIS, revised 05 Jun 2015.

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    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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