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The Variance Profile

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  • Luati, Alessandra
  • Proietti, Tommaso
  • Reale, Marco

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 non-decreasing 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 characterisation of a stochastic processes; we focus in particular on the class of fractionally integrated processes. Moreover, it enables a direct and immediate derivation of the Szego-Kolmogorov formula and the interpolation error variance formula. The paper proposes a non-parametric 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

  • Luati, Alessandra & Proietti, Tommaso & Reale, Marco, 2011. "The Variance Profile," MPRA Paper 30378, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:30378
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    References listed on IDEAS

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    1. Francis X. Diebold & Lutz Kilian, 2001. "Measuring predictability: theory and macroeconomic applications," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(6), pages 657-669.
    2. 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.
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    5. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    6. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    7. R. J. Bhansali, 1993. "Estimation Of The Prediction Error Variance And An R2 Measure By Autoregressive Model Fitting," Journal of Time Series Analysis, Wiley Blackwell, vol. 14(2), pages 125-146, March.
    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.
    9. Hannan, E J & Terrell, R D & Tuckwell, N E, 1970. "The Seasonal Adjustment of Economic Time Series," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 11(1), pages 24-52, February.
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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

    Keywords

    Predictability; Interpolation; Non-parametric spectral estimation; Long memory.;
    All these keywords.

    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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