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The Variance Profile


  • Alessandra Luati
  • Tommaso Proietti
  • Marco Reale


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

    1. 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.
    2. James H. Stock & Mark W. Watson, 2003. "Has the Business Cycle Changed and Why?," NBER Chapters,in: NBER Macroeconomics Annual 2002, Volume 17, pages 159-230 National Bureau of Economic Research, Inc.
    3. 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.
    4. 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.
    5. Kasahara, Yukio & Pourahmadi, Mohsen & Inoue, Akihiko, 2009. "Duals of random vectors and processes with applications to prediction problems with missing values," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1637-1646, July.
    6. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    7. 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.
    8. 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. 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.
    2. 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.
    3. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.

    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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