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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.

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File URL: http://hdl.handle.net/10.1080/01621459.2012.682832
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Article provided by Taylor & Francis Journals in its journal Journal of the American Statistical Association.

Volume (Year): 107 (2012)
Issue (Month): 498 (June)
Pages: 607-621

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Handle: RePEc:taf:jnlasa:v:107:y:2012:i:498:p:607-621

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  1. Francis X. Diebold & Lutz Kilian, 1997. "Measuring Predictability: Theory and Macroeconomic Applications," NBER Technical Working Papers 0213, National Bureau of Economic Research, Inc.
  2. 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.
  3. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
  4. 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.
  5. Luati, Alessandra & Proietti, Tommaso, 2009. "Hyper-spherical and Elliptical Stochastic Cycles," MPRA Paper 15169, University Library of Munich, Germany.
  6. James H. Stock & Mark W. Watson, 2002. "Has the Business Cycle Changed and Why?," NBER Working Papers 9127, National Bureau of Economic Research, Inc.
  7. 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.
  8. 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.
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Cited by:
  1. Tommaso Proietti & Alessandra Luati, 2013. "The Generalised Autocovariance Function," CEIS Research Paper 276, Tor Vergata University, CEIS, revised 30 Apr 2013.
  2. Tommaso Proietti & Alessandra Luati, 2013. "The Exponential Model for the Spectrum of a Time Series: Extensions and Applications," CREATES Research Papers 2013-34, School of Economics and Management, University of Aarhus.

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