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Hyper-spherical and elliptical stochastic cycles

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  • Alessandra Luati
  • Tommaso Proietti

Abstract

A univariate first-order stochastic cycle can be represented as an element of a bivariate first-order vector autoregressive process, or VAR(1), where the transition matrix is associated with a rotation along a circle in the plane, and the reduced form is ARMA(2,1). This paper generalizes this representation in two directions. According to the first, the cyclical dynamics originate from the motion of a point along an ellipse. The reduced form is also ARMA(2,1), but the model can account for certain types of asymmetries. The second deals with the multivariate case: the cyclical dynamics result from the projection along one of the coordinate axis of a point moving in along an hyper-sphere. This is described by a VAR(1) process whose transition matrix is obtained by a sequence of n-dimensional Givens rotations. The reduced form of an element of the system is shown to be ARMA(n, n - 1). The properties of the resulting models are analysed in the frequency domain, and we show that this generalization can account for a multimodal spectral density. The illustrations show that the proposed generalizations can be fitted successfully to some well-known case studies of the time series literature. Copyright Copyright 2010 Blackwell Publishing Ltd

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Bibliographic Info

Article provided by Wiley Blackwell in its journal Journal of Time Series Analysis.

Volume (Year): 31 (2010)
Issue (Month): 3 (05)
Pages: 169-181

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Handle: RePEc:bla:jtsera:v:31:y:2010:i:3:p:169-181

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  1. James Morley & Charles Nelson & Eric Zivot, 2003. "Why are Beveridge-Nelson and Unobserved-component decompositions of GDP so Different?," Working Papers UWEC-2002-18-P, University of Washington, Department of Economics.
  2. Harvey, A.C. & Trimbur, T.M., 2001. "General Model-based Filters for Extracting Cycles and Trends in Economic Time Series," Cambridge Working Papers in Economics 0113, Faculty of Economics, University of Cambridge.
  3. ZELLNER, Arnold & PALM, Franz, . "Time series analysis and simultaneous equation econometric models," CORE Discussion Papers RP -173, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  4. James C. Morley & Charles R. Nelson & Eric Zivot, 2003. "Why Are the Beveridge-Nelson and Unobserved-Components Decompositions of GDP So Different?," The Review of Economics and Statistics, MIT Press, vol. 85(2), pages 235-243, May.
  5. Harvey, A C, 1985. "Trends and Cycles in Macroeconomic Time Series," Journal of Business & Economic Statistics, American Statistical Association, vol. 3(3), pages 216-27, June.
  6. Durbin, James & Koopman, Siem Jan, 2001. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, number 9780198523543.
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Cited by:
  1. Tommaso Proietti & Alessandra Luati, 2013. "The Exponential Model for the Spectrum of a Time Series: Extensions and Applications," CEIS Research Paper 272, Tor Vergata University, CEIS, revised 19 Apr 2013.
  2. Giacomo Sbrana & Andrea Silvestrini, 2012. "Temporal aggregation of cyclical models with business cycle applications," Statistical Methods and Applications, Springer, vol. 21(1), pages 93-107, March.
  3. Tommaso, Proietti & Alessandra, Luati, 2012. "Maximum likelihood estimation of time series models: the Kalman filter and beyond," MPRA Paper 39600, University Library of Munich, Germany.
  4. Luati, Alessandra & Proietti, Tommaso & Reale, Marco, 2011. "The Variance Profile," MPRA Paper 30378, University Library of Munich, Germany.
  5. Ledenyov, Dimitri O. & Ledenyov, Viktor O., 2013. "On the Stratonovich – Kalman - Bucy filtering algorithm application for accurate characterization of financial time series with use of state-space model by central banks," MPRA Paper 50235, University Library of Munich, Germany.
  6. Tommaso Proietti & Alessandra Luati, 2013. "Generalised Linear Spectral Models," CEIS Research Paper 290, Tor Vergata University, CEIS, revised 03 Oct 2013.

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