IDEAS home Printed from https://ideas.repec.org/a/bla/jtsera/v31y2010i3p169-181.html
   My bibliography  Save this article

Hyper‐spherical and elliptical stochastic cycles

Author

Listed:
  • 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.

Suggested Citation

  • 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.
  • Handle: RePEc:bla:jtsera:v:31:y:2010:i:3:p:169-181
    DOI: 10.1111/j.1467-9892.2010.00655.x
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/j.1467-9892.2010.00655.x
    Download Restriction: no

    File URL: https://libkey.io/10.1111/j.1467-9892.2010.00655.x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. 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.
    2. Wenyang Zhang & Qiwei Yao & Howell Tong & Nils Chr. Stenseth, 2003. "Smoothing for Spatiotemporal Models and Its Application to Modeling Muskrat-Mink Interaction," Biometrics, The International Biometric Society, vol. 59(4), pages 813-821, December.
    3. Harvey, A C, 1985. "Trends and Cycles in Macroeconomic Time Series," Journal of Business & Economic Statistics, American Statistical Association, vol. 3(3), pages 216-227, June.
    4. W.‐Y. T. Chan & Kenneth F. Wallis, 1978. "Multiple Time Series Modelling: Another Look at the Mink‐Muskrat Interaction," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(2), pages 168-175, June.
    5. Zellner, Arnold & Palm, Franz, 1974. "Time series analysis and simultaneous equation econometric models," Journal of Econometrics, Elsevier, vol. 2(1), pages 17-54, May.
    6. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178, December.
    7. Henry L. Gray & Nien‐Fan Zhang & Wayne A. Woodward, 1989. "On Generalized Fractional Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(3), pages 233-257, May.
    8. Andrew C. Harvey & Thomas M. Trimbur, 2003. "General Model-Based Filters for Extracting Cycles and Trends in Economic Time Series," The Review of Economics and Statistics, MIT Press, vol. 85(2), pages 244-255, May.
    9. Harvey, A C & Jaeger, A, 1993. "Detrending, Stylized Facts and the Business Cycle," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(3), pages 231-247, July-Sept.
    10. Thomas M. Trimbur, 2006. "Properties of higher order stochastic cycles," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 1-17, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Tommaso Proietti & Alessandra Luati, 2013. "Maximum likelihood estimation of time series models: the Kalman filter and beyond," Chapters, in: Nigar Hashimzade & Michael A. Thornton (ed.), Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 15, pages 334-362, Edward Elgar Publishing.
    2. Giacomo Sbrana & Andrea Silvestrini, 2012. "Temporal aggregation of cyclical models with business cycle applications," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(1), pages 93-107, March.
    3. 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.
    4. 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.
    5. Ledenyov, Dimitri O. & Ledenyov, Viktor O., 2015. "Wave function method to forecast foreign currencies exchange rates at ultra high frequency electronic trading in foreign currencies exchange markets," MPRA Paper 67470, University Library of Munich, Germany.
    6. 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.
    7. Tommaso Proietti & Alessandra Luati, 2013. "Generalised Linear Spectral Models," CEIS Research Paper 290, Tor Vergata University, CEIS, revised 03 Oct 2013.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Oh, Kum Hwa & Zivot, Eric & Creal, Drew, 2008. "The relationship between the Beveridge-Nelson decomposition and other permanent-transitory decompositions that are popular in economics," Journal of Econometrics, Elsevier, vol. 146(2), pages 207-219, October.
    2. Kum Hwa Oh & Eric Zivot & Drew Creal, 2006. "The Relationship between the Beveridge-Nelson Decomposition andUnobserved Component Models with Correlated Shocks," Working Papers UWEC-2006-16-FC, University of Washington, Department of Economics.
    3. repec:ebl:ecbull:v:3:y:2008:i:60:p:1-9 is not listed on IDEAS
    4. Omar H. M. N. Bashar, 2015. "The Trickle‐down Effect of the Mining Boom in Australia: Fact or Myth?," The Economic Record, The Economic Society of Australia, vol. 91(S1), pages 94-108, June.
    5. L.A. Gil-Alana, 2005. "Fractional Cyclical Structures & Business Cycles in the Specification of the US Real Output," European Research Studies Journal, European Research Studies Journal, vol. 0(1-2), pages 99-126.
    6. Marian Vavra, 2016. "Testing the Validity of Assumptions of UC-ARIMA Models for Trend-Cycle Decompositions," Working and Discussion Papers WP 4/2016, Research Department, National Bank of Slovakia.
    7. Luis Uzeda, 2022. "State Correlation and Forecasting: A Bayesian Approach Using Unobserved Components Models," Advances in Econometrics, in: Essays in Honour of Fabio Canova, volume 44, pages 25-53, Emerald Group Publishing Limited.
    8. Xiaoshan Chen & Terence Mills, 2012. "Measuring the Euro area output gap using a multivariate unobserved components model containing phase shifts," Empirical Economics, Springer, vol. 43(2), pages 671-692, October.
    9. Tobias Hartl & Rolf Tschernig & Enzo Weber, 2020. "Fractional trends and cycles in macroeconomic time series," Papers 2005.05266, arXiv.org, revised May 2020.
    10. Guisinger, Amy Y. & Owyang, Michael T. & Soques, Daniel, 2024. "Industrial Connectedness and Business Cycle Comovements," Econometrics and Statistics, Elsevier, vol. 29(C), pages 132-149.
    11. Cayen, Jean-Philippe & van Norden, Simon, 2005. "The reliability of Canadian output-gap estimates," The North American Journal of Economics and Finance, Elsevier, vol. 16(3), pages 373-393, December.
    12. Álvarez, Luis J. & Gómez-Loscos, Ana, 2018. "A menu on output gap estimation methods," Journal of Policy Modeling, Elsevier, vol. 40(4), pages 827-850.
    13. Olivier Darné & Amélie Charles, 2008. "The impact of outliers on transitory and permanent components in macroeconomic time series," Economics Bulletin, AccessEcon, vol. 3(60), pages 1-9.
    14. Siem Jan Koopman & Rutger Lit & Andre Lucas, 2016. "Model-based Business Cycle and Financial Cycle Decomposition for Europe and the U.S," Tinbergen Institute Discussion Papers 16-051/IV, Tinbergen Institute.
    15. Joanne S. Ercolani, 2007. "Cyclical Trends in Continuous Time Models," Discussion Papers 07-13, Department of Economics, University of Birmingham.
    16. Drew Creal & Siem Jan Koopman & Eric Zivot, 2008. "The Effect of the Great Moderation on the U.S. Business Cycle in a Time-varying Multivariate Trend-cycle Model," Tinbergen Institute Discussion Papers 08-069/4, Tinbergen Institute.
    17. Tommaso Proietti, 2002. "Some Reflections on Trend-Cycle Decompositions with Correlated Components," Econometrics 0209002, University Library of Munich, Germany.
    18. Nicholas Sly & Caroline Weber, 2015. "Global tax policy and the synchronization of business cycles," Research Working Paper RWP 15-7, Federal Reserve Bank of Kansas City.
    19. Günes Kamber & James Morley & Benjamin Wong, 2018. "Intuitive and Reliable Estimates of the Output Gap from a Beveridge-Nelson Filter," The Review of Economics and Statistics, MIT Press, vol. 100(3), pages 550-566, July.
    20. Omar H. M. N. Bashar & Omar K. M. R. Bashar, 2020. "Resource abundance, financial crisis and economic growth: did resource‐rich countries fare better during the global financial crisis?," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 64(2), pages 376-395, April.
    21. Sbrana, Giacomo, 2013. "The exact linkage between the Beveridge–Nelson decomposition and other permanent-transitory decompositions," Economic Modelling, Elsevier, vol. 30(C), pages 311-316.

    More about this item

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:jtsera:v:31:y:2010:i:3:p:169-181. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=0143-9782 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.