IDEAS home Printed from https://ideas.repec.org/p/syd/wpaper/2024-01.html
   My bibliography  Save this paper

The general solution to an autoregressive law of motion

Author

Listed:
  • Brendan K Beare
  • Massimo Franchi
  • Phil Howlett

Abstract

In this article we provide a complete description of the set of all solutions to an autoregressive law of motion in a finite-dimensional complex vector space. Every solution is shown to be the sum of three parts, each corresponding to a directed flow of time. One part flows forward from the arbitrarily distant past; one flows backwards from the arbitrarily distant future; and one flows outward from time zero. The three parts are obtained by applying three complementary spectral projections to the solution, these corresponding to a separation of the eigenvalues of the autoregressive operator according to whether they are inside, outside or on the unit circle. We provide a finite-dimensional parametrization of the set of all solutions.

Suggested Citation

  • Brendan K Beare & Massimo Franchi & Phil Howlett, 2024. "The general solution to an autoregressive law of motion," Working Papers 2024-01, University of Sydney, School of Economics.
    • Handle: RePEc:syd:wpaper:2024-01
    as

    Download full text from publisher

    File URL: http://econ-wpseries.com/2024/202401.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Engle, Robert & Granger, Clive, 2015. "Co-integration and error correction: Representation, estimation, and testing," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 39(3), pages 106-135.
    2. Franchi, Massimo & Paruolo, Paolo, 2020. "Cointegration In Functional Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 36(5), pages 803-839, October.
    3. Massimo Franchi & Paolo Paruolo, 2021. "Cointegration, Root Functions and Minimal Bases," Econometrics, MDPI, vol. 9(3), pages 1-27, August.
    4. Lanne, Markku & Saikkonen, Pentti, 2013. "Noncausal Vector Autoregression," Econometric Theory, Cambridge University Press, vol. 29(3), pages 447-481, June.
    5. Gregoir, Stephane & Laroque, Guy, 1994. "Polynomial cointegration estimation and test," Journal of Econometrics, Elsevier, vol. 63(1), pages 183-214, July.
    Full references (including those not matched with items on IDEAS)

    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. Snell, Andy, 1998. "Testing for r versus r-1 cointegrating vectors," Journal of Econometrics, Elsevier, vol. 88(1), pages 151-191, November.
    2. Massimo Franchi & Paolo Paruolo, 2021. "Cointegration, Root Functions and Minimal Bases," Econometrics, MDPI, vol. 9(3), pages 1-27, August.
    3. Dietmar Bauer & Martin Wagner, 2002. "A Canonical Form for Unit Root Processes in the State Space Framework," Diskussionsschriften dp0204, Universitaet Bern, Departement Volkswirtschaft.
    4. Gianluca Cubadda & Alain Hecq & Elisa Voisin, 2023. "Detecting Common Bubbles in Multivariate Mixed Causal–Noncausal Models," Econometrics, MDPI, vol. 11(1), pages 1-16, March.
    5. Gourieroux, Christian & Jasiak, Joann, 2017. "Noncausal vector autoregressive process: Representation, identification and semi-parametric estimation," Journal of Econometrics, Elsevier, vol. 200(1), pages 118-134.
    6. Beare, Brendan K. & Seo, Won-Ki, 2020. "Representation Of I(1) And I(2) Autoregressive Hilbertian Processes," Econometric Theory, Cambridge University Press, vol. 36(5), pages 773-802, October.
    7. Lutkepohl, Helmut & Saikkonen, Pentti, 2000. "Testing for the cointegrating rank of a VAR process with a time trend," Journal of Econometrics, Elsevier, vol. 95(1), pages 177-198, March.
    8. Martin Wagner, 2010. "Cointegration analysis with state space models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 94(3), pages 273-305, September.
    9. Anindya Banerjee & Lynne Cockerell & Bill Russell, 2001. "An I(2) analysis of inflation and the markup," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(3), pages 221-240.
    10. Massimo Franchi & Iliyan Georgiev & Paolo Paruolo, 2024. "Canonical correlation analysis of stochastic trends via functional approximation," Papers 2411.19572, arXiv.org.
    11. Haldrup, Niels & Salmon, Mark, 1998. "Representations of I(2) cointegrated systems using the Smith-McMillan form," Journal of Econometrics, Elsevier, vol. 84(2), pages 303-325, June.
    12. Bilal Mehmood & Syed Hassan Raza & Mahwish Rana & Huma Sohaib & Muhammad Azhar Khan, 2014. "Triangular Relationship between Energy Consumption, Price Index and National Income in Asian Countries: A Pooled Mean Group Approach in Presence of Structural Breaks," International Journal of Energy Economics and Policy, Econjournals, vol. 4(4), pages 610-620.
    13. Law, Siong Hook & Tan, Hui & baharumshah, ahmad, 1999. "Financial Liberalization in ASEAN and the Fisher Hypothesis," Jurnal Ekonomi Malaysia, Faculty of Economics and Business, Universiti Kebangsaan Malaysia, vol. 33, pages 65-86.
    14. Shahbaz, Muhammad & Hoang, Thi Hong Van & Mahalik, Mantu Kumar & Roubaud, David, 2017. "Energy consumption, financial development and economic growth in India: New evidence from a nonlinear and asymmetric analysis," Energy Economics, Elsevier, vol. 63(C), pages 199-212.
    15. Levent, Korap, 2007. "Modeling purchasing power parity using co-integration: evidence from Turkey," MPRA Paper 19584, University Library of Munich, Germany.
    16. Ranjan Aneja & Umer J. Banday & Tanzeem Hasnat & Mustafa Koçoglu, 2017. "Renewable and Non-renewable Energy Consumption and Economic Growth: Empirical Evidence from Panel Error Correction Model," Jindal Journal of Business Research, , vol. 6(1), pages 76-85, June.
    17. Yih-Ing Hser & Haikang Shen & Chih-Ping Chou & Stephen C. Messer & M. Douglas Anglin, 2001. "Analytic Approaches for Assessing Long-Term Treatment Effects," Evaluation Review, , vol. 25(2), pages 233-262, April.
    18. Zamani, Mehrzad, 2007. "Energy consumption and economic activities in Iran," Energy Economics, Elsevier, vol. 29(6), pages 1135-1140, November.
    19. Muhammad Zia Ullah Khan & Muhammad Illyas & Muqqadas Rahman & Chaudhary Abdul Rahman, 2015. "Money Monetization and Economic Growth in Pakistan," International Journal of Economics and Empirical Research (IJEER), The Economics and Social Development Organization (TESDO), vol. 3(4), pages 184-192, April.
    20. Muhammad Shafiullah & Ravinthirakumaran Navaratnam, 2016. "Do Bangladesh and Sri Lanka Enjoy Export-Led Growth? A Comparison of Two Small South Asian Economies," South Asia Economic Journal, Institute of Policy Studies of Sri Lanka, vol. 17(1), pages 114-132, March.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:syd:wpaper:2024-01. 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: Vanessa Holcombe (email available below). General contact details of provider: https://edirc.repec.org/data/deusyau.html .

    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.