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The general solution to an autoregressive law of motion

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  • Brendan K. Beare
  • Massimo Franchi
  • Phil Howlett

Abstract

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 backward 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," Papers 2402.01966, arXiv.org, revised Sep 2024.
    • Handle: RePEc:arx:papers:2402.01966
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    References listed on IDEAS

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    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. Lanne, Markku & Saikkonen, Pentti, 2013. "Noncausal Vector Autoregression," Econometric Theory, Cambridge University Press, vol. 29(3), pages 447-481, June.
    4. Massimo Franchi & Paolo Paruolo, 2021. "Cointegration, Root Functions and Minimal Bases," Econometrics, MDPI, vol. 9(3), pages 1-27, August.
    5. Gregoir, Stephane & Laroque, Guy, 1994. "Polynomial cointegration estimation and test," Journal of Econometrics, Elsevier, vol. 63(1), pages 183-214, July.
    6. Engle, R. F. & Granger, C. W. J. & Hallman, J. J., 1989. "Merging short-and long-run forecasts : An application of seasonal cointegration to monthly electricity sales forecasting," Journal of Econometrics, Elsevier, vol. 40(1), pages 45-62, January.
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