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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 a vector autoregressive law of motion. Every solution is shown to be the sum of three components, each corresponding to a directed flow of time. One component flows forward from the arbitrarily distant past; one flows backward from the arbitrarily distant future; and one flows outward from time zero. The three components are obtained by applying three complementary spectral projections to the solution, these corresponding to a separation of the eigenvalues of the autoregressive coefficient matrix according to whether they are inside, outside or on the unit circle. We establish a one-to-one correspondence between the set of all solutions and a finite-dimensional space of initial conditions.

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 Jan 2026.
    • Handle: RePEc:arx:papers:2402.01966
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    References listed on IDEAS

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    1. Lanne, Markku & Saikkonen, Pentti, 2013. "Noncausal Vector Autoregression," Econometric Theory, Cambridge University Press, vol. 29(3), pages 447-481, June.
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    3. Franchi, Massimo & Paruolo, Paolo, 2020. "Cointegration In Functional Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 36(5), pages 803-839, October.
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