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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

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
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