{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T08:22:03Z","timestamp":1783498923033,"version":"3.55.0"},"reference-count":60,"publisher":"The Econometric Society","issue":"3","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/legalcode"}],"funder":[{"DOI":"10.13039\/501100021856","name":"Ministero dell'Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["QE"],"abstract":"<jats:p>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\u2010to\u2010one correspondence between the set of all solutions and a finite\u2010dimensional space of initial conditions.<\/jats:p>","DOI":"10.3982\/qe2719","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T11:13:34Z","timestamp":1783077214000},"page":"791-826","source":"Crossref","is-referenced-by-count":1,"title":["The general solution to an autoregressive law of motion"],"prefix":"10.3982","volume":"17","author":[{"given":"Brendan K.","family":"Beare","sequence":"first","affiliation":[{"name":"School of Economics, University of Sydney"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Massimo","family":"Franchi","sequence":"additional","affiliation":[{"name":"Department of Statistical Science, Sapienza University of Rome"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Phil","family":"Howlett","sequence":"additional","affiliation":[{"name":"Centre for Industrial and Applied Mathematics, University 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