The general solution to an autoregressive law of motion

• 成果类型: Article
• 署名作者: Beare, Brendan K.; Franchi, Massimo; Howlett, Phil
• 署名单位: University of Sydney; Sapienza University Rome; Adelaide University; University of South Australia
• 刊物名称: QUANTITATIVE ECONOMICS
• ISSN/ISSBN: 1759-7323
• DOI: 10.3982/QE2719
• 发表日期: 2026
• 关键词: multivariate time-series; HIDDEN UNIT ROOTS; error-correction; linear-systems; REPRESENTATION; cointegration; form; inversion; THEOREM

摘要

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.