The general solution to an autoregressive law of motion

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.