Are autoregressive models sensitive to initial conditions?



The answer is, it depends. If the poles of the autoregressive (AR) model all lie inside the unit circle on the complex z-plane, then it is not sensitive to the initial condition. Otherwise, it will be.

The question in the title arises from a discussion between Gray and Woodward [1992] and Tsonis [1991, 1992] concerning the sensitivity of climate models to initial conditions. It is an interesting, well-posed mathematical question which, unfortunately, was not treated mathematically in their comments. The answer given above is closely related to whether the AR process is stationary or minimum-phase; here I would like to offer a simple, analytic, intuitively clear proof, the essence of which can be traced back to a work by Prony [1795], which has been known for nearly 200 years.