Properties of estimated characteristic roots

Estimated characteristic roots in stationary autoregressions are shown to give rather noisy information about their population equivalents.  This is remarkable given the central role of the characteristic roots in the theory of autoregressive processes.  In the asymptotic analysis the problems appear when multiple roots are present as this implies a non-differentiablity so the δ-method does not apply, convergence rates are slow, and the asymptotic distribution is non-normal.  In finite samples this has a considerable influence on the finite sample distribution unless the roots are far apart.  With increasing order of the autoregressions it becomes increasingly difficult to place the roots far apart giving a very noisy signal from the characteristic roots.