Notes on poles of autoregressive type model, I: Non-robust singular pole (Q1100837)

Let \(x(t)=[x_ 1(t),x_ 2(t),x_ 3(t)]'\) be a 3-dimensional AR(1) process. Assume that only \(x_ 1(t)\) is observable. Then \(x_ 1(t)\) is described by an ARMA (3,2) model, which can be written in an AR(\(\infty)\) form. If only a finite number of members of this AR(\(\infty)\) expression is considered, we get a truncated AR(TAR) model. Using Rouché's theorem, the authors investigate location of poles of the TAR model and their relations to the poles of the original ARMA(3,2) model. It is also shown that the zero of the ARMA(3,2) model closest to the unit circle has an important role in the convergence of spectral density functions of the TAR models to that of the ARMA(3,2) model.