Counterexamples to general convergence of a commonly used recursive identification method

A recursive algorithm for parametric identification of discrete-time systems known as Panuska's method, the approximate maximum likelihood method or the extended matrix method, is analyzed. Making use of recently developed theory for asymptotic analysis of recursive stochastic algorithms, dynamic systems, and autoregressive moving average (ARMA) processes are constructed for which this algorithm does not converge. The manner in which the counterexamples are constructed yields insight into the algorithm and provides ideas how to improve the convergence properties.