Signal Extraction in Nonstationary Series

The state-space method is applied to the problem of separating an autoregressive (AR) signal from composite AR and white normal noise. In the stationary case, for which the Wiener filter exists, we show explicitly its equavalence to the steady-state Kalman filter. Existing results for difference-stationary processes are generalised to the explosive AR case, with careful attention paid to initial conditions, the limiting filter is shown to be stable. Conditions are given for convergence of the signal extraction erroe variance, and these are seen to exclude the existence of an unstable common factor in signal and noise autoregressions, but not nonstationarity. The general argument is illustrated with simple examples and the role of controllability and detectability is explored in an appendix.