Signal Extraction for Nonstationary Time Series

The nonstationary signal extraction problem is to estimate $s_t$ given observations on $z_t = s_t + n_t$ (signal plus noise) when either $s_t$ or $n_t$ or both is nonstationary. Homogeneous or explosive nonstationary time series described by models of the form $\delta(B)z_t = w_t$ where $\delta(B)$ has zeroes on or inside the unit circle and $w_t$ is stationary are considered. For certain cases, approximate solutions to the nonstationary signal extraction problem have been given by Hannan (1967), Sobel (1967), and Cleveland and Tiao (1976). The paper gives exact solutions in the forms of expressions for $E(s_t\mid\{z_t\})$ and $\operatorname{Var}(s_t\mid\{z_t\})$ (assuming normality) under two sets of alternative assumptions regarding the generation of $z_t, s_t$, and $n_t$. Extensions to signal extraction with a finite number of observations, to the nonGaussian case, and to the multivariate case are discussed.