An iterated parametric approach to nonstationary signal extraction

Consider the three-component time series model that decomposes observed data (Y) into the sum of seasonal (S), trend (T), and irregular (I) portions. Assuming that S and T are nonstationary and that I is stationary, it is demonstrated that widely used Wiener–Kolmogorov signal extraction estimates of S and T can be obtained through an iteration scheme applied to optimal estimates derived from reduced two-component models for S plus I and T plus I. This “bootstrapping” signal extraction methodology is reminiscent of the iterated nonparametric approach of the US Census Bureau's X-11 program. The analysis of the iteration scheme provides insight into the algebraic relationship between full model and reduced model signal extraction estimates.