Signal Extraction Error in Nonstationary Time Series

Abstract

It is supposed that an observable time series $\{x_t\}$ is representable as the sum of a "signal" $s_t$ and a "noise" $n_t$, and that it is desired to extract the signal $s_t$, i.e., to obtain an estimate $\hat{s}_t$ of $s_t$. Corresponding to any such estimate is a signal extraction error, $\delta_t = s_t - \hat{s}_t$, which for nondeterministic stochastic processes possesses a nonzero mean square. For the class of homogeneously nonstationary processes, characterizations of the extraction error process are given, and it is shown that the mean square of the error does not exist unless the nonstationary autoregressive operators in the $s$- and $n$-processes have distinct roots. The MSE, autocorrelations and spectrum of the error, when it is stationary, are illustrated for some special cases, including two stochastic-model approximations to the Census $X$-11 seasonal adjustment procedure.