Detection of Compatible Turning-Points and Signal-Extraction of Non-Stationary Time Series

Summary

Traditionally, trend-estimation or -extraction methods for non-stationary time series (ARIMA-model based, structural models, Census X12,…) make use of a stochastic modeling procedure which not only determines an optimal symmetric MA(∞)-extraction filter (this is not true for Census XI2) but also supplies missing values at both ends of a finite sample by optimal fore- and backcasts, hence minimizing the unconditional final revision variance. In this paper we propose a new trend estimation procedure based on a direct filtering approach. We generalize the class of time invariant filters by including explicit time dependence towards the end of a sample and optimizing in each time point a corresponding filter with respect to a conditional final revision variance minimization. The condition corresponds to a time delay restriction and this will generalize usual unconditional optimization procedures. It is shown that this optimization underlies an uncertainty-principle (APUP) which is best solved by general IIR- or ARMA-filters instead of the usual MA-designs. This direct IIR-filter-method may be used either for traditional trend extraction or for detection of compatible turning-points of a series (to be defined below). In the latter case it is shown that the theoretical extraction filter has a transferfunction taking the form of an indicator function.