Finite approximations to linear filters and the monitoring of revisions (Q1584172)

Statistical offices involved in the production of seasonal adjusted series face the problem of revising preliminary seasonal adjusted figures. The process of revising until final estimators are obtained can last relatively long, and, in general, the number of additional periods needed to get into a final figure cannot be controlled. Two main types of signal estimation procedures are available: empirical, through the use of ad hoc linear filters like \(X11,\) and model-based signal extraction. If the first approach uses linear filters of finite dimensions, optimal model-based decompositions involve the Wiener-Kolmogorov filter which in general is of infinite length. In this paper, using a distance measure between linear filters, it is shown how infinite filters can be optimally approximated by finite ones. Two applications of this result are developed. First, a procedure allowing practitioners to control the length of the revision period is proposed. Given an observed series, a stochastic linear model describing its second moments and the specification of the length of the revision period found acceptable, an optimal decomposition is derived. This procedure is illustrated on two monthly economic time series. Second, some insights into the understanding of the links between the optimal Wiener-Kolmogorov filter and the \(X11\) filters are provided. Also, it is shown that the \(X11\) estimators are better seen as finite approximations to the optimal signal extraction estimators.