Time Path of Kernels Asymmetric Filters for Nonstationary Mean Prediction of Seasonally Adjusted Series

In recent studies, Dagum and Luati (2002) found that restricted to 13 weights, the Gaussian Kernel (GK) and Loess of degree 2 (L2) approximate closely the symmetric 13-term Henderson filter. This latter is widely applied for non-stationary mean (trend-cycle) estimation by seasonal adjustment methods. In the context of current analysis, it is very important to get sound estimates of current trend-cycle. For the more recent data points, the above function estimators use time varying asymmetric filters, the properties of which differ from those of their symmetric filters. The purpose of this study is to analyze the time path of each estimator asymmetric filters from the viewpoint of monotone convergence and distance to their respective symmetric ones. We measure the total distance of each filter with respect to the symmetric one, and the consecutive distances between asymmetric filters by means of their gain and phase shift functions. For each distance measure, we distinguish between the frequency bands associated with the signal and the noise. The monotone convergence is given by a systematic decrease of the consecutive asymmetric filters distances for signal and noise as they approach to the symmetric one. It should be noted that the total distance is a measure of the total revision of the corresponding trend-cycle estimate due to filter changes. Another important aspect investigated in this paper is the loss in accuracy incurred by stopping estimation of the last data point before the symmetric filter can be applied.