A linear transformation and its properties with special applications in time series filtering

The main purpose of this paper is to introduce a linear transformation, called t, and to derive its algebraic properties by means of permutation matrices that represent it. To demonstrate the importance of the t-transformation for the estimation of latent variables in time series decomposition, we obtain a general expression for smoothing matrices characterized by symmetric and asymmetric weighting systems. We show that the submatrix of the symmetric weights (to be applied to central observations) is t-invariant whereas the submatrices of the asymmetric weights (to be applied to initial and final observations) are the t-transform of each other. By virtue of this relation, the properties of the t-transformation provide useful information on the smoothing of time series data. Finally, we illustrate the role of the t-transformation on the weighting systems of several smoothers often applied for trend-cycle estimation, such as the locally weighted regression smoother (loess), the cubic smoothing spline, the Gaussian kernel and the 13-term trend-cycle Henderson filter.