A note on the statistical properties of nonparametric trend estimators by means of smoothing matrices

Smoothing matrices associated with linear filters for the estimation of time series unobserved components differ from those used in linear regression or generalized additive models due to asymmetry. In fact, while projection smoother matrices are in general symmetric, filtering matrices are not. It follows that many inferential properties developed for symmetric projection matrices no longer hold for time series smoothing matrices. However, the latter have a well defined algebraic structure that allows one to derive many properties useful for inference in smoothing problems. In this note, some properties of symmetric smoother matrices are extended to centrosymmetric smoothing matrices. A decomposition of smoothing matrices in submatrices associated with the symmetric and asymmetric components of a filter enables to consider the different assumptions that characterize estimation in the interior and at the boundaries of a finite time series. Matrix-based measures are defined to approximate the bias and the variance of a trend estimator, both in the interior and at the boundaries. These measures do not depend on the data and provide useful information on the bias-variance trade-off that affects nonparametric estimators.