Evaluating the statistical properties of time series non parametric estimators by means of smoothing matrices

Smoothing matrices associated to linear filters for the estimation of time series unobserved component differ from those used in linear regression or generalized additive models due to asymmetry. In fact, while projection smoothing matrices are in general symmetric, filtering matrices are not. It follows that many inferential properties developed for symmetric projection matrices no longer hold for smoothing matrices. However, the latter have a well defined algebraic structure: they are centrosymmetric and invariant with respect to a linear transformation which results from pre- and post-multiplication by permutation matrices. This allows to derive many properties useful for inference in smoothing problems. In this paper, algebraic properties of smoothing matrices are interpreted in terms of those of fitting and smoothing of a linear filter and the theory of smoothing matrices is extended to non symmetric ones.