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Abstract: This paper is concerned with the spectral properties of matrices associatedwith linear filters for the estimation of the underlying trend of a timeseries. The interest lies in the fact that the eigenvectors can be interpretedas the latent components of any time series that the filter smooths through thecorresponding eigenvalues. A difficulty arises because matrices associated withtrend filters are finite approximations of Toeplitz operators and thereforevery little is known about their eigenstructure, which also depends on theboundary conditions or, equivalently, on the filters for trend estimation atthe end of the sample. Assuming reflecting boundary conditions, we derive atime series decomposition in terms of periodic latent components andcorresponding smoothing eigenvalues. This decomposition depends on the localpolynomial regression estimator chosen for the interior. Otherwise, theeigenvalue distribution is derived with an approximation measured by the sizeof the perturbation that different boundary conditions apport to theeigenvalues of matrices belonging to algebras with known spectral properties,such as the Circulant or the Cosine. The analytical form of the eigenvectors isthen derived with an approximation that involves the extremes only. A furthertopic investigated in the paper concerns a strategy for a filter design in thetime domain. Based on cut-off eigenvalues, new estimators are derived, that areless variable and almost equally biased as the original estimator, based on allthe eigenvalues. Empirical examples illustrate the effectiveness of the method.



Autor: Alessandra Luati, Tommaso Proietti

Fuente: https://arxiv.org/







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