On the Spectral Properties of Matrices Associated with Trend Filters

Luati, Alessandra and Proietti, Tommaso (2008): On the Spectral Properties of Matrices Associated with Trend Filters.

Abstract

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties, such as the Circulant and the generalised Tau. These results are used to characterise the properties of a smoother in terms of an approximate eigen-decomposition of the associated smoothing matrix.

• Item Type: MPRA Paper
• Language: English
• Keywords: Signal extraction; Smoothing; Boundary conditions; Matrix algebras
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Bauer F., Fike C. (1960), Norms and Exclusion Theorems, Numerische Mathematik, 2, 137-141.

Baxter, M., King, R.G. (1999). Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series. The Review of Economics and Statistics, 81, 575-593.

Bini D., Capovani M. (1983), Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra and its Applications, 52/53, pp. 99-126.

Bini D., Favati P. (1993), On a matrix algebra related to the discrete Hartley transform. SIMAX, 14, 2, 500-507.

Bottcher A., Grudsky, S.M. (2005), Spectral Properties of Banded Toeplitz Matrices, Siam.

Bozzo E., Di Fiore C. (1995), On the use of certain matrix algebras associated with discrete trigonometric transforms in matrix displacement decomposition, Siam J. Matrix anal. Appl., 16, 1, 312-326.

Buja A., Hastie T.J., Tibshirani R.J.(1989), Linear Smoothers and Additive Models, The Annals of Statistics, 17, 2, 453-555.

Christiano L.J., Fitzgerald T.J., (2003), The band pass filter, International Economic Review, 44, 435-465.

Cleveland, W.S., Loader, C.L. (1996). Smoothing by Local Regression: Principles and Methods. In W. H¨ardle and M. G. Schimek, editors, Statistical Theory and Computational Aspects of Smoothing, 1049. Springer, New York.

Dagum, E.B., Luati, A. (2004). A Linear Transformation and its Properties with Special Applications in Time Series, Linear Algebra and its Applications, 338, 107-117.

Davis, P.J., (1979). Circulant matrices, Wiley, New York. Fan J. and Gjibels I. (1996). Local Polynomial Modelling and its Applications, Chapman and Hall, New York.

Gray R.M. (2006) Toeplitz and Circulant Matrices: A review, Foundations and Trends in Communications and Information Theory, Vol 2, Issue 3, pp 155-239.

Green P.J. and Silverman, B.V. (1994) Nonparametric Regression and Generalized Linear Models: a Roughness Penalty Approach. Chapman & Hall, London.

Grenander, U., Szeg¨o, G. (1958). Toeplitz Forms and Their Applications, University of California Press.

Hastie T.J. and Tibshirani R.J.(1990), Generalized Additive Models, Chapman and Hall, London.

Hodrick, R., and Prescott, E.C. (1997). Postwar U.S. Business Cycle: an Empirical Investigation, Journal of Money, Credit and Banking, 29, 1, 1-16.

Kailath T., Sayed A. (1995) Displacement Structure: Theory and Applications, SIAM Review, 37, 3, 297-386.

Leser, C.E.V. (1961). A Simple Method of Trend Construction, Journal of the Royal Statistical Society B, 23, 91-107.

Percival, D.B.,Walden, A.T. (2000). Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge.

Pollock, D.S.G. (2002), Circulant matrices and time-series analysis, International Journal of Mathematical Education in Science and Technology, 33, 2, 1, 213-230(18)

Proietti, T., Luati, A. (2009). Real Time Estimation in Local Polynomial Regression, with an Application to Trend-Cycle Analysis, Annals of Applied Statistics, forthcoming.

Ruppert D., Wand, M.J., Carroll R.J. (2003). Semiparametric regression, Cambridge University Press.

Weaver J.R. (1985), Centrosymmetric (Cross-Symmetric) Matrices, their Basic Properties, Eigenvalues, Eigenvectors, Amer. Math. Monthly, 92, 711-717.

Whittle P. (1983). Prediction and Regulation by Linear Least Squares Methods, Second edition. Basil Blackwell, Oxford.