Summary
Let f be a given function on the unit circle such that f(e iθ) = | 1−e iθ |2α f 1(e iθ) with | α |< 1/2 and f 1 a strictly positive function that will be supposed to be sufficiently smooth. We give the asymptotic behavior of the first column of the inverse of T N(f), the (N +1) × (N + 1) Toeplitz matrix with elements (f i−j )0≤i,j≤N where \( f_k = \tfrac{1} {{2\pi }}\int_0^{2\pi } {f(e^{ - i\theta } )e^{ - ik\theta } d\theta } \). We shall compare our numerical results with those given by the Durbin-Levinson algorithm, with particular emphasis on problems of predicting either stationary stochastic long-range dependent processes, or processes with a long-range dependent component.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abry, P., Flandrin, P., Taqqu, M. and Veitch, D. (2003). Self-similarity and long-range dependence through the wavelet lens. In: Doukhan, P., Oppenheim, G. and Taqqu, M.S. (Eds.), Long-Range Dependence: Theory and Applications. Birkhauser, Boston, pp. 527–556.
Bardet, J-M., Lang, G., Oppenheim, G., Philippe, A. and Taqqu, M.S. (2003). Generators of Long-range Dependent Processes: A Survey. In: Doukhan, P., Oppenheim, G. and Taqqu, M.S. (Eds.), Long-Range Dependence: Theory and Applications, 579–623. Birkhauser, Boston.
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman & Hall, New York.
Bleher P.M. (1981). Inversion of Toeplitz matrices Trans. Moscow Math. Soc., 2, 201–224.
Bollerslev, T. and Mikkelsen, H.-O. (1996). Modelling and pricing long-memory in stock market volatility. Journal of Econometrics, 73, 151–184.
Breidt, J., Crato, N. and de Lima, P. (1998). On the detection and estimation of long memory in stochastic volatility. Journal of Econometrics, 83, 325–348.
Brockwell, P.J. and Dalhaus, R. (2004). Generalized Levinson-Durbin and Burg algorithms. Journal of Econometrics, 118, 129–149.
Chen, W., Hurvich, C.M. and Lu, Y. (2004). On the correlation matrix of the discrete Fourier transform and the fast solution of large Toeplitz systems for long-memory time series. Preprint.
Chen, W. and Deo, R. (2004). A generalized portmanteau goodness-of-fit test for time series models. Econometric Theory, 20, 382–416.
Christoffersen, P.F. and Diebold, F.X. (2000). How relevant is volatility forecasting for financial management. Review of Economics and Statistics, 82, 12–22.
Deo, R., Hurvich, C.M. and Lu, Y. (2003). Forecasting realized volatility using a long memory stochastic volatility model: estimation, prediction and seasonal adjustment. Preprint.
Diebold, F.X., Gunter, T. and Tay, A. (1998). Evaluating density forecasts, with application to financial risk management. International Economic Review, 39, 863–883.
Ehrhardt, T. and Silbermann, B. (1997). Toeplitz determinants with one Fisher-Hartwig singularity, Journal of Functional Analysis, 148, 229–256.
Fay, G. and Philippe, A. (2002). Goodness-of-fit test for long range dependent processes. ESAIM: Probability and Statistics, 6, 239–258.
Fisher, M.E. and Hartwig, R.E. (1968). Toeplitz determinants: some applications, theorems and conjectures. Adv. Chem. Phys, 15, 333–353.
Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4, 221–238.
Granger, C.W.J. (2002). Long memory, volatility, risk and distribution. Preprint.
Granger, C.W.J. (2000). Current perspectives on long memory processes. Academia Economic Papers, 28, 1–16.
Granger, C.W.J. and Hyung, N. (2004). Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance, 11, 399–421.
Granger, C.W.J. and Ding, Z. (1995). Some properties of absolute returns, an alternative measure of risk. Annales d’ Économie et de Statistique, 40, 67–91.
Henry, M. (2005). Bandwidth choice, optimal rates and adaptivity in semiparametric estimation of long memory. In: Teyssière, G., Kirman, A. (Eds.), Long-Memory in Economics. Springer Verlag, Berlin. Appears in this volume.
Henry, M. (2001). Robust automatic bandwidth for long-memory. Journal of Time Series Analysis, 22, 293–316.
Henry, M. and Robinson, P.M. (1996). Bandwidth choice in Gaussian semiparametric estimation of long range dependence. In: Robinson, P.M., Rosenblatt, M. (Eds.), Athens Conference on Applied Probability and Time Series Analysis, Time Series Analysis, In Memory of E.J. Hannan. Lecture Notes in Statistics, Vol. 115. Springer Verlag, New York, pp. 220–232.
Kateb, D., Rambour, P. and Seghier, A. (2004). The inversion of Toeplitz matrices: the singular case Prépublications de l’Université Paris-Sud 2004-10.
Künsch, H.R. (1987). Statistical Aspects of Self-Similar Processes In Yu. Prohorov and V. V. Sazanov editors, Proceedings of the First World Congress of the Bernoulli Society, 1, 67–74. VNU Science Press, Utrecht.
Lavielle, M. and Teyssière, G. (2005). Semiparametric adaptive detection of multiple change-points in asset price volatility. In: Teyssière, G., Kirman, A. (Eds.), Long-Memory in Economics. Springer Verlag, Berlin. Appears in this volume.
Lux, T. (2003). The multifractal model of asset returns: its estimation via GMM and its use for volatility forecasting. Preprint.
Mandelbrot, B.B., Fisher, A. and Calvet, L. (1997). The Multifractal model of asset returns. Cowles Foundation Discussion Paper 1164.
Mikosch, T. and Stărică, C. (2004). Non-stationarities in financial time series: the long range dependence and the IGARCH effects. Review of Economics and Statistics, 86, 278–290.
Mikosch, T. and Stărică, C. (2003). Long-range dependence effects and ARCH modeling. In: Doukhan, P., Oppenheim, G. and Taqqu, M.S. (Eds.), Long-Range Dependence: Theory and Applications. Birkhauser, Boston, pp. 439–459.
Mikosch, T. and Stărică, C. (1999). Change of structure in financial time series, long range dependence and the GARCH model. Preprint.
Rambour, P. and Seghier, A. (2003). Inversion des matrices de Toeplitz à symboles singuliers. Extension d’un résultat de Keysten. Submitted to Annales de l’Institut de Fourier.
Robinson, P.M. (1995). Gaussian semiparametric estimation of long-range dependence. The Annals of Statistics, 23, 1630–1661.
Robinson, P.M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics, 47, 67–84.
Stărică, C. and Granger, C.W.J. (2001). Non-stationarities in stock returns. Review of Economics and Statistics, forthcoming.
Teyssière, G. and Abry, P. (2005). Wavelet analysis of nonlinear long-range dependent processes. Applications to financial time series. In: Teyssière, G. and Kirman, A., (Eds.), Long-Memory in Economics. Springer Verlag, Berlin. Appears in this volume.
Veitch, D. and Abry, P. (1999). A wavelet based joint estimator of the parameters of long-range dependence. IEEE Transactions on Information Theory, 45, 878–897.
Wu, T.T. (1966). Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising Model. I. Physical Review, 149, 380–401.
Zygmund, A. (1959). Trigonomic Series, second edition. Cambridge University Press.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer Berlin · Heidelberg
About this chapter
Cite this chapter
Kateb, D., Seghier, A., Teyssière, G. (2007). Prediction, Orthogonal Polynomials and Toeplitz Matrices. A Fast and Reliable Approximation to the Durbin-Levinson Algorithm. In: Teyssière, G., Kirman, A.P. (eds) Long Memory in Economics. Springer, Berlin, Heidelberg . https://doi.org/10.1007/978-3-540-34625-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-34625-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22694-9
Online ISBN: 978-3-540-34625-8
eBook Packages: Business and EconomicsEconomics and Finance (R0)