Error Bounds and Asymptotic Expansions for Toeplitz Product Functionals of Unbounded Spectra

Offer Lieberman and Peter Phillips
Additional contact information
Offer Lieberman: Technion-Israel Institute of Technology

No 1374, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

Abstract: This paper establishes error orders for integral limit approximations to traces of powers to the pth order) of products of Toeplitz matrices. Such products arise frequently in the analysis of stationary time series and in the development of asymptotic expansions. The elements of the matrices are Fourier transforms of functions which we allow to be bounded, unbounded, or even to vanish on [-pi,pi], thereby including important cases such as the spectral functions of fractional processes. Error rates are also given in the case in which the matrix product involves inverse matrices. The rates are sharp up to an arbitrarily small epsilon > 0. The results improve on the o(1) rates obtained in earlier work for analogous products. For the p = 1 case, an explicit second order asymptotic expansion is found for a quadratic functional of the autocovariance sequences of stationary long memory time series. The order of magnitude of the second term in this expansion is shown to depend on the long memory parameters. It is demonstrated that the pole in the first order approximation is removed by the second order term, which provides a substantially improved approximation to the original functional.

Keywords: Asymptotic expansion; higher cumulants; long memory; singularity; spectral density; Toeplitz matrix (search for similar items in EconPapers)
JEL-codes: C22 (search for similar items in EconPapers)
Pages: 22 pages
Date: 2002-05
Note: CFP 1141.
References: Add references at CitEc
Citations:

Published in Journal of Time Series Analysis (2004), 25(5): 733-753

Downloads: (external link)
https://cowles.yale.edu/sites/default/files/files/pub/d13/d1374.pdf (application/pdf)
Our link check indicates that this URL is bad, the error code is: 404 Not Found

Related works:
Journal Article: Error bounds and asymptotic expansions for toeplitz product functionals of unbounded spectra (2004)
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:cwl:cwldpp:1374

Ordering information: This working paper can be ordered from
Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA
The price is None.

Access Statistics for this paper

More papers in Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University Yale University, Box 208281, New Haven, CT 06520-8281 USA. Contact information at EDIRC.
Bibliographic data for series maintained by Brittany Ladd ().