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HIGHER-ORDER ACCURATE, POSITIVE SEMIDEFINITE ESTIMATION OF LARGE-SAMPLE COVARIANCE AND SPECTRAL DENSITY MATRICES

Published online by Cambridge University Press:  03 March 2011

Dimitris N. Politis
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Abstract

A new class of large-sample covariance and spectral density matrix estimators is proposed based on the notion of flat-top kernels. The new estimators are shown to be higher-order accurate when higher-order accuracy is possible. A discussion on kernel choice is presented as well as a supporting finite-sample simulation. The problem of spectral estimation under a potential lack of finite fourth moments is also addressed. The higher-order accuracy of flat-top kernel estimators typically comes at the sacrifice of the positive semidefinite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semidefinite (even strictly positive definite) while maintaining its higher-order accuracy. In addition, an easy (and consistent) procedure for optimal bandwidth choice is given; this procedure estimates the optimal bandwidth associated with each individual element of the target matrix, automatically sensing (and adapting to) the underlying correlation structure.

Type
Research Article
Information
Econometric Theory , Volume 27 , Issue 4 , August 2011 , pp. 703 - 744
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

This research was partially supported by NSF grants SES-04-18136 and DMS-07-06732. Many thanks are due to Arthur Berg and Tucker McElroy for numerous helpful interventions, to Peter Robinson for a critical reading and suggestions of some key early references, and to Dimitrios Gatzouras for his help with the proof of Lemma B.1. The S+ software for the practical computation of the different spectral density estimators was compiled with the invaluable help of Isheeta Nargis and Arif Dowla of www.stochasticlogic.com, and is now publicly available from www.math.ucsd.edu/∼politis/SOFT/SfunctionsFLAT-TOPS.html. Finally, the author is grateful to the co-editor and three anonymous reviewers for their insightful suggestions.

References

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