Higher-order accurate, positive semi-definite estimation of large-sample covariance and spectral density matrices

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 semi-definite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semi-definite (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. ∗Research partially supported by NSF grant SES-04-18136 funded jointly by the Economics and Statistics Divisions of NSF. Many thanks are due to Arthur Berg 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 9.1. 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.