Higher-Order Accurate, Positive Semidefinite Estimation of Large-Sample Covariance and Spectral Density Matrices

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 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 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 10.1. 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.

Fixed-Smoothing Asymptotics in a Two-Step GMM Framework

“…Many nonparametric estimators of the variance matrix are available in the literature. In this paper, we consider a class of quadratic variance estimators, which includes the conventional kernel variance estimators of Andrews (1991), Newey and West (1987), Politis (2011), sharp and steep kernel variance estimators of Phillips, Jin (2006, 2007), and the orthonormal series (OS) variance estimators of Phillips (2005), Müller (2007), and Sun (2011Sun ( , 2013 as special cases. Following Phillips, Jin (2006, 2007), we refer to the conventional kernel estimators as contracted kernel estimators and the sharp and steep kernel estimators as exponentiated kernel estimators.…”

Design-free estimation of variance matrices

“…Ledoit and Wolf (2003) propose an optimal estimation procedure for the shrinkage parameter, where the chosen metric is the Frobenius norm between the variance and the shrinkage matrix. An alternative approach whereby offdiagonal elements are downweighted towards zero is given in McMurry and Politis (2010) and Politis (2011) in the context of time series. See also an approach to shrinkage via condition-number regularization in Won, Lim, Kim, and Rajaratnam (2013).…”

Aggregation of spectral density estimators

“…The simplest flat-top lag-window is the trapezoid proposed by Politis and Romano (1995); for the definition and properties of general flat-top lag windows see Politis (2001Politis ( , 2005Politis ( and 2011.…”