A Nonparametric Prewhitened Covariance Estimator

This paper proposes a new nonparametric spectral density estimator for time series models with general autocorrelation. The conventional nonparametric estimator that uses a positive kernel has mean squared error no better than n. We show that the best implementation of our estimator has mean squared error of order n, provided there is sufficient smoothness present in the spectral density. This is, of course, achieved by bias reduction; however, unlike most other bias reduction methods, like the kernel method with higher-order kernels, our procedure ensures a positive definite estimate. Our method is a generalization of the well-known prewhitening method of spectral estimation; we argue that this can best be interpreted as multiplicative bias reduction. Higher-order expansions for the proposed estimator are derived, providing an improved bandwidth choice that minimizes the mean squared error to the second order. A simulation study shows that the recommended prewhitened kernel estimator reduces bias and mean squared error in spectral density estimation