Higher‐Order Accurate Spectral Density Estimation of Functional Time Series

Abstract

Under the frequency domain framework for weakly dependent functional time series, a key element is the spectral density kernel which encapsulates the second‐order dynamics of the process. We propose a class of spectral density kernel estimators based on the notion of a flat‐top kernel. The new class of estimators employs the inverse Fourier transform of a flat‐top function as the weight function employed to smooth the periodogram. It is shown that using a flat‐top kernel yields a bias reduction and results in a higher‐order accuracy in terms of optimizing the integrated mean square error (IMSE). Notably, the higher‐order accuracy of flat‐top estimation 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) in finite samples while retaining its favorable asymptotic properties. In addition, we introduce a data‐driven bandwidth selection procedure realized by an automatic inspection of the estimated correlation structure. Our asymptotic results are complemented by a finite‐sample simulation where the higher‐order accuracy of flat‐top estimators is manifested in practice.

Keywords: Functional time series, spectral density kernel, spectral density estimation flat‐top kernel, positive semi‐definite estimation