Higher‐Order Accurate Spectral Density Estimation of Functional Time Series

18 Pages Posted: 29 May 2020

See all articles by Tingyi Zhu

Tingyi Zhu

University of California, San Diego (UCSD)

Dimitris N. Politis

University of California, San Diego (UCSD) - Department of Mathematics

Date Written: January 2020

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

Suggested Citation

Zhu, Tingyi and Politis, Dimitris, Higher‐Order Accurate Spectral Density Estimation of Functional Time Series (January 2020). Journal of Time Series Analysis, Vol. 41, Issue 1, pp. 3-20, 2020, Available at SSRN: https://ssrn.com/abstract=3613057 or http://dx.doi.org/10.1111/jtsa.12473

Tingyi Zhu (Contact Author)

University of California, San Diego (UCSD)

9500 Gilman Drive
La Jolla, CA 92093
United States

Dimitris Politis

University of California, San Diego (UCSD) - Department of Mathematics ( email )

9500 Gilman Drive
La Jolla, CA 92093-0112
United States
858-534-5861 (Phone)
858-534-5273 (Fax)

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