Estimating the Spectral Density at Frequencies Near Zero

Abstract

Estimating the spectral density function f(w) for some $w\in [-\pi ,\pi ]$ has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, that is, approximating f(w) by a constant over a window of small width. Although f(w) is uniformly continuous and periodic with period $2\pi$, in this article we recognize the fact that w = 0 effectively acts as a boundary point in the underlying kernel smoothing problem, and the same is true for $w=\pm \pi$. It is well-known that local averaging may be suboptimal in kernel regression at (or near) a boundary point. As an alternative, we propose a local polynomial regression of the periodogram or log-periodogram when w is at (or near) the points 0 or $\pm \pi$. The case w = 0 is of particular importance since f(0) is the large-sample variance of the sample mean; hence, estimating f(0) is crucial in order to conduct any sort of inference on the mean. Supplementary materials for this article are available online.

Keywords:

• Flat-top lag-windows
• Function estimation
• Kernel smoothing
• Local polynomials
• Long-run variance
• Sample mean

Supplementary Materials

Supplementary material can be found at the GitHub repo https://github.com/tuckermcelroy/SpecLocal; this includes the following: (i) code: R functions and scripts needed for simulations and data analyses; (ii) output: text and zip files with simulation output; (iii) data: two data sets used in the paper; and (iv) manuscript: main paper (latex and pdf), supplement (pdf), and figure files.

Acknowledgments

This report is released to inform interested parties of research and to encourage discussion. The views expressed on statistical issues are those of the authors and not those of the U.S. Census Bureau. Many thanks are due to the Editor, Associate Editor, and two anonymous reviewers for their constructive comments.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

¹ U.S. Bureau of Economic Analysis, Gross Domestic Product [GDP], retrieved from FRED, Federal Reserve Bank of St. Louis; https://urldefense.com/v3/__https://fred.stlouisfed.org/series/GDP__;!!Mih3wA!VcvF56vPVXyxZNQZTiLhlZAYfL6m6vIdVZ4DS7pJ1Q60-k3FOtf9LQmDql_gcN4Zz1Q$, November 13, 2020.

² The definition of annual growth rates by the U.S. Bureau of Economic Analysis is given in https://urldefense.com/v3/__https://www.bea.gov/help/faq/463__;!!Mih3wA!VcvF56vPVXyxZNQZTiLhlZAYfL6m6vIdVZ4DS7pJ1Q60-k3FOtf9LQmDql_gAGofiBQ$.

³ Data source: NASA’s Goddard Institute for Space Studies, retrieved from https://urldefense.proofpoint.com/v2/url?u=https-3A__climate.nasa.gov_vital-2Dsigns_global-2Dtemperature_&d=DwIGAg&c=-35OiAkTchMrZOngvJPOeA&r=ArlQHWCRCNUBzA5G5_1B9GdyataLP0jyqZ2bB0rc–I&m=nJu0dJ8ItbpQqfaMV57vykhCYpgsE4ZFw-lCzJZjvzXh5KhdF30_iojNCT47iDed&s=fctvW9MgNu0kDnpzI2DITEgYI3hZfLJf30WboaljAAk&e=on December 28, 2021.

Additional information

Funding

The research of the second author was partially supported by NSF grant DMS 19-14556.