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Asymptotics Of Spectral Density Estimates

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  • Liu, Weidong
  • Wu, Wei Biao

Abstract

We consider nonparametric estimation of spectral densities of stationary processes, a fundamental problem in spectral analysis of time series. Under natural and easily verifiable conditions, we obtain consistency and asymptotic normality of spectral density estimates. Asymptotic distribution of maximum deviations of the spectral density estimates is also derived. The latter result sheds new light on the classical problem of tests of white noises.

Suggested Citation

  • Liu, Weidong & Wu, Wei Biao, 2010. "Asymptotics Of Spectral Density Estimates," Econometric Theory, Cambridge University Press, vol. 26(4), pages 1218-1245, August.
    • Handle: RePEc:cup:etheor:v:26:y:2010:i:04:p:1218-1245_99
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    1. Politis, Dimitris, 2012. "On The Behavior Of Nonparametric Density And Spectral Density Estimators At Zero Points Of Their Support," University of California at San Diego, Economics Working Paper Series qt40g0z0tz, Department of Economics, UC San Diego.
    2. Stefan Birr & Stanislav Volgushev & Tobias Kley & Holger Dette & Marc Hallin, 2017. "Quantile spectral analysis for locally stationary time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1619-1643, November.
    3. Forni, Mario & Hallin, Marc & Lippi, Marco & Zaffaroni, Paolo, 2017. "Dynamic factor models with infinite-dimensional factor space: Asymptotic analysis," Journal of Econometrics, Elsevier, vol. 199(1), pages 74-92.
    4. Deb, Soudeep & Karmakar, Sayar, 2023. "A novel spatio-temporal clustering algorithm with applications on COVID-19 data from the United States," Computational Statistics & Data Analysis, Elsevier, vol. 188(C).
    5. Loubaton, Philippe & Rosuel, Alexis & Vallet, Pascal, 2023. "On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    6. Horváth, Lajos & Rice, Gregory & Whipple, Stephen, 2016. "Adaptive bandwidth selection in the long run covariance estimator of functional time series," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 676-693.
    7. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2014. "Multivariate Variance Ratio Statistics," Cambridge Working Papers in Economics 1459, Faculty of Economics, University of Cambridge.
    8. Zhang, Rongmao & Chan, Ngai Hang & Chi, Changxiong, 2023. "Nonparametric testing for the specification of spatial trend functions," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    9. Steland, Ansgar, 2024. "Flexible nonlinear inference and change-point testing of high-dimensional spectral density matrices," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    10. Alexander Braumann & Jens‐Peter Kreiss & Marco Meyer, 2021. "Simultaneous inference for autocovariances based on autoregressive sieve bootstrap," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 534-553, September.
    11. Berkes, István & Horváth, Lajos & Rice, Gregory, 2016. "On the asymptotic normality of kernel estimators of the long run covariance of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 150-175.
    12. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2015. "An investigation into Multivariate Variance Ratio Statistics and their application to Stock Market Predictability," Cambridge Working Papers in Economics 1552, Faculty of Economics, University of Cambridge.
    13. Horváth, Lajos & Liu, Zhenya & Rice, Gregory & Wang, Shixuan, 2020. "Sequential monitoring for changes from stationarity to mild non-stationarity," Journal of Econometrics, Elsevier, vol. 215(1), pages 209-238.
    14. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2014. "Multivariate variance ratio statistics," CeMMAP working papers 29/14, Institute for Fiscal Studies.
    15. van Delft, Anne, 2020. "A note on quadratic forms of stationary functional time series under mild conditions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4206-4251.
    16. Horváth, Lajos & Rice, Gregory & Zhao, Yuqian, 2023. "Testing for changes in linear models using weighted residuals," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    17. Panxu Yuan & Xiao Guo, 2022. "High-dimensional inference for linear model with correlated errors," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(1), pages 21-52, January.
    18. Horváth, Lajos & Reeder, Ron, 2012. "Detecting changes in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 310-334.
    19. Lajos Horváth & Gregory Rice, 2014. "Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 219-255, June.
    20. Kin Wai Chan & Chun Yip Yau, 2017. "High-order Corrected Estimator of Asymptotic Variance with Optimal Bandwidth," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(4), pages 866-898, December.
    21. Efstathios Paparoditis & Dimitris N. Politis, 2016. "A Note on the Behaviour of Nonparametric Density and Spectral Density Estimators at Zero Points of their Support," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(2), pages 182-194, March.
    22. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2015. "An investigation into multivariate variance ratio statistics and their application to stock market predictability," CeMMAP working papers 13/15, Institute for Fiscal Studies.

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