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Locally adaptive fitting of semiparametric models to nonstationary time series

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  • Dahlhaus, Rainer
  • Neumann, Michael H.

Abstract

We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function [mu](·) and a p-dimensional function [theta](·)=([theta](1)(·),...,[theta](p)(·))' that parametrizes the time-varying spectral density f[theta](·)([lambda]). Whereas the mean function is estimated by a usual kernel estimator, each component of [theta](·) is estimated by a nonlinear wavelet method. According to a truncated wavelet series expansion of [theta](i)(·), we define empirical versions of the corresponding wavelet coefficients by minimizing an empirical version of the Kullback-Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of [theta](i)(·). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor.

Suggested Citation

  • Dahlhaus, Rainer & Neumann, Michael H., 2001. "Locally adaptive fitting of semiparametric models to nonstationary time series," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 277-308, February.
    • Handle: RePEc:eee:spapps:v:91:y:2001:i:2:p:277-308
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    References listed on IDEAS

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    1. Dahlhaus, R., 1996. "On the Kullback-Leibler information divergence of locally stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 139-168, March.
    2. Michael H. Neumann, 1996. "Spectral Density Estimation Via Nonlinear Wavelet Methods For Stationary Non‐Gaussian Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(6), pages 601-633, November.
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    1. Gabe Chandler & Wolfgang Polonik, 2017. "Residual Empirical Processes and Weighted Sums for Time-Varying Processes with Applications to Testing for Homoscedasticity," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(1), pages 72-98, January.
    2. Olsen, Lena Ringstad & Chaudhuri, Probal & Godtliebsen, Fred, 2008. "Multiscale spectral analysis for detecting short and long range change points in time series," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3310-3330, March.
    3. Dahlhaus, Rainer, 2009. "Local inference for locally stationary time series based on the empirical spectral measure," Journal of Econometrics, Elsevier, vol. 151(2), pages 101-112, August.
    4. Roueff, François & von Sachs, Rainer, 2011. "Locally stationary long memory estimation," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 813-844, April.
    5. von Sachs, Rainer, 2019. "Spectral Analysis of Multivariate Time Series," LIDAM Discussion Papers ISBA 2019008, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Euan T. McGonigle & Rebecca Killick & Matthew A. Nunes, 2022. "Trend locally stationary wavelet processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(6), pages 895-917, November.
    7. Marios Sergides & Efstathios Paparoditis, 2009. "Frequency Domain Tests of Semiparametric Hypotheses for Locally Stationary Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 800-821, December.

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