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Frequency Domain Tests of Semiparametric Hypotheses for Locally Stationary Processes

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  • MARIOS SERGIDES
  • EFSTATHIOS PAPARODITIS

Abstract

. Many time series in applied sciences obey a time‐varying spectral structure. In this article, we focus on locally stationary processes and develop tests of the hypothesis that the time‐varying spectral density has a semiparametric structure, including the interesting case of a time‐varying autoregressive moving‐average (tvARMA) model. The test introduced is based on a L2‐distance measure of a kernel smoothed version of the local periodogram rescaled by the time‐varying spectral density of the estimated semiparametric model. The asymptotic distribution of the test statistic under the null hypothesis is derived. As an interesting special case, we focus on the problem of testing for the presence of a tvAR model. A semiparametric bootstrap procedure to approximate more accurately the distribution of the test statistic under the null hypothesis is proposed. Some simulations illustrate the behaviour of our testing methodology in finite sample situations.

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  • 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.
    • Handle: RePEc:bla:scjsta:v:36:y:2009:i:4:p:800-821
    DOI: 10.1111/j.1467-9469.2009.00652.x
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    References listed on IDEAS

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    1. Sébastien Van Bellegem & Rainer Dahlhaus, 2006. "Semiparametric estimation by model selection for locally stationary processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(5), pages 721-746, November.
    2. Fryzlewicz, Piotr & van Bellegem, Sébastien & von Sachs, Rainer, 2003. "Forecasting non-stationary time series by wavelet process modelling," LSE Research Online Documents on Economics 25830, London School of Economics and Political Science, LSE Library.
    3. Kenji Sakiyama & Masanobu Taniguchi, 2003. "Testing Composite Hypotheses for Locally Stationary Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(4), pages 483-504, July.
    4. Piotr Fryzlewicz & Sébastien Bellegem & Rainer Sachs, 2003. "Forecasting non-stationary time series by wavelet process modelling," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 737-764, December.
    5. 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.
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    7. Davis, Richard A. & Lee, Thomas C.M. & Rodriguez-Yam, Gabriel A., 2006. "Structural Break Estimation for Nonstationary Time Series Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 223-239, March.
    8. Marios Sergides & Efstathios Paparoditis, 2008. "Bootstrapping the Local Periodogram of Locally Stationary Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(2), pages 264-299, March.
    9. Ombao, Hernando & von Sachs, Rainer & Guo, Wensheng, 2005. "SLEX Analysis of Multivariate Nonstationary Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 519-531, June.
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    Cited by:

    1. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 361-411, September.
    2. Chen, Yen-Hung & Hsu, Nan-Jung, 2014. "A frequency domain test for detecting nonstationary time series," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 179-189.
    3. Franziska Häfner & Claudia Kirch, 2017. "Moving Fourier Analysis for Locally Stationary Processes with the Bootstrap in View," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(6), pages 895-922, November.
    4. Kawka, Rafael, 2022. "Convergence of spectral density estimators in the locally stationary framework," Econometrics and Statistics, Elsevier, vol. 24(C), pages 94-115.
    5. Philip Preuss & Mathias Vetter & Holger Dette, 2013. "Testing Semiparametric Hypotheses in Locally Stationary Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 417-437, September.
    6. Ruprecht Puchstein & Philip Preuß, 2016. "Testing for Stationarity in Multivariate Locally Stationary Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(1), pages 3-29, January.

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