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A note on testing hypotheses for stationary processes in the frequency domain

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  • Dette, Holger
  • Hildebrandt, Thimo

Abstract

In a recent paper, Eichler (2008)Â [11] considered a class of non- and semiparametric hypotheses in multivariate stationary processes, which are characterized by a functional of the spectral density matrix. The corresponding statistics are obtained using kernel estimates for the spectral distribution and are asymptotically normally distributed under the null hypothesis and local alternatives. In this paper, we derive the asymptotic properties of these test statistics under fixed alternatives. In particular, we also show weak convergence but with a different rate compared to the null hypothesis. We also discuss potential statistical applications of the asymptotic theory by means of a small simulation study.

Suggested Citation

  • Dette, Holger & Hildebrandt, Thimo, 2012. "A note on testing hypotheses for stationary processes in the frequency domain," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 101-114, February.
    • Handle: RePEc:eee:jmvana:v:104:y:2012:i:1:p:101-114
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    References listed on IDEAS

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    1. Holger Dette & Ingrid Spreckelsen, 2003. "A Note on a Specification Test for Time Series Models Based on Spectral Density Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(3), pages 481-491, September.
    2. Marc Hallin & Abdessamad Saidi, 2005. "Testing non-correlation and non-causality between two multivariate ARMA time series," ULB Institutional Repository 2013/2129, ULB -- Universite Libre de Bruxelles.
    3. Holger Dette & Efstathios Paparoditis, 2009. "Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(4), pages 831-857, September.
    4. Efstathios Paparoditis, 2000. "Spectral Density Based Goodness‐of‐Fit Tests for Time Series Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(1), pages 143-176, March.
    5. Peter J. Diggle & Nicholas I. Fisher, 1991. "Nonparametric Comparison of Cumulative Periodograms," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 40(3), pages 423-434, November.
    6. Michael Eichler, 2007. "A Frequency-domain Based Test for Non-correlation between Stationary Time Series," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 65(2), pages 133-157, February.
    7. Eichler, Michael, 2008. "Testing nonparametric and semiparametric hypotheses in vector stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 968-1009, May.
    8. Marc Hallin & Abdessamad Saidi, 2005. "Testing Non‐Correlation and Non‐Causality between Multivariate ARMA Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(1), pages 83-105, January.
    9. Chafik Bouhaddioui & Roch Roy, 2006. "A Generalized Portmanteau Test For Independence Of Two Infinite‐Order Vector Autoregressive Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(4), pages 505-544, July.
    10. Taniguchi, Masanobu & Puri, Madan L. & Kondo, Masao, 1996. "Nonparametric Approach for Non-Gaussian Vector Stationary Processes," Journal of Multivariate Analysis, Elsevier, vol. 56(2), pages 259-283, February.
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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. Shibin Zhang, 2023. "A copula spectral test for pairwise time reversibility," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(5), pages 705-729, October.
    3. Fontaine, Charles & Frostig, Ron D. & Ombao, Hernando, 2020. "Modeling non-linear spectral domain dependence using copulas with applications to rat local field potentials," Econometrics and Statistics, Elsevier, vol. 15(C), pages 85-103.
    4. Xu Guo & Wangli Xu & Lixing Zhu, 2015. "Model checking for parametric regressions with response missing at random," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 229-259, April.

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