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Generalised likelihood ratio tests for spectral density


  • Jianqing Fan


There are few techniques available for testing whether or not a family of parametric times series models fits a set of data reasonably well without serious restrictions on the forms of alternative models. In this paper, we consider generalised likelihood ratio tests of whether or not the spectral density function of a stationary time series admits certain parametric forms. We propose a bias correction method for the generalised likelihood ratio test of Fan et al. (2001). In particular, our methods can be applied to test whether or not a residual series is white noise. Sampling properties of the proposed tests are established. A bootstrap approach is proposed for estimating the null distribution of the test statistics. Simulation studies investigate the accuracy of the proposed bootstrap estimate and compare the power of the various ways of constructing the generalised likelihood ratio tests as well as some classic methods like the Cramer--von Mises and Ljung--Box tests. Our results favour the newly proposed bias reduction method using the local likelihood estimator. Copyright Biometrika Trust 2004, Oxford University Press.

Suggested Citation

  • Jianqing Fan, 2004. "Generalised likelihood ratio tests for spectral density," Biometrika, Biometrika Trust, vol. 91(1), pages 195-209, March.
  • Handle: RePEc:oup:biomet:v:91:y:2004:i:1:p:195-209

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    References listed on IDEAS

    1. Kai-tai Fang & Rahul Mukerjee, 2005. "Expected lengths of confidence intervals based on empirical discrepancy statistics," Biometrika, Biometrika Trust, vol. 92(2), pages 499-503, June.
    2. Smith, Richard J, 1997. "Alternative Semi-parametric Likelihood Approaches to Generalised Method of Moments Estimation," Economic Journal, Royal Economic Society, vol. 107(441), pages 503-519, March.
    3. J. N. K. Rao & Changbao Wu, 2010. "Bayesian pseudo-empirical-likelihood intervals for complex surveys," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(4), pages 533-544.
    4. Nicole A. Lazar, 2003. "Bayesian empirical likelihood," Biometrika, Biometrika Trust, vol. 90(2), pages 319-326, June.
    5. Zellner, Arnold, 1988. "Bayesian analysis in econometrics," Journal of Econometrics, Elsevier, vol. 37(1), pages 27-50, January.
    6. J. Chen, 2002. "Using empirical likelihood methods to obtain range restricted weights in regression estimators for surveys," Biometrika, Biometrika Trust, vol. 89(1), pages 230-237, March.
    7. Susanne M. Schennach, 2005. "Bayesian exponentially tilted empirical likelihood," Biometrika, Biometrika Trust, vol. 92(1), pages 31-46, March.
    8. Jiang, Jiming & Lahiri, P., 2006. "Estimation of Finite Population Domain Means: A Model-Assisted Empirical Best Prediction Approach," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 301-311, March.
    9. Susanne M. Schennach, 2007. "Point estimation with exponentially tilted empirical likelihood," Papers 0708.1874,
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    Cited by:

    1. repec:spr:annopr:v:260:y:2018:i:1:d:10.1007_s10479-017-2659-0 is not listed on IDEAS
    2. Yongmiao Hong, 2013. "Serial Correlation and Serial Dependence," WISE Working Papers 2013-10-14, Wang Yanan Institute for Studies in Economics (WISE), Xiamen University.
    3. repec:wyi:journl:002087 is not listed on IDEAS
    4. Aït-Sahalia, Yacine & Fan, Jianqing & Peng, Heng, 2009. "Nonparametric Transition-Based Tests for Jump Diffusions," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1102-1116.
    5. Zhang, Wenyang & Peng, Heng, 2010. "Simultaneous confidence band and hypothesis test in generalised varying-coefficient models," Journal of Multivariate Analysis, Elsevier, vol. 101(7), pages 1656-1680, August.
    6. Sonia Díaz & José Vilar, 2010. "Comparing Several Parametric and Nonparametric Approaches to Time Series Clustering: A Simulation Study," Journal of Classification, Springer;The Classification Society, vol. 27(3), pages 333-362, November.
    7. Zhang, Riquan & Huang, Zhensheng & Lv, Yazhao, 2010. "Statistical inference for the index parameter in single-index models," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 1026-1041, April.
    8. Ip, Wai-Cheung & Wong, Heung & Zhang, Riquan, 2007. "Generalized likelihood ratio test for varying-coefficient models with different smoothing variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4543-4561, May.
    9. 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.
    10. Zhu, Ke & Li, Wai Keung, 2015. "A bootstrapped spectral test for adequacy in weak ARMA models," Journal of Econometrics, Elsevier, vol. 187(1), pages 113-130.
    11. 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.

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