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Frequency domain bootstrap for ratio statistics under long-range dependence

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Abstract

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates - especially normalized periodogram ordinates - by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.

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References

  • Adenstedt, R. K. (1974). On large-sample estimation for the mean of a stationary sequence. The Annals of Statistics, 2, 1095–1107.

    Article  MathSciNet  MATH  Google Scholar 

  • Beran, J. (1994). Statistical methods for long memory processes. London: Chapman & Hall.

    MATH  Google Scholar 

  • Beran, J., Feng, Y., Ghosh, S., & Kulik, R. (2013). Long-memory processes: probabilistic properties and statistical methods. Berlin Heidelberg: Springer-Verlag.

    Book  MATH  Google Scholar 

  • Bhansali, R. J., Giraitis, L., & Kokoszka, P. S. (2006). Estimation of the memory parameter by fitting fractionally differenced autoregressive models. Journal of Multivariate Analysis, 94, 2101–2130.

    Article  MathSciNet  MATH  Google Scholar 

  • Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli, 3, 123–148.

    Article  MathSciNet  MATH  Google Scholar 

  • Carlstein, E. (1986). The use of subseries methods for estimating the variance of a general statistic from a stationary time series. The Annals of Statistics, 14, 1171–1179.

    Article  MathSciNet  MATH  Google Scholar 

  • Dahlhaus, R. (1983). Spectral analysis with tapered data. Journal of Time Series Analysis, 4, 163–175.

    Article  MathSciNet  MATH  Google Scholar 

  • Dahlhaus, R., & Janas, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis. The Annals of Statistics, 24, 1934–1963.

    Article  MathSciNet  MATH  Google Scholar 

  • Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics, 7, 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  • Fox, R., & Taqqu, M. S. (1986). Large sample properties of parameter estimates for strongly dependent stationary gaussian time series. The Annals of Statistics, 14, 517–532.

    Article  MathSciNet  MATH  Google Scholar 

  • Franke, J., & Härdle, W. (1992). On bootstrapping kernel spectral estimates. The Annals of Statistics, 20, 121–145.

    Article  MathSciNet  MATH  Google Scholar 

  • Giraitis, L., & Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probability Theory and Related Field, 86, 87–104.

    Article  MathSciNet  MATH  Google Scholar 

  • Granger, C. W. J., & Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis, 1, 15–29.

    Article  MathSciNet  MATH  Google Scholar 

  • Hannan, E. J. (1973). The asymptotic theory of linear time-series models. Journal of Applied Probability, 10, 130–145.

    Article  MathSciNet  MATH  Google Scholar 

  • Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68, 165–176.

    Article  MathSciNet  MATH  Google Scholar 

  • Hurvich, C. M., & Brodsky, J. (2001). Broadband semiparametric estimation of the memory parameter of a long-memory time series using fractional exponential models. Journal of Time Series Analysis, 22, 221–249.

    Article  MathSciNet  MATH  Google Scholar 

  • Hyndman, R. J. (2015). Discussion of high-dimensional autocovariance matrices and optimal linear prediction. Electronic Journal of Statistics, 9, 792–796.

    Article  MathSciNet  MATH  Google Scholar 

  • Jones, P. D., & Briffa, K. R. (1992). Global surface air temperature variations during the twentieth century, part 1. The Holocene, 2, 165–179.

    Article  Google Scholar 

  • Kapetanios, G., & Psaradakis, Z. (2006). Sieve bootstrap for strongly dependent stationary processes. In Working paper 552. Dept. of Economics, Queen Mary, University of London.

    Google Scholar 

  • Kim, Y.-M., Lahiri, S. N., & Nordman, D. J. (2018). Nonparametric spectral density estimation under long-range dependence. Journal of Time Series Analysis, 39(3), 380–401. http://dx.doi.org/10.1111/jtsa.12284.

    Article  MathSciNet  MATH  Google Scholar 

  • Kim, Y.-M., & Nordman, D. J. (2011). Properties of a block bootstrap under long-range dependence. Sankhya, 73(1), 79–109.

    Article  MathSciNet  MATH  Google Scholar 

  • Kim, Y.-M., & Nordman, D. J. (2013). A frequency domain bootstrap on whittle estimation under long-range dependence. Journal of Multivariate Analysis, 115, 405–420.

    Article  MathSciNet  MATH  Google Scholar 

  • Künsch, H. R. (1989). The jackknife and bootstrap for general stationary observations. The Annals of Statistics, 17, 1217–1261.

    Article  MathSciNet  MATH  Google Scholar 

  • Lahiri, S. N. (1993). On the moving block bootstrap under long range dependence. Statistics & Probability Letters, 18, 405–413.

    Article  MathSciNet  MATH  Google Scholar 

  • Mandelbrot, B. B., & Van Ness, J. W. (1968). Fractional brownian motions, fractional noises and applications. SIAM Review, 10, 422–437.

    Article  MathSciNet  MATH  Google Scholar 

  • McMurry, T. L., & Politis, D. N. (2010). Banded and tapered estimates for autocovariance matrices and the linear process bootstrap. Journal of Time Series Analysis, 31(6), 471–482.

    Article  MathSciNet  MATH  Google Scholar 

  • Moulines, E., & Soulier, P. (1999). Broadband log-periodogram regression of time series with long-range dependence. The Annals of Statistics, 27, 1415–1439.

    Article  MathSciNet  MATH  Google Scholar 

  • Moulines, E., & Soulier, P. (2000). Data driven order selection for projection estimator of the spectral density of time series with long range dependence. Journal of Time Series Analysis, 21, 193–218.

    Article  MathSciNet  MATH  Google Scholar 

  • Narukawa, M., & Matsuda, Y. (2011). Broadband semi-parametric estimation of long-memory time series by fractional exponential models. Journal of Time Series Analysis, 32, 175–193.

    Article  MathSciNet  MATH  Google Scholar 

  • Nordgaard, A. (1992). Lecture notes in economic and mathematical systems: Vol. 376, Resampling stochastic processes using a bootstrap approach (pp. 181–185).

    Google Scholar 

  • Nordman, D. J., & Lahiri, S. N. (2006). A frequency domain empirical likelihood for short- and long-range dependence. The Annals of Statistics, 34, 3019–3050.

    Article  MathSciNet  MATH  Google Scholar 

  • Palma, W. (2007). Long-memory time series: theory and methods. New Jersey: Wiley.

    Book  MATH  Google Scholar 

  • Paparoditis, E. (2002). Frequency domain bootstrap for time series. In Empirical process techniques for dependent data (pp. 365–381). Springer.

    Chapter  MATH  Google Scholar 

  • Pollak, M., Croarkin, C., & Hagewood, C. (1993). Surveillance schemes with applications to mass calibration, Vol. 5158. U.S. Department of Commerce, National Institute of Standards and Technology.

    Google Scholar 

  • Poskitt, D. S. (2008). Properties of the sieve bootstrap for fractionally integrated and non-invertible processes. Journal of Time Series Analysis, 2992, 224–250.

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, P. M. (1995a). Gaussian semiparametric estimation of long range dependence. The Annals of Statistics, 23, 1630–1661.

    Article  MathSciNet  MATH  Google Scholar 

  • Singh, K. (1981). On the asymptotic accuracy of the Efron’s bootstrap. The Annals of Statistics, 9, 1187–1195.

    Article  MathSciNet  MATH  Google Scholar 

Further reading

  • Andrews, D. W. K., & Guggenberger, P. (2003). A bias-reduced log-periodogram regression estimator for the long-memory parameter. Econometrica, 71, 675–712.

    Article  MathSciNet  MATH  Google Scholar 

  • Andrews, D. W. K., & Sun, Y. (2004). Adaptive local polynomial whittle estimation of long-range dependence. Econometrica, 72, 569–614.

    Article  MathSciNet  MATH  Google Scholar 

  • Geweke, J., & Porter-Hudak, S. (1983). The estimation and application of long-memory time series models. Journal of Time Series Analysis, 4, 221–238.

    Article  MathSciNet  MATH  Google Scholar 

  • Lahiri, S. N. (2003a). A necessary and sufficient condition for asymptotic independence of discrete fourier transforms under short- and long-range dependence. The Annals of Statistics, 31, 613–641.

    Article  MathSciNet  MATH  Google Scholar 

  • Lahiri, S. N. (2003b). Resampling methods for dependent data. New York: Springer.

    Book  MATH  Google Scholar 

  • Robinson, P. M. (1995b). Log-periodogram regression of time series with long range dependence. The Annals of Statistics, 23, 1048–1072.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Kyungpook National University, Yonsei University, Korea

    Young Min Kim & Jongho Im

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  1. Young Min Kim
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  2. Jongho Im
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Correspondence to Young Min Kim.

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Kim, Y.M., Im, J. Frequency domain bootstrap for ratio statistics under long-range dependence. J. Korean Stat. Soc. 48, 547–560 (2019). https://doi.org/10.1016/j.jkss.2019.03.001

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