Electronic Journal of Statistics

Some properties of the autoregressive-aided block bootstrap

Tobias Niebuhr, Jens-Peter Kreiss, and Efstathios Paparoditis

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We investigate properties of a hybrid bootstrap procedure for general, strictly stationary sequences, called the autoregressive-aided block bootstrap which combines a parametric autoregressive bootstrap with a nonparametric moving block bootstrap. The autoregressive-aided block bootstrap consists of two main steps, namely an autoregressive model fit and an ensuing (moving) block resampling of residuals. The linear parametric model-fit prewhitenes the time series so that the dependence structure of the remaining residuals gets closer to that of a white noise sequence, while the moving block bootstrap applied to these residuals captures nonlinear features that are not taken into account by the linear autoregressive fit. We establish validity of the autoregressive-aided block bootstrap for the important class of statistics known as generalized means which includes many commonly used statistics in time series analysis as special cases. Numerical investigations show that the hybrid bootstrap procedure considered in this paper performs quite well, it behaves as good as or it outperforms in many cases the ordinary moving block bootstrap and it is robust against mis-specifications of the autoregressive order, a substantial advantage over the autoregressive bootstrap.

Article information

Electron. J. Statist., Volume 11, Number 1 (2017), 725-751.

Received: April 2016
First available in Project Euclid: 8 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F40: Bootstrap, jackknife and other resampling methods 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62P20: Applications to economics [See also 91Bxx] 91B84: Economic time series analysis [See also 62M10]

Block bootstrap weak ARMA CARMA low-frequency observations

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Niebuhr, Tobias; Kreiss, Jens-Peter; Paparoditis, Efstathios. Some properties of the autoregressive-aided block bootstrap. Electron. J. Statist. 11 (2017), no. 1, 725--751. doi:10.1214/17-EJS1239. https://projecteuclid.org/euclid.ejs/1488964115

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  • [1] Akaike, H. (1973): Information theory and an extension of the maximum likelihood principle. In: B. N. Petrov and F. Csaki (eds.), 2nd International Symposium on Information Theory, pp. 267–281. Akademiai Kiado, Budapest.
  • [2] Akaike, H. (1974): A new look at the statistical model identification., IEEE Transactions on Automatic Control, Vol. 19, pp. 716–723.
  • [3] Billingsley, P. (1979):, Probability and Measure. John Wiley and Sons, New York.
  • [4] Bose, A. (1988): Edgeworth correction by bootstrap in autoregressions., The Annals of Statistics, Vol. 16, pp. 1709–1722.
  • [5] Brockwell, P.J. and Davis, R.A. (1991):, Time Series: Theory and Methods (2nd Ed.). Springer, New York.
  • [6] Brockwell, P.J., Davis, R.A. and Yang, Y. (2010): Estimation for Non-Negative Lévy-Driven CARMA Processes., Journal of Business and Economic Statistics, Vol. 29, pp. 250–259.
  • [7] Broersen, P.M.T. and de Waele, S. (2004): Finite Sample Properties of ARMA order selection., IEEE, Vol. 53, No. 3.
  • [8] Bühlmann, P. (1994): Blockwise bootstrapped empirical processes for stationary sequences., The Annals of Statistics, Vol. 22, pp. 995–1012.
  • [9] Bühlmann, P. (1997): Sieve Bootstrap for Time Series., Bernoulli, Vol. 3, pp. 123–148.
  • [10] Bühlmann, P. and Künsch, H.R. (1995). The blockwise bootstrap for general parameters of a stationary time series., Scandinavian Journal of Statistics, Vol. 22, pp. 35–54.
  • [11] Bühlmann, P. and Künsch, H.R. (1999). Block length selection in the bootstrap for time series., Computational Statistics and Data Analysis, Vol. 31, pp. 295–310.
  • [12] Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary time series., The Annals of Statistics, Vol. 14, pp. 1171–1179.
  • [13] Carlstein, E., Do, K.-A., Hall, P. Hesterberg, T. and Künsch, H.R. (1998). Matched-block bootstrap for dependent data., Bernoulli, Vol. 4, pp. 305–328.
  • [14] Davison, A.C. and Hinkley, D.V. (1997):, Bootstrap methods and their application. Cambridge University Press, Cambridge.
  • [15] Efron, B. (1979). Bootstrap methods: another look at the jackknife., The Annals of Statistics, Vol. 7, pp. 1–26.
  • [16] Efron, B. and Tibshirani, R.J. (1986). Bootstrap methods for standard errors, confidence intervals and other measures of statistical accuracy (with discussion)., Statistical Science, Vol. 1, pp. 54–77.
  • [17] Fischer, W. and Lieb, I. (1988):, Funktionentheorie (5th Ed.). Vieweg, Wiesbaden.
  • [18] Freedman, D. (1984): On bootstrapping two-stage least squares estimates in stationary linear models., The Annals of Statistics, Vol. 12, pp. 827–842.
  • [19] Götze, F., and Künsch, H. (1996):, Second-order correctness of the blockwise bootstrap for stationary observations. The Annals of Statistics, Vol. 24, pp. 1914–1933.
  • [20] Hall, P., Horowitz, J. L., and Jing, B.-Y. (1995):, On blocking rules for the bootstrap with dependent data. Biometrika, Vol. 82, pp.561–574.
  • [21] Hannan, E.J. (1980): The estimation of the order of an ARMA process., The Annals of Statistics, Vol. 8, pp. 1071–1081.
  • [22] Kreiss, J.-P. and Franke, J. (1992): Bootstrapping stationary autoregressive moving average models., Journal of Time Series Analysis, Vol. 13, pp. 297–319.
  • [23] Kreiss, J.-P. und Neuhaus, G. (2006):, Einführung in die Zeitreihenanalyse. Springer, Heidelberg.
  • [24] Kreiss, J.-P. and Paparoditis, E. (2012): The hybrid wild bootstrap for time series., Journal of the American Statistical Association, Vol. 107, No. 499, pp. 1073–1084.
  • [25] Kreiss, J.-P. and Paparoditis, E. (2003): Autoregressive-aided periodogram bootstrap for time series., The Annals of Statistics, Vol. 31, pp. 1923–1955.
  • [26] Künsch, H.R. (1989). The jackknife and the bootstrap for general stationary observations., The Annals of Statistics, Vol. 17, pp. 1217–1241.
  • [27] Lahiri, S.N. (1993). On the moving block bootstrap under long range dependence., Statistics and Probability Letters, Vol. 18, pp. 405–413.
  • [28] Lahiri, S.N. (1999). Theoretical comparisons of block bootstrap methods., The Annals of Statistics, Vol. 27, pp. 386–404.
  • [29] Lahiri, S.N. (2003). Resampling methods for dependent data. New York:, Springer.
  • [30] Lahiri, S.N., Furukawa, K., and Lee, Y.-D. (2007). A nonparametric plug-in method for selecting the optimal block length., Statistical Methodology, Vol. 4, pp. 292–321.
  • [31] Liang, G., Wilkes, D.M. and Cadzow, J.A. (1993). ARMA model order estimation based on the eigenvalues of the covariance matrix., IEEE transactions on signal processing, Vol. 41, pp. 3003–3009.
  • [32] Liu, R.Y. and Singh, K. (1992). Moving blocks Jackknife and Bootstrap capture weak dependence. In, Exploring the Limits of Bootstrap, eds. R. lePage and L. Billard. New York, Wiley, pp. 225–248.
  • [33] Naik-Nimbalkar, U.V. and Rajarshi, M.B. (1994): Validity of blockwise bootstrap for empirical processes with stationary observations., The Annals of Statistics, Vol. 22, pp. 980–994.
  • [34] Niebuhr, T. and Kreiss, J.-P. (2014): Asymptotics for Autocovariances and Integrated Periodograms for Linear Processes Observed at Lower Frequencies., The International Statistical Review, Vol. 82, No. 1, pp. 123–140.
  • [35] Nordman, D. and Lahiri, S. (2014): Convergence rates of empirical block length selectors for block bootstrap., Bernoulli, Vol. 20, No. 2, pp. 958–978.
  • [36] Paparoditis, E. and Politis, D.N. (2001): The tapered block bootstrap., Biometrika, Vol. 88, pp. 1105–1119.
  • [37] Patton, A., Politis, D.N. and White, H. (2009): Correction to ’Automatic block length selection for the dependent bootstrap by D.N. Politis and H. White’., Econometric Reviews, Vol. 28, pp. 372–375.
  • [38] Politis, D.N. and Romano, J.P. (1992): A circular block resampling procedure for stationary data. In, Exploring the Limits of Bootstrap, eds. R. lePage and L. Billard. New York, Wiley, pp. 263–270.
  • [39] Politis, D.N. and White, H. (2004): Automatic block length selection for the dependent bootstrap., Econometric Reviews, Vol. 23, pp. 53–70.
  • [40] Press, H. and Tukey, J.W. (1956): Power spectral methods of analysis and their application to problems in airplane dynamics., Bell System Monographs 2606.
  • [41] Romano, J.P. and Wolf, M. (2000): A more general central limit theorem for $m$-dependent random variables with unbounded $m$., Statistics and Probability Letters, Vol. 47, pp. 115–124.
  • [42] Shao, Q.M. and Yu, H. (1993): Bootstrapping the sample means for stationary mixing sequences., Stoch. Processes Applic., Vol. 48, pp. 175–190.
  • [43] Shao, X. (2010): The dependent wild bootstrap., Journal of the American Statistical Association, Vol. 105, pp. 218–235.