The Annals of Statistics

On the range of validity of the autoregressive sieve bootstrap

Jens-Peter Kreiss, Efstathios Paparoditis, and Dimitris N. Politis

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We explore the limits of the autoregressive (AR) sieve bootstrap, and show that its applicability extends well beyond the realm of linear time series as has been previously thought. In particular, for appropriate statistics, the AR-sieve bootstrap is valid for stationary processes possessing a general Wold-type autoregressive representation with respect to a white noise; in essence, this includes all stationary, purely nondeterministic processes, whose spectral density is everywhere positive. Our main theorem provides a simple and effective tool in assessing whether the AR-sieve bootstrap is asymptotically valid in any given situation. In effect, the large-sample distribution of the statistic in question must only depend on the first and second order moments of the process; prominent examples include the sample mean and the spectral density. As a counterexample, we show how the AR-sieve bootstrap is not always valid for the sample autocovariance even when the underlying process is linear.

Article information

Ann. Statist., Volume 39, Number 4 (2011), 2103-2130.

First available in Project Euclid: 26 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M15: Spectral analysis
Secondary: 62G09: Resampling methods

Autoregression bootstrap time series


Kreiss, Jens-Peter; Paparoditis, Efstathios; Politis, Dimitris N. On the range of validity of the autoregressive sieve bootstrap. Ann. Statist. 39 (2011), no. 4, 2103--2130. doi:10.1214/11-AOS900.

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  • Andrews, B., Davis, R. A. and Breidt, F. J. (2007). Rank-based estimation for all-pass time series models. Ann. Statist. 35 844–869.
  • Baxter, G. (1962). An asymptotic result for the finite predictor. Math. Scand. 10 137–144.
  • Baxter, G. (1963). A norm inequality for a “finite-section” Wiener–Hopf equation. Illinois J. Math. 7 97–103.
  • Bickel, P. J. and Bühlmann, P. (1999). A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli 5 413–446.
  • Breidt, F. J. and Davis, R. A. (1992). Time-reversibility, identifiability and independence of innovations for stationary time series. J. Time Series Anal. 13 377–390.
  • Breidt, F. J., Davis, R. A. and Dunsmuir, W. T. M. (1995). Improved bootstrap prediction intervals for autoregressions. J. Time Series Anal. 16 177–200.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Bühlmann, P. (1995). Sieve bootstrap for time series. Technical Report 431, Dept. Statistics, Univ. California, Berkeley.
  • Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli 3 123–148.
  • Bühlmann, P. (2002). Bootstraps for time series. Statist. Sci. 17 52–72.
  • Choi, E. and Hall, P. (2000). Bootstrap confidence regions computed from autoregressions of arbitrary order. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 461–477.
  • Dahlhaus, R. (1985). Asymptotic normality of spectral estimates. J. Multivariate Anal. 16 412–431.
  • Dahlhaus, R. and Janas, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis. Ann. Statist. 24 1934–1963.
  • Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 313–342.
  • Franke, J. and Härdle, W. (1992). On bootstrapping kernel spectral estimates. Ann. Statist. 20 121–145.
  • Freedman, D. (1984). On bootstrapping two-stage least-squares estimates in stationary linear models. Ann. Statist. 12 827–842.
  • Gröchenig, K. (2007). Weight functions in time–frequency analysis. In Pseudo-differential Operators: Partial Differential Equations and Time–Frequency Analysis (L. Rodino et al., eds.). Fields Inst. Commun. 52 343–366. Amer. Math. Soc., Providence, RI.
  • Kirch, C. and Politis, D. N. (2011). TFT-bootstrap: Resampling time series in the frequency domain to obtain replicates in the time domain. Ann. Statist. 39 1427–1470.
  • Kokoszka, P. and Politis, D. N. (2011). Nonlinearity of ARCH and stochastic volatility models and Bartlett’s formula. Probab. Math. Statist. To appear.
  • Kreiss, J. P. (1988). Asymptotical inference for a class of stochastic processes. Habilitationsschrift, Univ. Hamburg.
  • Kreiss, J.-P. (1992). Bootstrap procedures for AR(∞)-processes. In Bootstrapping and Related Techniques (Trier, 1990) (K. H. Jöckel, G. Rothe and W. Sendler, eds.). Lecture Notes in Econom. and Math. Systems 376 107–113. Springer, Berlin.
  • Kreiss, J. P. and Neuhaus, G. (2006). Einführung in die Zeitreihenanalyse. Springer, Heidelberg.
  • Kreiss, J.-P. and Paparoditis, E. (2003). Autoregressive-aided periodogram bootstrap for time series. Ann. Statist. 31 1923–1955.
  • Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217–1241.
  • Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer, New York.
  • Lii, K. S. and Rosenblatt, M. (1982). Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes. Ann. Statist. 10 1195–1208.
  • Lii, K.-S. and Rosenblatt, M. (1996). Maximum likelihood estimation for non-Gaussian nonminimum phase ARMA sequences. Statist. Sinica 6 1–22.
  • Paparoditis, E. (1996). Bootstrapping autoregressive and moving average parameter estimates of infinite order vector autoregressive processes. J. Multivariate Anal. 57 277–296.
  • Paparoditis, E. and Politis, D. N. (2009). Resampling and subsampling for financial time series. In Handbook of Financial Time Series (T. G. Andersen, R. A. Davis, J.-P. Kreiss and T. Mikosch, eds.) 983–999. Springer, Berlin.
  • Paparoditis, E. and Streitberg, B. (1992). Order identification statistics in stationary autoregressive moving-average models: Vector autocorrelations and the bootstrap. J. Time Series Anal. 13 415–434.
  • Politis, D. N. (2003). The impact of bootstrap methods on time series analysis: Silver anniversary of the bootstrap. Statist. Sci. 18 219–230.
  • Politis, D., Romano, J. P. and Wolf, M. (1999). Weak convergence of dependent empirical measures with application to subsampling in function spaces. J. Statist. Plann. Inference 79 179–190.
  • Poskitt, D. S. (2008). Properties of the sieve bootstrap for fractionally integrated and non-invertible processes. J. Time Series Anal. 29 224–250.
  • Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory. Wiley, New York.
  • Priestley, M. B. (1981). Spectral Analysis and Time Series. Academic Press, New York.
  • Romano, J. P. and Thombs, L. A. (1996). Inference for autocorrelations under weak assumptions. J. Amer. Statist. Assoc. 91 590–600.
  • Shao, X. and Wu, W. B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773–1801.
  • Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Springer, New York.