zbMATH — the first resource for mathematics

On the range of validity of the autoregressive sieve bootstrap. (English) Zbl 1227.62067
Summary: 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.

MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62M15 Inference from stochastic processes and spectral analysis 62G09 Nonparametric statistical resampling methods 62G20 Asymptotic properties of nonparametric inference
Keywords:
autoregression; time series
Full Text:
References:
 [1] Andrews, B., Davis, R. A. and Breidt, F. J. (2007). Rank-based estimation for all-pass time series models. Ann. Statist. 35 844-869. · Zbl 1117.62089 [2] Baxter, G. (1962). An asymptotic result for the finite predictor. Math. Scand. 10 137-144. · Zbl 0112.09401 [3] Baxter, G. (1963). A norm inequality for a “finite-section” Wiener-Hopf equation. Illinois J. Math. 7 97-103. · Zbl 0113.09101 [4] 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. · Zbl 0954.62102 [5] 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. · Zbl 0753.62058 [6] 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. · Zbl 0813.62084 [7] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods , 2nd ed. Springer, New York. · Zbl 0709.62080 [8] Bühlmann, P. (1995). Sieve bootstrap for time series. Technical Report 431, Dept. Statistics, Univ. California, Berkeley. [9] Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli 3 123-148. · Zbl 0874.62102 [10] Bühlmann, P. (2002). Bootstraps for time series. Statist. Sci. 17 52-72. · Zbl 1013.62048 [11] 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. · Zbl 0966.62027 [12] Dahlhaus, R. (1985). Asymptotic normality of spectral estimates. J. Multivariate Anal. 16 412-431. · Zbl 0579.62082 [13] Dahlhaus, R. and Janas, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis. Ann. Statist. 24 1934-1963. · Zbl 0867.62072 [14] Doukhan, P. and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 313-342. · Zbl 0996.60020 [15] Franke, J. and Härdle, W. (1992). On bootstrapping kernel spectral estimates. Ann. Statist. 20 121-145. · Zbl 0757.62048 [16] Freedman, D. (1984). On bootstrapping two-stage least-squares estimates in stationary linear models. Ann. Statist. 12 827-842. · Zbl 0542.62051 [17] 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. · Zbl 1132.42313 [18] 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. · Zbl 1220.62107 [19] Kokoszka, P. and Politis, D. N. (2011). Nonlinearity of ARCH and stochastic volatility models and Bartlett’s formula. Probab. Math. Statist. · Zbl 1260.62068 [20] Kreiss, J. P. (1988). Asymptotical inference for a class of stochastic processes. Habilitationsschrift, Univ. Hamburg. [21] Kreiss, J.-P. (1992). Bootstrap procedures for AR(\infty )-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. [22] Kreiss, J. P. and Neuhaus, G. (2006). Einführung in die Zeitreihenanalyse . Springer, Heidelberg. · Zbl 1099.62101 [23] Kreiss, J.-P. and Paparoditis, E. (2003). Autoregressive-aided periodogram bootstrap for time series. Ann. Statist. 31 1923-1955. · Zbl 1042.62081 [24] Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217-1241. · Zbl 0684.62035 [25] Lahiri, S. N. (2003). Resampling Methods for Dependent Data . Springer, New York. · Zbl 1028.62002 [26] 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. · Zbl 0512.62090 [27] Lii, K.-S. and Rosenblatt, M. (1996). Maximum likelihood estimation for non-Gaussian nonminimum phase ARMA sequences. Statist. Sinica 6 1-22. · Zbl 0839.62085 [28] Paparoditis, E. (1996). Bootstrapping autoregressive and moving average parameter estimates of infinite order vector autoregressive processes. J. Multivariate Anal. 57 277-296. · Zbl 0863.62078 [29] 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. · Zbl 1178.62046 [30] 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. · Zbl 0752.62066 [31] Politis, D. N. (2003). The impact of bootstrap methods on time series analysis: Silver anniversary of the bootstrap. Statist. Sci. 18 219-230. · Zbl 1332.62340 [32] 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. · Zbl 0943.60003 [33] Poskitt, D. S. (2008). Properties of the sieve bootstrap for fractionally integrated and non-invertible processes. J. Time Series Anal. 29 224-250. · Zbl 1164.62053 [34] Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory . Wiley, New York. · Zbl 0982.62074 [35] Priestley, M. B. (1981). Spectral Analysis and Time Series . Academic Press, New York. · Zbl 0537.62075 [36] Romano, J. P. and Thombs, L. A. (1996). Inference for autocorrelations under weak assumptions. J. Amer. Statist. Assoc. 91 590-600. · Zbl 0868.62071 [37] Shao, X. and Wu, W. B. (2007). Asymptotic spectral theory for nonlinear time series. Ann. Statist. 35 1773-1801. · Zbl 1147.62076 [38] Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series . Springer, New York. · Zbl 0955.62088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.