## Statistical Science

### The Impact of Bootstrap Methods on Time Series Analysis

Dimitris N. Politis

#### Abstract

Sparked by Efron's seminal paper, the decade of the 1980s was a period of active research on bootstrap methods for independent data--mainly i.i.d. or regression set-ups. By contrast, in the 1990s much research was directed towards resampling dependent data, for example, time series and random fields. Consequently, the availability of valid nonparametric inference procedures based on resampling and/or subsampling has freed practitioners from the necessity of resorting to simplifying assumptions such as normality or linearity that may be misleading.

#### Article information

Source
Statist. Sci., Volume 18, Issue 2 (2003), 219-230.

Dates
First available in Project Euclid: 19 September 2003

https://projecteuclid.org/euclid.ss/1063994977

Digital Object Identifier
doi:10.1214/ss/1063994977

Mathematical Reviews number (MathSciNet)
MR2026081

Zentralblatt MATH identifier
1332.62340

#### Citation

Politis, Dimitris N. The Impact of Bootstrap Methods on Time Series Analysis. Statist. Sci. 18 (2003), no. 2, 219--230. doi:10.1214/ss/1063994977. https://projecteuclid.org/euclid.ss/1063994977

#### References

• Arcones, M. A. (2001). On the asymptotic accuracy of the bootstrap under arbitrary resampling size. Ann. Inst. Statist. Math. To appear.
• Babu, G. J. and Singh, K. (1983). Inference on means using the bootstrap. Ann. Statist. 11 999--1003.
• Bartlett, M. S. (1946). On the theoretical specification and sampling properties of autocorrelated time-series. Suppl. J. Roy. Statist. Soc. 8 27--41.
• Bertail, P. and Politis, D. N. (2001). Extrapolation of subsampling distribution estimators: The i.i.d. and strong mixing cases. Canad. J. Statist. 29 667--680.
• Bickel, P. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196--1217.
• Bickel, P., Götze, F. and van Zwet, W. R. (1997). Resampling fewer than $n$ observations: Gains, losses, and remedies for losses. Statist. Sinica 7 1--32.
• Bollerslev, T., Chou, R. and Kroner, K. (1992). ARCH modelling in finance: A review of the theory and empirical evidence. J. Econometrics 52 5--59.
• Booth, J. G. and Hall, P. (1993). An improvement of the jackknife distribution function estimator. Ann. Statist. 21 1476--1485.
• Bose, A. (1988). Edgeworth correction by bootstrap in autoregressions. Ann. Statist. 16 1709--1722.
• Brockwell, P. and Davis, R. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
• Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli 3 123--148.
• Bühlmann, P. (2002). Bootstraps for time series. Statist. Sci. 17 52--72.
• Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary time series. Ann. Statist. 14 1171--1179.
• 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. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1--37.
• Davison, A. C. and Hall, P. (1993). On Studentizing and blocking methods for implementing the bootstrap with dependent data. Austral. J. Statist. 35 215--224.
• Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1--26.
• Efron, B. and Tibshirani, R. J. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy (with discussion). Statist. Sci. 1 54--77.
• Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York.
• Engle, R., ed. (1995). ARCH: Selected Readings. Oxford Univ. Press.
• Franke, J., Kreiss, J.-P. and Mammen, E. (2002). Bootstrap of kernel smoothing in nonlinear time series. Bernoulli 8 1--37.
• Freedman, D. A. (1981). Bootstrapping regression models. Ann. Statist. 9 1218--1228.
• Freedman, D. A. (1984). On bootstrapping two-stage least-squares estimates in stationary linear models. Ann. Statist. 12 827--842.
• Fuller, W. A. (1996). Introduction to Statistical Time Series, 2nd ed. Wiley, New York.
• Giné, E. and Zinn, J. (1990). Necessary conditions for the bootstrap of the mean. Ann. Statist. 17 684--691.
• Götze, F. and Künsch, H. (1996). Second-order correctness of the blockwise bootstrap for stationary observations. Ann. Statist. 24 1914--1933.
• Granger, C. and Andersen, A. (1978). An Introduction to Bilinear Time Series Models. Vandenhoeck und Ruprecht, Göttingen.
• Grenander, U. and Rosenblatt, M. (1957). Statistical Analysis of Stationary Time Series. Wiley, New York.
• Hall, P. (1985). Resampling a coverage pattern. Stochastic Process. Appl. 20 231--246.
• Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
• Hall, P., DiCiccio, T. J. and Romano, J. P. (1989). On smoothing and the bootstrap. Ann. Statist. 17 692--704.
• Hall, P., Horowitz, J. L. and Jing, B.-Y. (1995). On blocking rules for the bootstrap with dependent data. Biometrika 82 561--574.
• Hamilton, J. D. (1994). Time Series Analysis. Princeton Univ. Press.
• Härdle, W. and Bowman, A. (1988). Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bands. J. Amer. Statist. Assoc. 83 102--110.
• Horowitz, J. L. (2003). Bootstrap methods for Markov processes. Econometrica 71 1049--1082.
• Kreiss, J.-P. (1988). Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Faculty of Mathematics, Univ. Hamburg, Germany.
• Kreiss, J.-P. (1992). Bootstrap procedures for AR($\infty$) processes. In Bootstrapping and Related Techniques (K. H. Jöckel, G. Rothe and W. Sendler, eds.) 107--113. Springer, Berlin.
• Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217--1241.
• Lahiri, S. N. (1991). Second order optimality of stationary bootstrap. Statist. Probab. Lett. 11 335--341.
• Lahiri, S. N. (1999). Theoretical comparisons of block bootstrap methods. Ann. Statist. 27 386--404.
• Liu, R. Y. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap (R. LePage and L. Billard, eds.) 225--248. Wiley, New York.
• Masry, E. and Tjøstheim, D. (1995). Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory 11 258--289.
• Neumann, M. and Kreiss, J.-P. (1998). Regression-type inference in nonparametric autoregression. Ann. Statist. 26 1570--1613.
• Paparoditis, E. (1992). Bootstrapping some statistics useful in identifying ARMA models. In Bootstrapping and Related Techniques (K. H. Jöckel, G. Rothe and W. Sendler, eds.) 115--119. Springer, Berlin.
• Paparoditis, E. and Politis, D. N. (2000). The local bootstrap for kernel estimators under general dependence conditions. Ann. Inst. Statist. Math. 52 139--159.
• Paparoditis, E. and Politis, D. N. (2001a). Tapered block bootstrap. Biometrika 88 1105--1119.
• Paparoditis, E. and Politis, D. N. (2001b). A Markovian local resampling scheme for nonparametric estimators in time series analysis. Econometric Theory 17 540--566.
• Paparoditis, E. and Politis, D. N. (2001c). The continuous-path block-bootstrap. In Asymptotics in Statistics and Probability (M. Puri, ed.) 305--320. VSP Publications, Zeist, The Netherlands.
• Paparoditis, E. and Politis, D. N. (2002a). The local bootstrap for Markov processes. J. Statist. Plann. Inference 108 301--328.
• Paparoditis, E. and Politis, D. N. (2002b). The tapered block bootstrap for general statistics from stationary sequences. Econom. J. 5 131--148.
• Paparoditis, E. and Politis, D. N. (2002c). Local block bootstrap. C. R. Math. Acad. Sci. Paris. 335 959--962.
• Paparoditis, E. and Politis, D. N. (2003). Residual-based block bootstrap for unit root testing. Econometrica 71 813--855.
• Politis, D. N. (2001a). Resampling time series with seasonal components. In Frontiers in Data Mining and Bioinformatics: Proceedings of the 33rd Symposium on the Interface of Computing Science and Statistics.
• Politis, D. N. (2001b). Adaptive bandwidth choice. J. Nonparametr. Statist. To appear.
• Politis, D. N. and Romano, J. P. (1992a). A general resampling scheme for triangular arrays of $\alpha$-mixing random variables with application to the problem of spectral density estimation. Ann. Statist. 20 1985--2007.
• Politis, D. N. and Romano, J. P. (1992b). A circular block-resampling procedure for stationary data. In Exploring the Limits of Bootstrap (R. LePage and L. Billard, eds.) 263--270. Wiley, New York.
• Politis, D. N. and Romano, J. P. (1992c). A general theory for large sample confidence regions based on subsamples under minimal assumptions. Technical Report 399, Dept. Statistics, Stanford Univ.
• Politis, D. N. and Romano, J. P. (1993). Estimating the distribution of a Studentized statistic by subsampling. Bull. Internat. Statist. Inst. 2 315--316.
• Politis, D. N. and Romano, J. P. (1994a). The stationary bootstrap. J. Amer. Statist. Assoc. 89 1303--1313.
• Politis, D. N. and Romano, J. P. (1994b). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031--2050.
• Politis, D. N. and Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Ser. Anal. 16 67--103.
• Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
• Politis, D. N. and White, H. (2001). Automatic block-length selection for the dependent bootstrap. Econometric Rev. To appear.
• Quenouille, M. (1949). Approximate tests of correlation in time-series. J. Roy. Statist. Soc. Ser. B 11 68--84.
• Quenouille, M. (1956). Notes on bias in estimation. Biometrika 43 353--360.
• Radulovic, D. (1996). The bootstrap of the mean for strong mixing sequences under minimal conditions. Statist. Probab. Lett. 28 65--72.
• Rajarshi, M. B. (1990). Bootstrap in Markov sequences based on estimates of transition density. Ann. Inst. Statist. Math. 42 253--268.
• Romano, J. P. and Thombs, L. (1996). Inference for autocorrelations under weak assumptions. J. Amer. Statist. Assoc. 91 590--600.
• Sakov, A. and Bickel, P. (2000). An Edgeworth expansion for the $m$ out of $n$ bootstrapped median. Statist. Probab. Lett. 49 217--223.
• Shao, J. and Wu, C.-F. J. (1989). A general theory for jackknife variance estimation. Ann. Statist. 17 1176--1197.
• Sherman, M. and Carlstein, E. (1996). Replicate histograms. J. Amer. Statist. Assoc. 91 566--576.
• Shi, S. G. (1991). Local bootstrap. Ann. Inst. Statist. Math. 43 667--676.
• Shibata, R. (1976). Selection of the order of an autoregressive model by Akaike's information criterion. Biometrika 63 117--126.
• Singh, K. (1981). On the asymptotic accuracy of Efron's bootstrap. Ann. Statist. 9 1187--1195.
• Subba Rao, T. and Gabr, M. (1984). An Introduction to Bispectral Analysis and Bilinear Time Series Models. Lecture Notes in Statist. 24. Springer, New York.
• Swanepoel, J. W. H. (1986). A note on proving that the (modified) bootstrap works. Comm. Statist. Theory Methods. 15 3193--3203.
• Swanepoel, J. W. H. and van Wyk, J. W. J. (1986). The bootstrap applied to power spectral density function estimation. Biometrika 73 135--141.
• Tong, H. (1990). Non-linear Time Series: A Dynamical Systems Approach. Oxford Univ. Press.
• Tukey, J. W. (1958). Bias and confidence in not-quite large samples (abstract). Ann. Math. Statist. 29 614.
• Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis (with discussion). Ann. Statist. 14 1261--1350.