Skip to main content
×
×
Home

BLOCK BOOTSTRAP CONSISTENCY UNDER WEAK ASSUMPTIONS

  • Gray Calhoun (a1)
Abstract

This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e., the moving blocks, circular blocks, and stationary bootstraps) to be valid for the sample mean of Near-Epoch-Dependent (NED) functions of mixing processes; they are consistent under the weakest conditions that ensure the original NED process obeys a central limit theorem (CLT), established by De Jong (1997, Econometric Theory 13(3), 353–367). In doing so, this paper extends De Jong’s method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a functional CLT (FCLT) under the same conditions.

Copyright
Corresponding author
*Address correspondence to Gray Calhoun, Economics Department, Iowa State University, Ames, IA 50011, USA; e-mail: gcalhoun@iastate.edu, url: http://gray.clhn.org.
Footnotes
Hide All

I would like to thank Helle Bunzel, Dimitris Politis, Robert Taylor, and three anonymous referees for their comments and feedback on earlier versions of this paper.

Footnotes
References
Hide All
Bai J. (1994) Least squares estimation of a shift in linear processes. Journal of Time Series Analysis 15(5), 453472.
Chow Y., Robbins H., & Teicher H. (1965) Moments of randomly stopped sums. The Annals of Mathematical Statistics 36(3), 789799.
Dahlhaus R. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 379381.
Davidson J. (1993) An L 1-convergence theorem for heterogeneous mixingale arrays with trending moments. Statistics & Probability Letters 16(4), 301304.
Davidson J. (1994) Stochastic Limit Theory: An Introduction for Econometricians. Advanced Texts in Econometrics. Oxford University Press.
De Jong R.M. (1997) Central limit theorems for dependent heterogeneous random variables. Econometric Theory 13(3), 353367.
De Jong R.M. & Davidson J. (2000) The functional central limit theorem and weak convergence to stochastic integrals I: Weakly dependent processes. Econometric Theory 16(5), 621642.
Dowla A., Paparoditis E., & Politis D.N. (2003) Locally stationary processes and the Local Block Bootstrap. In Akritas M.G. & Politis D.N. (eds.), Recent Advances and Trends in Nonparametric Statistics, pp. 437444. Elsevier (North Holland).
Gonçalves S. & De Jong R.M. (2003) Consistency of the stationary bootstrap under weak moment conditions. Economics Letters 81(2), 273278.
Gonçalves S. & Politis D. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 383386.
Gonçalves S. & White H. (2002) The bootstrap of the mean for dependent heterogeneous arrays. Econometric Theory 18(6), 13671384.
Hall P. & Heyde C. (1980) Martingale Limit Theory and its Application. Academic Press.
Horowitz J.L. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 387389.
Jentsch C. & Mammen E. (2011) Discussion: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 391392.
Kreiss J.-P. & Paparoditis E. (2011a) Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 357378.
Kreiss J.-P. & Paparoditis E. (2011b) Rejoinder: Bootstrap methods for dependent data: A review. Journal of the Korean Statistical Society 40(4), 393395.
Kunsch H.R. (1989) The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17(3), 12171241.
Lahiri S. (1999) Theoretical comparisons of block bootstrap methods. The Annals of Statistics 27(1), 386404.
Liu R.Y. & Singh K. (1992) Moving blocks Jackknife and Bootsrap capture weak dependence. In LePage R. & Billard L. (eds.), Exploring the Limits of Bootstrap, pp. 225248. John Wiley.
McLeish D. (1975) Invariance principles for dependent variables. Probability Theory and Related Fields 32, 165178.
McLeish D. (1977) On the invariance principle for nonstationary mixingales. The Annals of Probability 5(4), 616621.
Nordman D. (2009) A note on the stationary bootstrap’s variance. The Annals of Statistics 37(1), 359370.
Paparoditis E. & Politis D. (2002) Local block bootstrap. Comptes Rendus Mathematique 335(11), 959962.
Politis D.N. & Romano J.P. (1992) A circular block resampling procedure for stationary data. In Page R. & LePage R. (eds.), Exploring the Limits of Bootstrap, pp. 263270. John Wiley.
Politis D.N. & Romano J.P. (1994) The stationary bootstrap. Journal of the American Statistical Association 89(428), 13031313.
Radulović D. (1996) The bootstrap of the mean for strong mixing sequences under minimal conditions. Statistics & Probability Letters 28(1), 6572.
Van der Vaart A.W. (2000) Asymptotic Statistics. Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
Type Description Title
PDF
Supplementary materials

Calhoun supplementary material
Appendix

 PDF (320 KB)
320 KB
UNKNOWN
Supplementary materials

Calhoun supplementary material
Appendix

 Unknown (21 KB)
21 KB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 12 *
Loading metrics...

Abstract views

Total abstract views: 52 *
Loading metrics...

* Views captured on Cambridge Core between 1st February 2018 - 25th February 2018. This data will be updated every 24 hours.