Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-02T12:37:46.319Z Has data issue: false hasContentIssue false
  • English
  • Français

THE CONTINUITY OF THE LIMIT DISTRIBUTION IN THE PARAMETER OF INTEREST IS NOT ESSENTIAL FOR THE VALIDITY OF THE BOOTSTRAP

Published online by Cambridge University Press:  24 September 2003

Atsushi Inoue  and
Lutz Kilian
Atsushi Inoue
Affiliation:
North Carolina State University
Lutz Kilian
Affiliation:
University of Michigan, European Central Bank, and CEPR
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that the unrestricted bootstrap estimator of the slope parameter in the random walk model without drift converges to a random distribution. This bootstrap failure is commonly attributed to the discontinuity of the limit distribution of the least-squares estimator in the parameter of interest. We demonstrate by counterexample that this type of continuity is not essential for the validity of the bootstrap nor is it essential that the rate of convergence of the estimator remain constant over the whole parameter space.We thank Don Andrews, Shinichi Sakata, Jonathan Wright, and two anonymous referees for very helpful comments. The views expressed in this paper do not necessarily reflect those of the European Central Bank or its members.

Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 6 , December 2003 , pp. 944 - 961
Copyright
© 2003 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D.W.K. (2000) Inconsistency of the bootstrap when a parameter is on the boundary of the parameter space. Econometrica 68, 399–405.Google Scholar
Basawa, I.V., A.K. Mallik, W.P. McCormick, J.H. Reeves, & R.L. Taylor (1991a) Bootstrapping unstable first-order autoregressive processes. Annals of Statistics 19, 1098–1101.Google Scholar
Basawa, I.V., A.K. Mallik, W.P. McCormick, J.H. Reeves, & R.L. Taylor (1991b) Bootstrap test of significance and sequential bootstrap estimation for unstable first-order autoregressive processes. Communications in Statistical Theory and Methods 20, 1015–1026.Google Scholar
Benkwitz, A., H. Lütkepohl, & M.H. Neumann (2000) Problems related to confidence intervals for impulse responses of autoregressive processes. Econometric Reviews 19, 69–103.Google Scholar
Beran, R. (1997) Diagnosing bootstrap success. Annals of the Institute of Statistical Mathematics 49, 1–24.Google Scholar
Beran, R. & G.R. Ducharme (1991) Asymptotic Theory for Bootstrap Methods in Statistics. Les Publications CRM, Centre de recherches mathematiques. Montreal: University of Montreal.
Bickel, P.J. & D.A. Freedman (1981) Some asymptotic theory for the bootstrap. Annals of Statistics 9, 1196–1217.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.
Bose, A. (1988) Edgeworth correction by bootstrap in autoregression. Annals of Statistics 16, 1709–1722.Google Scholar
Datta, S. (1996) On asymptotic properties of bootstrap for AR(1) processes. Journal of Statistical Planning and Inference 53, 361–374.Google Scholar
Dickey, D.A. & W.A. Fuller (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Efron, B. & R.J. Tibshirani (1993) An Introduction to the Bootstrap. New York: Chapman and Hall.
Hamilton, J.D. (1994) Time Series Analysis. Princeton: Princeton University Press.
Inoue, A. & L. Kilian (2002) Bootstrapping autoregressions with possible unit roots. Econometrica 70, 377–391.Google Scholar
Kilian, L. (1998) Small-sample confidence intervals for impulse response functions. Review of Economics and Statistics 80, 218–230.Google Scholar
Parthasarathy, K.R. (1967) Probability Measures on Metric Spaces. New York: Academic Press.
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.Google Scholar
Politis, D.N., J.P. Romano, & M. Wolf (1999) Subsampling. New York: Springer-Verlag.
Romano, J.P & M. Wolf (2001) Subsampling intervals in autoregressive models with linear time trend. Econometrica 69, 1283–1314.Google Scholar
Zaman, A. (1996) Statistical Foundations for Econometric Techniques. San Diego: Academic Press.