Hostname: page-component-89b8bd64d-sd5qd Total loading time: 0 Render date: 2026-05-12T04:03:01.976Z Has data issue: false hasContentIssue false
  • English
  • Français

THE BOOTSTRAP IN THRESHOLD REGRESSION

Published online by Cambridge University Press:  01 April 2014

Ping Yu
Ping Yu*
Affiliation:
University of Auckland
*
*Address correspondence and reprint requests to Ping Yu, Department of Economics, 12 Grafton Road, University of Auckland, Auckland, New Zealand. e-mail: p.yu@auckland.ac.nz.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper develops a general procedure to check the bootstrap validity in M-estimation. We apply the procedure in discontinuous threshold regression to show the inconsistency of the nonparametric bootstrap for inference on the threshold point. Especially, the conditional weak limit of the nonparametric bootstrap is shown not to exist. By comparing with two other boundaries in the literature, we show the fact that the threshold point is a boundary of the covariate that makes its bootstrap inference so different. The remedies to the bootstrap failure in the literature are summarized, and the nonparametric posterior interval is suggested by some simulation studies.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 3 , June 2014 , pp. 676 - 714
Copyright
Copyright © Cambridge University Press 2014 

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Abadie, A. & Imbens, G.W. (2008) On the failure of the bootstrap for matching estimators. Econometrica 76, 1537–1557.Google Scholar
Abrevaya, J. & Huang, J. (2005) On the bootstrap of the maximum score estimator. Econometrica 73, 1175–1204.Google Scholar
Aldous, D.J. (1978) Stopping times and tightness. The Annals of Probability 6, 335–340.Google Scholar
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
Antoch, J., et al. . (1995) Change-point problem and bootstrap. Journal of Nonparametric Statistics 5, 123–144.Google Scholar
Arcones, M.A. & Giné, E. (1992) On the bootstrap of M-estimators and other statistical functionals. In LePage, R. & Billard, L. (eds.), Exploring the Limits of Bootstrap, pp. 13–47. Wiley.Google Scholar
Beran, R. (1997) Diagnosing bootstrap success. Annals of the Institute of Statistical Mathematics 49, 1–24.CrossRefGoogle Scholar
Bickel, P.J., et al. . (1997) Resampling fewer than n observations: Gains, losses, and remedies for losses. Statistica Sinica 7, 1–31.Google Scholar
Bickel, P.J. & Freedman, D.A. (1981) Some asymptotic theory for the bootstrap. The Annals of Statistics 9, 1196–1217.Google Scholar
Billingsley, P. (1995) Probability and Measure. Wiley.Google Scholar
Bose, A. & Chatterjee, S. (2001) Generalised bootstrap in non-regular M-estimation problems. Statistics and Probability Letters 55, 319–328.Google Scholar
Botev, Z.I., et al. . (2010) Kernel density estimation via diffusion. The Annals of Statistics 38, 2916–2957.Google Scholar
Chan, K.S. (1993) Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. The Annals of Statistics 21, 520–533.Google Scholar
Cheng, G. & Huang, J.Z. (2010) Bootstrap consistency for general semiparametric M -estimation. The Annals of Statistics 38, 2884–2915.Google Scholar
Chernozhukov, V. & Hong, H. (2004) Likelihood estimation and inference in a class of nonregular econometric models. Econometrica 72, 1445–1480.Google Scholar
Delgado, M., et al. . (2001) Subsampling inference in cube root asymptotics with an application to Manski’s maximum score estimator. Economics Letters 73, 241–250.Google Scholar
Dümbgen, L. (1991) The asymptotic behavior of some nonparametric change-point estimators. The Annals of Statistics 19, 1471–1495.Google Scholar
Efron, B. (1979) Bootstrap methods: Another look at the jackknife. The Annals of Statistics 7, 1–26.Google Scholar
Gijbels, I., et al. . (2004) Interval and band estimation for curves with jumps. Journal of Applied Probability (Special Issue “Stochastic Methods and Their Applications”, in honor of Chris Heyde) 41A, 65–79.Google Scholar
Gonzalo, J. & Wolf, M. (2005) Subsampling inference in threshold autoregressive models. Journal of Econometrics 127, 201–224.Google Scholar
Hall, P., et al. . (1989) On smoothing and the bootstrap. The Annals of Statistics 17, 692–704.Google Scholar
Hansen, B.E. (2000) Sample splitting and threshold estimation. Econometrica 68, 575–603.Google Scholar
Hansen, B.E. (2008) Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726–748.Google Scholar
Hansen, B.E. (2011) Threshold autoregression in economics. Statistics and Its Interface 4, 123–127.Google Scholar
Hirano, K. & Porter, J.R. (2003) Asymptotic efficiency in parametric structural models with parameter-dependent support. Econometrica 71, 1307–1338.Google Scholar
Horowitz, J.L. (2001) The bootstrap. In Heckman, J.J. & Leamer, E.E. (eds.), Handbook of Econometrics, vol. 5, pp. 3159–3228. Elsevier Science B.V.Google Scholar
Hušková, M. & Kirch, C. (2008) Bootstrapping confidence intervals for the change-point of time series. Journal of Time Series Analysis 29, 947–972.CrossRefGoogle Scholar
Huang, J. & Wellner, J.A. (1995) Estimation of a monotone density or monotone hazard under random censoring. Scandinavian Journal of Statistics 22, 3–33.Google Scholar
Kac, M. (1949) On deviations between theoretical and empirical distributions. Proceedings of the National Academy of Sciences USA 35, 252–257.Google Scholar
Kim, J. & Pollard, D. (1990) Cube root asymptotics. The Annals of Statistics 18, 191–219.Google Scholar
Klaassen, C.A. & Wellner, J.A. (1992) Kac empirical processes and the bootstrap. In Hahn, M. & Kuelbs, J. (eds.), Proceedings of the Eighth International Conference on Probability in Banach Spaces, pp. 411–429. Birkhäuser.Google Scholar
Kosorok, M.R. (2008) Bootstrapping in Grenander estimator. In Beyond Parametrics in Interdisciplinary Research; Festschrift in Honor of Professor Pranab K. Sen, IMS Collections, vol. 1,pp. 282–292. IMS.Google Scholar
Kosorok, M.R. & Song, R. (2007) Inference under right censoring for transformation models with a change-point based on a covariate threshold. The Annals of Statistics 35, 957–989.Google Scholar
Lee, S.M.S & Pun, M.C. (2006) On m out of n bootstrapping for nonstandard M-estimation with nuisance parameters. Journal of the American Statistical Association 101, 1185–1197.Google Scholar
Lee, S. & Seo, M.H. (2008) Semiparametric estimation of a binary response model with a change-point due to a covariate threshold. Journal of Econometrics 144, 492–499.Google Scholar
Léger, C. & MacGibbon, B. (2006) On the bootstrap in cube root asymptotics. The Canadian Journal of Statistics 34, 29–44.Google Scholar
Li, D. & Ling, S.Q. (2012) On the least squares estimation of multiple-regime threshold autoregressive models. Journal of Econometrics 167, 240–253.Google Scholar
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057.Google Scholar
Praestgaard, J. & Wellner, J.A. (1993) Exchangeably weighted bootstraps of the general empirical process. The Annals of Probability 21, 2053–2086.Google Scholar
Politis, D., et al. . (1999) Subsampling. Springer-Verlag.Google Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag.Google Scholar
Romano, J.P. & Shaikh, A.M. (2008) Inference for identifiable parameters in partially identified econometric models. Journal of Statistical Planning and Inference 138, 2786–2807.Google Scholar
Rubin, D.B. (1981) The Bayesian bootstrap. The Annals of Statistics 9, 130–134.Google Scholar
Seijo, E. & Sen, B. (2011) Change-point in stochastic design regression and the bootstrap. The Annals of Statistics 39, 1580–1607.Google Scholar
Sen, B., et al. . (2010) Inconsistency of bootstrap: The Grenander estimator. The Annals of Statistics 38, 1953–1977.Google Scholar
Seo, M.H. & Linton, O. (2007) A smoothed least squares estimator for threshold regression models. Journal of Econometrics 141, 704–735.Google Scholar
Shao, J. & Tu, D. (1995) The Jackknife and Bootstrap. Springer.Google Scholar
Silverman, B.W. (1978) Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. The Annals of Statistics 6, 177–184.Google Scholar
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall.Google Scholar
Silverman, B.W. & Young, G.A. (1987) The bootstrap: To smooth or not to smooth? Biometrika 74, 469–479.Google Scholar
Tong, H. (1978) On a threshold model. In Chen, C.H. (ed.), Pattern Recognition and Signal Processing, pp. 575–586. Sijthoff & Noordhoff.Google Scholar
Tong, H. (1983) Threshold Models in Nonlinear Time Series Analysis, Lecture Notes in Statistics. Springer-Verlag.Google Scholar
Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Clarendon Press.Google Scholar
Tong, H. (2011) Threshold models in time series analysis – 30 years on. Statistics and Its Interface 4, 107–118.Google Scholar
Van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes: With Applications to Statistics. Springer.Google Scholar
Wellner, J.A. & Zhan, Y. (1996) Bootstrapping Z-Estimators. Unpublished manuscript, Department of Statistics, University of Washington.Google Scholar
Yu, P. (2008) Adaptive Estimation of the Threshold Point in Threshold Regression. Unpublished manuscript, Department of Economics, University of Auckland.Google Scholar
Yu, P. (2012) Likelihood estimation and inference in threshold regression. Journal of Econometrics 167, 274–294.Google Scholar
Yu, P. & Zhao, Y. (2009) Asymptotics for Threshold Regression Under General Conditions. Unpublished manuscript, Department of Economics, University of Auckland.Google Scholar
Supplementary material: PDF

Yu Supplementary Material

Supplementary Material

Download Yu Supplementary Material(PDF)
PDF 463.8 KB