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Statistics and Computing

, Volume 21, Issue 3, pp 395–414 | Cite as

A comparison of estimators for regression models with change points

  • Cathy W. S. ChenEmail author
  • Jennifer S. K. Chan
  • Richard Gerlach
  • William Y. L. Hsieh
Article

Abstract

We consider two problems concerning locating change points in a linear regression model. One involves jump discontinuities (change-point) in a regression model and the other involves regression lines connected at unknown points. We compare four methods for estimating single or multiple change points in a regression model, when both the error variance and regression coefficients change simultaneously at the unknown point(s): Bayesian, Julious, grid search, and the segmented methods. The proposed methods are evaluated via a simulation study and compared via some standard measures of estimation bias and precision. Finally, the methods are illustrated and compared using three real data sets. The simulation and empirical results overall favor both the segmented and Bayesian methods of estimation, which simultaneously estimate the change point and the other model parameters, though only the Bayesian method is able to handle both continuous and dis-continuous change point problems successfully. If it is known that regression lines are continuous then the segmented method ranked first among methods.

Keywords

Change point Jump discontinuities MCMC Grid-search Segmented regression 

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References

  1. Andrews, D.W.K.: Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  2. Andrews, D.W.K., Ploberger, W.: Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bacon, D.W., Watts, D.G.: Estimating the transition between two intersecting straight lines. Biometrika 58, 525–534 (1971) zbMATHCrossRefGoogle Scholar
  4. Bai, J.: Estimation of a change point in multiple regressions models. Rev. Econ. Stat. 79, 551–563 (1997) CrossRefGoogle Scholar
  5. Carlin, B.P., Gelfand, A.E., Smith, A.F.M.: Hierarchical Bayesian analysis of change point problems. Appl. Stat. 41, 389–405 (1992) zbMATHCrossRefGoogle Scholar
  6. Chen, C.W.S., Lee, J.C.: Bayesian inference of threshold autoregressive models. J. Time Ser. Anal. 16, 483–492 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  7. Chen, C.W.S., Gerlach, R., Lin, A.M.H.: Falling and explosive, dormant and rising markets via multiple-regime financial time series models. Appl. Stoch. Models Bus. Ind. 26, 28–49 (2010). MathSciNetzbMATHCrossRefGoogle Scholar
  8. Chib, S.: Bayes regression with autoregressive errors: a Gibbs sampling approach. J. Econom. 58, 275–294 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  9. Chow, G.: Tests of equality between sets of coefficients in two linear regressions. Econometrica 28, 591–605 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  10. Fearnhead, P.: Exact and efficient Bayesian inference for multiple changepoint problems. Stat. Comput. 16, 203–213 (2006) MathSciNetCrossRefGoogle Scholar
  11. Ferreira, P.E.: A Bayesian analysis of a switching regression model: a known number of regimes. J. Am. Stat. Assoc. 70, 370–374 (1975) zbMATHCrossRefGoogle Scholar
  12. Hastings, W.K.: Monte-Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970) zbMATHCrossRefGoogle Scholar
  13. Hinkley, D.: Bootstrap methods (with discussion). J. R. Stat. Soc., Ser. B. 50, 321–337 (1988) MathSciNetzbMATHGoogle Scholar
  14. Hinkley, D., Schechtman, E.: Conditional bootstrap methods in the mean-shift model. Biometrika 74, 85–93 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  15. Julious, S.A.: Inference and estimation in a changepoint regression problem. J. R. Stat. Soc. Ser. D, Stat. 50, 51–61 (2001) MathSciNetCrossRefGoogle Scholar
  16. Kim, H.J., Siegmund, D.: The likelihood ratio test for a changepoint in simple linear regression. Biometrika 76, 409–423 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  17. Lerman, P.M.: Fitting segmented regression models by grid search. Appl. Stat. 29, 77–84 (1980) CrossRefGoogle Scholar
  18. Loader, C.R.: Change point estimation using nonparametric regression. Ann. Stat. 24, 1667–1678 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  19. MacNeill, I.B., Mao, Y.: Change-point analysis for mortality and morbidity rate. In: Sinha, B., Rukhin, A., Ahsanullah, M. (eds.) Applied Change Point Problems in Statistics, pp. 37–55 (1995) Google Scholar
  20. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091 (1953) CrossRefGoogle Scholar
  21. Muggeo, V.M.R.: Estimating regression models with unknown break-points. Stat. Med. 22, 3055–3071 (2003) CrossRefGoogle Scholar
  22. Muggeo, V.M.R.: Segmented: an R package to fit regression models with broken-line relationships. News. R Proj. 8(1), 20–25 (2008) Google Scholar
  23. Pastor, R., Guallar, E.: Use of two-segmented logistic regression to estimate change-points in epidemiologic studies. Am. J. Epidemiol. 148, 631–642 (1998) CrossRefGoogle Scholar
  24. Quandt, R.E.: The estimation of the parameters of a linear regression system obeying two separate regimes. J. Am. Stat. Assoc. 53, 873–880 (1958) MathSciNetzbMATHCrossRefGoogle Scholar
  25. Quandt, R.E.: Tests of the hypotheses that a linear regression system obeys two separate regimes. J. Am. Stat. Assoc. 55, 324 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  26. Seber, G.A.F., Wild, C.J.: Nonlinear Regression. Wiley, New York (1989) zbMATHCrossRefGoogle Scholar
  27. Smith, A.F.M., Cook, D.G.: Straight lines with a change-point: a Bayesian analysis of some renal transplant data. Appl. Stat. 29, 180–189 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  28. Stephens, D.A.: Bayesian retrospective multiple-changepoint identification. Appl. Stat. 43, 159–178 (1994) zbMATHCrossRefGoogle Scholar
  29. Ulm, K.: A statistical method for assessing a threshold in epidemiological studies. Stat. Med. 10, 341–349 (1991) CrossRefGoogle Scholar
  30. Vostrikova, L.J.: Detecting “disorder” in multidimensional random process. Soviet Math. Dokl. 24, 55–59 (1981) zbMATHGoogle Scholar
  31. Zhou, H.L., Liang, K.Y.: On estimating the change point in generalized linear models. In: IMS Collections Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, vol. 1, pp. 305–320 (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Cathy W. S. Chen
    • 1
    Email author
  • Jennifer S. K. Chan
    • 2
  • Richard Gerlach
    • 3
  • William Y. L. Hsieh
    • 1
  1. 1.Graduate Institute of Statistics & Actuarial ScienceFeng Chia UniversityTaichungTaiwan
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  3. 3.Faculty of Economics and BusinessUniversity of SydneySydneyAustralia

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