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.
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References
Andrews, D.W.K.: Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856 (1993)
Andrews, D.W.K., Ploberger, W.: Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414 (1994)
Bacon, D.W., Watts, D.G.: Estimating the transition between two intersecting straight lines. Biometrika 58, 525–534 (1971)
Bai, J.: Estimation of a change point in multiple regressions models. Rev. Econ. Stat. 79, 551–563 (1997)
Carlin, B.P., Gelfand, A.E., Smith, A.F.M.: Hierarchical Bayesian analysis of change point problems. Appl. Stat. 41, 389–405 (1992)
Chen, C.W.S., Lee, J.C.: Bayesian inference of threshold autoregressive models. J. Time Ser. Anal. 16, 483–492 (1995)
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).
Chib, S.: Bayes regression with autoregressive errors: a Gibbs sampling approach. J. Econom. 58, 275–294 (1993)
Chow, G.: Tests of equality between sets of coefficients in two linear regressions. Econometrica 28, 591–605 (1960)
Fearnhead, P.: Exact and efficient Bayesian inference for multiple changepoint problems. Stat. Comput. 16, 203–213 (2006)
Ferreira, P.E.: A Bayesian analysis of a switching regression model: a known number of regimes. J. Am. Stat. Assoc. 70, 370–374 (1975)
Hastings, W.K.: Monte-Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
Hinkley, D.: Bootstrap methods (with discussion). J. R. Stat. Soc., Ser. B. 50, 321–337 (1988)
Hinkley, D., Schechtman, E.: Conditional bootstrap methods in the mean-shift model. Biometrika 74, 85–93 (1987)
Julious, S.A.: Inference and estimation in a changepoint regression problem. J. R. Stat. Soc. Ser. D, Stat. 50, 51–61 (2001)
Kim, H.J., Siegmund, D.: The likelihood ratio test for a changepoint in simple linear regression. Biometrika 76, 409–423 (1989)
Lerman, P.M.: Fitting segmented regression models by grid search. Appl. Stat. 29, 77–84 (1980)
Loader, C.R.: Change point estimation using nonparametric regression. Ann. Stat. 24, 1667–1678 (1996)
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)
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)
Muggeo, V.M.R.: Estimating regression models with unknown break-points. Stat. Med. 22, 3055–3071 (2003)
Muggeo, V.M.R.: Segmented: an R package to fit regression models with broken-line relationships. News. R Proj. 8(1), 20–25 (2008)
Pastor, R., Guallar, E.: Use of two-segmented logistic regression to estimate change-points in epidemiologic studies. Am. J. Epidemiol. 148, 631–642 (1998)
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)
Quandt, R.E.: Tests of the hypotheses that a linear regression system obeys two separate regimes. J. Am. Stat. Assoc. 55, 324 (1960)
Seber, G.A.F., Wild, C.J.: Nonlinear Regression. Wiley, New York (1989)
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)
Stephens, D.A.: Bayesian retrospective multiple-changepoint identification. Appl. Stat. 43, 159–178 (1994)
Ulm, K.: A statistical method for assessing a threshold in epidemiological studies. Stat. Med. 10, 341–349 (1991)
Vostrikova, L.J.: Detecting “disorder” in multidimensional random process. Soviet Math. Dokl. 24, 55–59 (1981)
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)
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Chen, C.W.S., Chan, J.S.K., Gerlach, R. et al. A comparison of estimators for regression models with change points. Stat Comput 21, 395–414 (2011). https://doi.org/10.1007/s11222-010-9177-0
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DOI: https://doi.org/10.1007/s11222-010-9177-0