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Least squares estimation of a shift in linear processes


  • Bai, Jushan


This paper considers a mean shift with an unknown shift point in a linear process and estimates the unknown shift point (change point) by the method of least squares. Pre-shift and post-shift means are estimated concurrently with the change point. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained when the magnitude of shift is small. It is shown that serial correlation affects the variance of the change point estimator via the sum of the coefficients (impulses) of the linear process. When the underlying process is an ARMA, a mean shift causes overestimation of its order. A simple procedure is suggested to mitigate the bias in order estimation.

Suggested Citation

  • Bai, Jushan, 1993. "Least squares estimation of a shift in linear processes," MPRA Paper 32878, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:32878

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    References listed on IDEAS

    1. Ploberger, W & Kramer, W & Alt, R, 1989. "A Modification of the CUSUM Test in the Linear Regression Model with Lagged Dependent Variables," Empirical Economics, Springer, vol. 14(2), pages 65-75.
    2. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-856, July.
    3. Bhattacharya, P.K., 1987. "Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 183-208, December.
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    Cited by:

    1. Rafael Di Tella & Ernesto Schargrodsky, 2004. "Do Police Reduce Crime? Estimates Using the Allocation of Police Forces After a Terrorist Attack," American Economic Review, American Economic Association, vol. 94(1), pages 115-133, March.
    2. Bruce E. Hansen, 2001. "The New Econometrics of Structural Change: Dating Breaks in U.S. Labour Productivity," Journal of Economic Perspectives, American Economic Association, vol. 15(4), pages 117-128, Fall.
    3. Kim, Tae-Hwan & Leybourne, Stephen & Newbold, Paul, 2002. "Unit root tests with a break in innovation variance," Journal of Econometrics, Elsevier, vol. 109(2), pages 365-387, August.

    More about this item


    Mean shift; linear processes; change point; rate of convergence; order estimation; generalized residuals;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes


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