Asymptotic and Bootstrap Tests for a Change in Autoregression Omitting Variability Estimation

  • Barbora Peštová
  • Michal PeštaEmail author
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


A sequence of time-ordered observations follows an autoregressive model of order one and its parameter is possibly subject to change at most once at some unknown time point. The aim is to test whether such an unknown change has occurred or not. A change-point method presented here rely on a ratio type test statistic based on the maxima of cumulative sums. The main advantage of the developed approach is that the variance of the observations neither has to be known nor estimated. Asymptotic distribution of the test statistic under the no-change null hypothesis is derived. Moreover, we prove the consistency of the test under the alternative. A bootstrap procedure is proposed in the way of a completely data-driven technique without any tuning parameters. The results are illustrated through a simulation study, which demonstrates the computational efficiency of the procedure. A practical application to real data is presented as well.


Change point Structural change Change in autoregression Hypothesis testing Bootstrap Ratio type statistic Variance estimation free test 



Institutional support to Barbora Peštová was provided by RVO:67985807. Michal Pešta was supported by the Czech Science Foundation project No. 18-01781Y.


  1. 1.
    Hušková, M., Prášková, Z., Steinebach, J.: On the detection of changes in autoregressive time series I. Asymptotics. J. Stat. Plan. Infer. 137(4), 1243–1259 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hušková, M., Kirch, C., Prášková, Z., Steinebach, J.: On the detection of changes in autoregressive time series, II. Resampling procedures. J. Stat. Plan. Infer. 138(6), 1697–1721 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Horváth, L., Horváth, Z., Hušková, M.: Ratio tests for change point detection. In: Balakrishnan, N., Peña, E.A., Silvapulle, M.J. (eds.) Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, vol. 1, pp. 293–304. IMS Collections, Beachwood, Ohio (2009)CrossRefGoogle Scholar
  4. 4.
    Peštová, B., Pešta, M.: Testing structural changes in panel data with small fixed panel size and bootstrap. Metrika 78(6), 665–689 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Peštová, B., Pešta, M.: Change point estimation in panel data without boundary issue. Risks 5(1), 7 (2017)CrossRefGoogle Scholar
  6. 6.
    Prague Stock Exchange: PX Index 2015. Updated April 30, 2015; Accessed April 30, 2015
  7. 7.
    Davidson, J.: Stochastic Limit Theory: An Introduction for Econometricians. Oxford University Press, New York (1994)CrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Institute of Computer ScienceThe Czech Academy of SciencesPragueCzech Republic
  2. 2.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

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