IDEAS home Printed from
   My bibliography  Save this paper

Changepoint detection in random coefficient autoregressive models


  • Lajos Horvath
  • Lorenzo Trapani


We propose a family of CUSUM-based statistics to detect the presence of changepoints in the deterministic part of the autoregressive parameter in a Random Coefficient AutoRegressive (RCA) sequence. In order to ensure the ability to detect breaks at sample endpoints, we thoroughly study weighted CUSUM statistics, analysing the asymptotics for virtually all possible weighing schemes, including the standardised CUSUM process (for which we derive a Darling-Erdos theorem) and even heavier weights (studying the so-called R\'enyi statistics). Our results are valid irrespective of whether the sequence is stationary or not, and no prior knowledge of stationarity or lack thereof is required. Technically, our results require strong approximations which, in the nonstationary case, are entirely new. Similarly, we allow for heteroskedasticity of unknown form in both the error term and in the stochastic part of the autoregressive coefficient, proposing a family of test statistics which are robust to heteroskedasticity, without requiring any prior knowledge as to the presence or type thereof. Simulations show that our procedures work very well in finite samples. We complement our theory with applications to financial, economic and epidemiological time series.

Suggested Citation

  • Lajos Horvath & Lorenzo Trapani, 2021. "Changepoint detection in random coefficient autoregressive models," Papers 2104.13440,
  • Handle: RePEc:arx:papers:2104.13440

    Download full text from publisher

    File URL:
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    1. Peter C. B. Phillips & Yangru Wu & Jun Yu, 2011. "EXPLOSIVE BEHAVIOR IN THE 1990s NASDAQ: WHEN DID EXUBERANCE ESCALATE ASSET VALUES?," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 52(1), pages 201-226, February.
    2. Horváth, Lajos & Liu, Zhenya & Rice, Gregory & Wang, Shixuan, 2020. "Sequential monitoring for changes from stationarity to mild non-stationarity," Journal of Econometrics, Elsevier, vol. 215(1), pages 209-238.
    3. Leybourne, S J & McCabe, B P M & Tremayne, A R, 1996. "Can Economic Time Series Be Differenced to Stationarity?," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(4), pages 435-446, October.
    4. Sangyeol Lee & Jeongcheol Ha & Okyoung Na & Seongryong Na, 2003. "The Cusum Test for Parameter Change in Time Series Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(4), pages 781-796, December.
    5. Chun Yip Yau & Zifeng Zhao, 2016. "Inference for multiple change points in time series via likelihood ratio scan statistics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 895-916, September.
    6. Lumsdaine, Robin L, 1996. "Consistency and Asymptotic Normality of the Quasi-maximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models," Econometrica, Econometric Society, vol. 64(3), pages 575-596, May.
    7. Patrick Bardsley & Lajos Horváth & Piotr Kokoszka & Gabriel Young, 2017. "Change point tests in functional factor models with application to yield curves," Econometrics Journal, Royal Economic Society, vol. 20(1), pages 86-117, February.
    8. Ke‐Li Xu, 2015. "Testing for structural change under non‐stationary variances," Econometrics Journal, Royal Economic Society, vol. 18(2), pages 274-305, June.
    9. 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.
    10. Harvey, David I. & Leybourne, Stephen J. & Sollis, Robert & Taylor, A.M. Robert, 2016. "Tests for explosive financial bubbles in the presence of non-stationary volatility," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 548-574.
    11. Horváth, Lajos & Trapani, Lorenzo, 2016. "Statistical inference in a random coefficient panel model," Journal of Econometrics, Elsevier, vol. 193(1), pages 54-75.
    12. Nagakura, Daisuke, 2009. "Asymptotic theory for explosive random coefficient autoregressive models and inconsistency of a unit root test against a stochastic unit root process," Statistics & Probability Letters, Elsevier, vol. 79(24), pages 2476-2483, December.
    13. Xu, Ke-Li & Phillips, Peter C.B., 2008. "Adaptive estimation of autoregressive models with time-varying variances," Journal of Econometrics, Elsevier, vol. 142(1), pages 265-280, January.
    14. Gombay, Edit & Horváth, Lajos, 1996. "On the Rate of Approximations for Maximum Likelihood Tests in Change-Point Models," Journal of Multivariate Analysis, Elsevier, vol. 56(1), pages 120-152, January.
    15. Aue, Alexander, 2004. "Strong approximation for RCA(1) time series with applications," Statistics & Probability Letters, Elsevier, vol. 68(4), pages 369-382, July.
    16. Górecki, Tomasz & Horváth, Lajos & Kokoszka, Piotr, 2018. "Change point detection in heteroscedastic time series," Econometrics and Statistics, Elsevier, vol. 7(C), pages 63-88.
    17. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. David I. Harvey & Stephen J. Leybourne & Yang Zu, 2019. "Testing explosive bubbles with time-varying volatility," Econometric Reviews, Taylor & Francis Journals, vol. 38(10), pages 1131-1151, November.
    2. Horváth, Lajos & Trapani, Lorenzo, 2019. "Testing for randomness in a random coefficient autoregression model," Journal of Econometrics, Elsevier, vol. 209(2), pages 338-352.
    3. W. K. Li & Shiqing Ling & Michael McAleer, 2001. "A Survey of Recent Theoretical Results for Time Series Models with GARCH Errors," ISER Discussion Paper 0545, Institute of Social and Economic Research, Osaka University.
    4. Yongheng Deng & Eric Girardin & Roselyne Joyeux & Shuping Shi, 2017. "Did bubbles migrate from the stock to the housing market in China between 2005 and 2010?," Pacific Economic Review, Wiley Blackwell, vol. 22(3), pages 276-292, August.
    5. Tao, Yubo & Phillips, Peter C.B. & Yu, Jun, 2019. "Random coefficient continuous systems: Testing for extreme sample path behavior," Journal of Econometrics, Elsevier, vol. 209(2), pages 208-237.
    6. Lorenzo Trapani, 2018. "Testing for strict stationarity in a random coefficient autoregressive model," Discussion Papers 18/02, University of Nottingham, Granger Centre for Time Series Econometrics.
    7. Lajos Horváth & Gregory Rice, 2014. "Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 219-255, June.
    8. Cizek, P., 2010. "Modelling Conditional Heteroscedasticity in Nonstationary Series," Other publications TiSEM a5a7b05f-5f1f-46ed-8ce8-5, Tilburg University, School of Economics and Management.
    9. Sam Astill & David I. Harvey & Stephen J. Leybourne & Robert Sollis & A. M. Robert Taylor, 2018. "Real‐Time Monitoring for Explosive Financial Bubbles," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 863-891, November.
    10. Cizek, P., 2010. "Modelling Conditional Heteroscedasticity in Nonstationary Series," Discussion Paper 2010-84, Tilburg University, Center for Economic Research.
    11. Bernard, Jean-Thomas & Idoudi, Nadhem & Khalaf, Lynda & Yelou, Clement, 2007. "Finite sample multivariate structural change tests with application to energy demand models," Journal of Econometrics, Elsevier, vol. 141(2), pages 1219-1244, December.
    12. Giraitis, Liudas & Leipus, Remigijus & Robinson, Peter M. & Surgailis, Donatas, 2004. "LARCH, leverage, and long memory," LSE Research Online Documents on Economics 294, London School of Economics and Political Science, LSE Library.
    13. Lombardi, Marco J. & Calzolari, Giorgio, 2009. "Indirect estimation of [alpha]-stable stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2298-2308, April.
    14. Offer Lieberman & Peter C. B. Phillips, 2014. "Norming Rates And Limit Theory For Some Time-Varying Coefficient Autoregressions," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(6), pages 592-623, November.
    15. Escobari, Diego & Garcia, Sergio & Mellado, Cristhian, 2017. "Identifying bubbles in Latin American equity markets: Phillips-Perron-based tests and linkages," Emerging Markets Review, Elsevier, vol. 33(C), pages 90-101.
    16. Carsoule, F. & Franses, Ph.H.B.F., 1999. "Monitoring structural change in variance," Econometric Institute Research Papers EI 9925A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    17. Clements, M.P. & Smith, J., 1997. "Forecasting Seasonal UK Consumption Components," The Warwick Economics Research Paper Series (TWERPS) 487, University of Warwick, Department of Economics.
    18. Cerrato, Mario & Crosby, John & Kim, Minjoo & Zhao, Yang, 2017. "Relation between higher order comoments and dependence structure of equity portfolio," Journal of Empirical Finance, Elsevier, vol. 40(C), pages 101-120.
    19. Moschini, GianCarlo & Myers, Robert J., 2002. "Testing for constant hedge ratios in commodity markets: a multivariate GARCH approach," Journal of Empirical Finance, Elsevier, vol. 9(5), pages 589-603, December.
    20. Bollerslev, Tim & Ghysels, Eric, 1996. "Periodic Autoregressive Conditional Heteroscedasticity," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(2), pages 139-151, April.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2104.13440. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.