Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Time Series Econometrics

Editor-in-Chief: Hidalgo, Javier

2 Issues per year

CiteScore 2017: 0.25

SCImago Journal Rank (SJR) 2017: 0.236
Source Normalized Impact per Paper (SNIP) 2017: 0.682

See all formats and pricing
More options …

Sequential Testing with Uniformly Distributed Size

Stanislav Anatolyev
  • Corresponding author
  • CERGE-EI, Politických vězňů 7, 11121 Prague 1, Czech Republic
  • New Economic School, Moscow, Russia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Grigory Kosenok
Published Online: 2018-02-06 | DOI: https://doi.org/10.1515/jtse-2017-0002


Sequential procedures for the testing for structural stability do not provide enough guidance on the shape of boundaries that are used to decide on acceptance or rejection, requiring only that the overall size of the test is asymptotically controlled. We introduce and motivate a reasonable criterion for the shape of boundaries which requires that the test size be uniformly distributed over the testing period. Under this criterion, we numerically construct boundaries for the most popular sequential tests that are characterized by a test statistic behaving asymptotically either as a Wiener process or Brownian bridge. We handle this problem both in the context of retrospecting a historical sample and in the context of monitoring newly arriving data. We tabulate the boundaries by fitting them to certain flexible yet parsimonious functional forms. Interesting patterns emerge in an illustrative application of sequential tests to the Phillips curve model.

Keywords: structural stability; sequential tests; CUSUM; retrospection; monitoring; boundaries; asymptotic size


  • Anatolyev, S. 2008. “Nonparametric Retrospection and Monitoring of Predictability of Financial Returns.” Journal of Business & Economic Statistics 27:149–160.Web of ScienceGoogle Scholar

  • Anatolyev, S., and G. Kosenok 2011. “Another Numerical Method of Finding Critical Values for the Andrews Stability Test.” Econometric Theory 28:239–246.Web of ScienceGoogle Scholar

  • Andreou, E., and E. Ghysels 2002. “Detecting Multiple Breaks in Financial Market Volatility Dynamics.” Journal of Applied Econometrics 17:579–600.CrossrefGoogle Scholar

  • Andreou, E., and E. Ghysels 2006. “Monitoring Disruptions in Financial Markets.” Journal of Econometrics 135:77–124.CrossrefGoogle Scholar

  • Andrews, D. W. K. 1993. “Tests for Parameter Instability and Structural Change with Unknown Change Point.” Econometrica 61:821–856.CrossrefGoogle Scholar

  • Aue, A., L. Horváth, M. Hušková, and P. Kokoszka. 2006. “Change-Point Monitoring in Linear Models.” Econometrics Journal 9:373–403.CrossrefGoogle Scholar

  • Bai, J., and P. Perron. 1998. “Estimating and Testing Linear Models with Multiple Structural Changes.” Econometrica 66:47–78.CrossrefGoogle Scholar

  • Bai, J., and P. Perron. 2003. “Computation and Analysis of Multiple Structural Change Models.” Journal of Applied Econometrics 18:1–22.CrossrefGoogle Scholar

  • Brown, R. L., Durbin, J., Evans, J. M., 1975. “Techniques for Testing the Constancy of Regression Relationships Over Time.” Journal of Royal Statistical Society B 37:149–163.Google Scholar

  • Brunner, H., and van der Houwen P. J. 1986. “The Numerical Solution of Volterra Equations. CWI Monographs, North–Holland, Amsterdam.Google Scholar

  • Chu, C. S. J., K. Hornik, and C. M. Kuan. 1995. “The Moving-Estimates Test for Parameter Stability.” Econometric Theory 11:669–720.Google Scholar

  • Chu, C.S.J., M. Stinchcombe, and H. White. 1996. “Monitoring Structural Change.” Econometrica 64:1045–1065.CrossrefGoogle Scholar

  • Deng, A., and P. Perron. 2008. “The Limit Distribution of the Cusum of Squares Test Under General Mixing Conditions.” Econometric Theory 24:809–822.Web of ScienceGoogle Scholar

  • Diogo, N., P. Lima and M. Rebelo. 2005. “Computational Methods for a Nonlinear Volterra Integral Equation.” Proceedings of HERCMA 2005:100–107.Google Scholar

  • Durbin, J. 1971. “Boundary-Crossing Probabilities for the Brownian Motion and Poisson Processes and Techniques for Computing the Power of the Kolmogorov–Smirnov Test.” Journal of Applied Probability 8:431–453.CrossrefGoogle Scholar

  • Ghysels, E., A. Guay and A.Hall. 1997. “Predictive Tests for Structural Change with Unknown Breakpoint.” Journal of Econometrics 82:209–233.Google Scholar

  • Groen, J., G. Kapetanios and S. Price. 2013. “Multivariate Methods for Monitoring Structural Change.” Journal of Applied Econometrics 28:250–274.CrossrefWeb of ScienceGoogle Scholar

  • Inclán, C., and G.C. Tiao. 1994. “Use of Cumulative Sums of Squares for Retrospective Detection of Changes in Variance.” Journal of American Statistical Association 89:913–923.Google Scholar

  • Inoue, A., and B. Rossi. 2005. “Recursive Predictability Tests for Real Time Data.” Journal of Business and Economic Statistics 23:336–345.CrossrefGoogle Scholar

  • Karatzas, I., and S.E. Shreve. 1988. Brownian Motion and Stochastic Calculus. New York: Springer-Verlag.Google Scholar

  • Kuan, C.-M., and K. Hornik. 1995. “The Generalized Fluctuation Test: A Unifying View.” Econometric Reviews 14: 135–161.Google Scholar

  • Kurozumi, E. 2017. “Monitoring Parameter Constancy with Endogenous Regressors.” Journal of Time Series Analysis 38:791–805.CrossrefWeb of ScienceGoogle Scholar

  • Leisch, F., Hornik, K., Kuan, C.M. 2000. “Monitoring Structural Changes with the Generalized Fluctuation Test.” Econometric Theory 16:835–854.CrossrefGoogle Scholar

  • Ploberger, W., and W. Krämer 1992. “The CUSUM test with OLS Residuals.” Econometrica 60:271–285.CrossrefGoogle Scholar

  • Ploberger, W., Krämer, W., Alt R. 1988. “Testing for Structural Change in Dynamic Models.” Econometrica 56:1355–1369.CrossrefGoogle Scholar

  • Ploberger, W., Krämer, W., Kontrus K. 1989. “A New Test for Structural Stability in the Linear Regression Model.” Journal of Econometrics 40:307–318.CrossrefGoogle Scholar

  • Robbins, H. and D. Siegmund. 1970. “Boundary Crossing Probabilities for the Wiener Process and Sample Sums.” Annals of Mathematical Statistics 41:1410–1429.CrossrefGoogle Scholar

  • Romano, J.P., A.M. Shaikh and M. Wolf. 2010. “Multiple Testing. In The New Palgrave Dictionary of Economics, edited by S.N. Durlauf, and L.E. Blume. Palgrave Macmillan.Google Scholar

  • Zeileis A. 2004. “Alternative Boundaries for CUSUM Tests.” Statistical Papers 45:123–131.CrossrefGoogle Scholar

  • Zeileis A., F. Leisch, C. Kleiber and K. Hornik. 2005. “Monitoring Structural Change in Dynamic Econometric Models.” Journal of Applied Econometrics 20:99–121.CrossrefGoogle Scholar

About the article

Published Online: 2018-02-06

Citation Information: Journal of Time Series Econometrics, Volume 10, Issue 2, 20170002, ISSN (Online) 1941-1928, DOI: https://doi.org/10.1515/jtse-2017-0002.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in