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Testing For Periodic Stationarity


  • Eiji Kurozumi


This paper proposes a test for the null hypothesis of periodic stationarity against the alternative hypothesis of periodic integration. We derive the limiting distribution of the test statistic and its characteristic function, which are the same as those of the test developed in Kwiatkowski, Phillips, Schmidt and Shin.[15] We find that some parameters, which we must assume under the alternative, have an important effect on the limiting power, so we should choose such parameters carefully. A Monte Carlo simulation reveals that the test has reasonable power but may be affected by the lag truncation parameter that is used for the correction of nuisance parameters.

Suggested Citation

  • Eiji Kurozumi, 2002. "Testing For Periodic Stationarity," Econometric Reviews, Taylor & Francis Journals, vol. 21(2), pages 243-270.
  • Handle: RePEc:taf:emetrv:v:21:y:2002:i:2:p:243-270 DOI: 10.1081/ETC-120014351

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    References listed on IDEAS

    1. Kwiatkowski, D. & Phillips, P.C.B. & Schmidt, P., 1992. "Testing for Stationarity in the Components Representation of a Time Series," Econometric Theory, Cambridge University Press, vol. 8(04), pages 586-591, December.
    2. Franses, Philip Hans, 1996. "Periodicity and Stochastic Trends in Economic Time Series," OUP Catalogue, Oxford University Press, number 9780198774549, June.
    3. Kwiatkowski, Denis & Phillips, Peter C. B. & Schmidt, Peter & Shin, Yongcheol, 1992. "Testing the null hypothesis of stationarity against the alternative of a unit root : How sure are we that economic time series have a unit root?," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 159-178.
    4. Canova, Fabio & Hansen, Bruce E, 1995. "Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(3), pages 237-252, July.
    5. repec:cup:etheor:v:8:y:1992:i:2:p:188-202 is not listed on IDEAS
    6. Osborn, Denise R., 1991. "The implications of periodically varying coefficients for seasonal time-series processes," Journal of Econometrics, Elsevier, vol. 48(3), pages 373-384, June.
    7. Caner, Mehmet, 1998. "A Locally Optimal Seaosnal Unit-Root Test," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 349-356, July.
    8. Hansen, Bruce E., 2000. "Testing for structural change in conditional models," Journal of Econometrics, Elsevier, vol. 97(1), pages 93-115, July.
    9. Andrews, Donald W K & Ploberger, Werner, 1994. "Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative," Econometrica, Econometric Society, vol. 62(6), pages 1383-1414, November.
    10. Franses, Philip Hans & Paap, Richard, 1994. "Model Selection in Periodic Autoregressions," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 56(4), pages 421-439, November.
    11. Osborn, Denise R & Smith, Jeremy P, 1989. "The Performance of Periodic Autoregressive Models in Forecasting Seasonal U. K. Consumption," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(1), pages 117-127, January.
    12. Peter C.B. Phillips, 1988. "Spectral Regression for Cointegrated Time Series," Cowles Foundation Discussion Papers 872, Cowles Foundation for Research in Economics, Yale University.
    13. Peter Boswijk, H. & Franses, Philip Hans, 1995. "Testing for periodic integration," Economics Letters, Elsevier, vol. 48(3-4), pages 241-248, June.
    14. Robert B. Davies, 2002. "Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case," Biometrika, Biometrika Trust, vol. 89(2), pages 484-489, June.
    15. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-1580, November.
    16. Boswijk, H. Peter & Franses, Philip Hans & Haldrup, Niels, 1997. "Multiple unit roots in periodic autoregression," Journal of Econometrics, Elsevier, vol. 80(1), pages 167-193, September.
    17. Hylleberg, Svend, 1995. "Tests for seasonal unit roots general to specific or specific to general?," Journal of Econometrics, Elsevier, vol. 69(1), pages 5-25, September.
    18. Richard Paap & Philip Hans Franses, 1999. "On trends and constants in periodic autoregressions," Econometric Reviews, Taylor & Francis Journals, vol. 18(3), pages 271-286.
    19. Johansen, Søren, 1992. "A Representation of Vector Autoregressive Processes Integrated of Order 2," Econometric Theory, Cambridge University Press, vol. 8(02), pages 188-202, June.
    20. Franses, Philip Hans, 1994. "A multivariate approach to modeling univariate seasonal time series," Journal of Econometrics, Elsevier, vol. 63(1), pages 133-151, July.
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    Cited by:

    1. Siem Jan Koopman & Marius Ooms & Irma Hindrayanto, 2009. "Periodic Unobserved Cycles in Seasonal Time Series with an Application to US Unemployment," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 71(5), pages 683-713, October.
    2. Hindrayanto, Irma & Koopman, Siem Jan & Ooms, Marius, 2010. "Exact maximum likelihood estimation for non-stationary periodic time series models," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2641-2654, November.

    More about this item


    Periodic stationarity; Periodic integration; Hypothesis testing; JEL Classification ; C22; C32;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models


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