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A Second Order Cumulant Spectrum Based Test for Strict Stationarity


  • Douglas Patterson
  • Melvin Hinich
  • Denisa Roberts


This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs present in the principal domain of the cumulant spectrum. The test uses a frame (window) averaged sample estimate of the second order cumulant spectrum to build a test statistic that has an asymptotic complex standard normal distribution. We derive the test statistic, study the size and power properties of the test, and demonstrate its implementation with intraday stock market return data. The test has conservative size properties and good power to detect varying variance and unit root in the presence of varying variance.

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  • Douglas Patterson & Melvin Hinich & Denisa Roberts, 2018. "A Second Order Cumulant Spectrum Based Test for Strict Stationarity," Papers 1801.06727,
  • Handle: RePEc:arx:papers:1801.06727

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    References listed on IDEAS

    1. Hinich , Melvin J. & Rothman, Philip, 1998. "Frequency-Domain Test Of Time Reversibility," Macroeconomic Dynamics, Cambridge University Press, vol. 2(01), pages 72-88, March.
    2. 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.
    3. John U. Farley & Melvin J. Hinich, 1970. "Detecting "Small" Mean Shifts in Time Series," Management Science, INFORMS, vol. 17(3), pages 189-199, November.
    4. Marsaglia, George & Tsang, Wai Wan & Wang, Jingbo, 2003. "Evaluating Kolmogorov's Distribution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 8(i18).
    5. Hinich, Melvin A. & Wild, Phillip, 2001. "Testing Time-Series Stationarity Against An Alternative Whose Mean Is Periodic," Macroeconomic Dynamics, Cambridge University Press, vol. 5(03), pages 380-412, June.
    6. Ashley, Richard A. & Patterson, Douglas M., 2010. "Apparent Long Memory In Time Series As An Artifact Of A Time-Varying Mean: Considering Alternatives To The Fractionally Integrated Model," Macroeconomic Dynamics, Cambridge University Press, vol. 14(S1), pages 59-87, May.
    7. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2005. "Stationarity Tests Under Time-Varying Second Moments," Econometric Theory, Cambridge University Press, vol. 21(06), pages 1112-1129, December.
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