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Using the HEGY Procedure When Not All Roots Are Present

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  • Tomas del Barrio Castro

Abstract

. Empirical studies have shown little evidence to support the presence of all unit roots present in the Δ4 filter in quarterly seasonal time series. This paper analyses the performance of the Hylleberg, Engle, Granger and Yoo [Journal of Econometrics (1990) Vol. 44, pp. 215–238] (HEGY) procedure when the roots under the null are not all present. We exploit the vector of quarters representation and cointegration relationship between the quarters when factors (1 − L), (1 + L), (1 + L2), (1 − L2) and (1 + L + L2 + L3) are a source of nonstationarity in a process in order to obtain the distribution of tests of the HEGY procedure when the underlying processes have a root at the zero, Nyquist frequency, two complex conjugates of frequency π/2 and two combinations of the previous cases. We show both theoretically and through a Monte Carlo analysis that the t‐ratios t and t and the F‐type tests used in the HEGY procedure have the same distribution as under the null of a seasonal random walk when the root(s) is (are) present, although this is not the case for the t‐ratio tests associated with unit roots at frequency π/2.

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  • Tomas del Barrio Castro, 2007. "Using the HEGY Procedure When Not All Roots Are Present," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(6), pages 910-922, November.
    • Handle: RePEc:bla:jtsera:v:28:y:2007:i:6:p:910-922
    DOI: 10.1111/j.1467-9892.2007.00539.x
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    1. Hylleberg, Svend & Jorgensen, Clara & Sorensen, Nils Karl, 1993. "Seasonality in Macroeconomic Time Series," Empirical Economics, Springer, vol. 18(2), pages 321-335.
    2. Smith, Richard J. & Taylor, A.M. Robert & del Barrio Castro, Tomas, 2009. "Regression-Based Seasonal Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 25(2), pages 527-560, April.
    3. Rodrigues, Paulo M. M. & Taylor, A. M. Robert, 2004. "Alternative estimators and unit root tests for seasonal autoregressive processes," Journal of Econometrics, Elsevier, vol. 120(1), pages 35-73, May.
    4. Smith, Richard J. & Taylor, A.M. Robert & del Barrio Castro, Tomas, 2009. "Regression-Based Seasonal Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 25(02), pages 527-560, April.
    5. Taylor, A.M. Robert, 2003. "On The Asymptotic Properties Of Some Seasonal Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 19(2), pages 311-321, April.
    6. Castro, Tomas del Barrio & Osborn, Denise R., 2008. "Testing For Seasonal Unit Roots In Periodic Integrated Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 24(4), pages 1093-1129, August.
    7. Hylleberg, S. & Engle, R. F. & Granger, C. W. J. & Yoo, B. S., 1990. "Seasonal integration and cointegration," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 215-238.
    8. Tomas del Barrio Castro, 2007. "Using the HEGY Procedure When Not All Roots Are Present," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(6), pages 910-922, November.
    9. H. Peter Boswijk & Philip Hans Franses, 1996. "Unit Roots In Periodic Autoregressions," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(3), pages 221-245, May.
    10. Smith, Richard J. & Taylor, A. M. Robert, 1998. "Additional critical values and asymptotic representations for seasonal unit root tests," Journal of Econometrics, Elsevier, vol. 85(2), pages 269-288, August.
    11. Burridge, Peter & Taylor, A M Robert, 2001. "On the Properties of Regression-Based Tests for Seasonal Unit Roots in the Presence of Higher-Order Serial Correlation," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(3), pages 374-379, July.
    12. Ghysels,Eric & Osborn,Denise R., 2001. "The Econometric Analysis of Seasonal Time Series," Cambridge Books, Cambridge University Press, number 9780521565882, January.
    13. Rodrigues, Paulo M.M. & Taylor, A.M. Robert, 2004. "Asymptotic Distributions For Regression-Based Seasonal Unit Root Test Statistics In A Near-Integrated Model," Econometric Theory, Cambridge University Press, vol. 20(4), pages 645-670, August.
    14. Ghysels, Eric & Lee, Hahn S. & Noh, Jaesum, 1994. "Testing for unit roots in seasonal time series : Some theoretical extensions and a Monte Carlo investigation," Journal of Econometrics, Elsevier, vol. 62(2), pages 415-442, June.
    15. Osborn, Denise R., 1993. "Seasonal cointegration," Journal of Econometrics, Elsevier, vol. 55(1-2), pages 299-303.
    16. Denise Osborn & Paulo Rodrigues, 2002. "Asymptotic Distributions Of Seasonal Unit Root Tests: A Unifying Approach," Econometric Reviews, Taylor & Francis Journals, vol. 21(2), pages 221-241.
    17. Phillips, Peter C B & Ouliaris, S, 1990. "Asymptotic Properties of Residual Based Tests for Cointegration," Econometrica, Econometric Society, vol. 58(1), pages 165-193, January.
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    1. Tomás del Barrio Castro & Gianluca Cubadda & Denise R. Osborn, 2022. "On cointegration for processes integrated at different frequencies," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 412-435, May.
    2. Tomás Del Barrio Castro & Denise R. Osborn, 2011. "HEGY Tests in the Presence of Moving Averages," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 73(5), pages 691-704, October.
    3. Tomas del Barrio Castro, 2007. "Using the HEGY Procedure When Not All Roots Are Present," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(6), pages 910-922, November.
    4. Ghassen El Montasser, 2011. "The overall seasonal integration tests under non-stationary alternatives: A methodological note," EERI Research Paper Series EERI_RP_2011_06, Economics and Econometrics Research Institute (EERI), Brussels.
    5. Zou, Nan & Politis, Dimitris N., 2021. "Bootstrap seasonal unit root test under periodic variation," Econometrics and Statistics, Elsevier, vol. 19(C), pages 1-21.
    6. del Barrio Castro, Tomás & Osborn, Denise R., 2023. "Periodic Integration and Seasonal Unit Roots," MPRA Paper 117935, University Library of Munich, Germany, revised 2023.

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    More about this item

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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