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On Augmented Hegy Tests For Seasonal Unit Roots

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  • Castro, Tomás del Barrio
  • Osborn, Denise R.
  • Taylor, A.M. Robert

Abstract

In this paper we extend the large-sample results provided for the augmented Dickey–Fuller test by Said and Dickey ( 1984 , Biometrika 71, 599–607) and Chang and Park ( 2002 , Econometric Reviews 21, 431–447) to the case of the augmented seasonal unit root tests of Hylleberg, Engle, Granger, and Yoo ( 1990 , Journal of Econometrics 44, 215–238), inter alia. Our analysis is performed under the same conditions on the innovations as in Chang and Park ( 2002 ), thereby allowing for general linear processes driven by (possibly conditionally heteroskedastic) martingale difference innovations. We show that the limiting null distributions of the t -statistics for unit roots at the zero and Nyquist frequencies and joint F -type statistics are pivotal, whereas those of the t -statistics at the harmonic seasonal frequencies depend on nuisance parameters that derive from the lag parameters characterizing the linear process. Moreover, the rates on the lag truncation required for these results to hold are shown to coincide with the corresponding rates given in Chang and Park ( 2002 ); in particular, an o ( T 1/2 ) rate is shown to be sufficient.

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  • Castro, Tomás del Barrio & Osborn, Denise R. & Taylor, A.M. Robert, 2012. "On Augmented Hegy Tests For Seasonal Unit Roots," Econometric Theory, Cambridge University Press, vol. 28(05), pages 1121-1143, October.
  • Handle: RePEc:cup:etheor:v:28:y:2012:i:05:p:1121-1143_00
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    1. P. C. B. Phillips & S. N. Durlauf, 1986. "Multiple Time Series Regression with Integrated Processes," Review of Economic Studies, Oxford University Press, vol. 53(4), pages 473-495.
    2. Hylleberg, S. & Engle, R. F. & Granger, C. W. J. & Yoo, B. S., 1990. "Seasonal integration and cointegration," Journal of Econometrics, Elsevier, pages 215-238.
    3. Serena Ng & Pierre Perron, 2001. "LAG Length Selection and the Construction of Unit Root Tests with Good Size and Power," Econometrica, Econometric Society, pages 1519-1554.
    4. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
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    6. Paulo M.M. Rodrigues & A.M. Robert Taylor, 2004. "Efficient Tests of the Seasonal Unit Root Hypothesis," Economics Working Papers ECO2004/29, European University Institute.
    7. Hylleberg, S. & Engle, R. F. & Granger, C. W. J. & Yoo, B. S., 1990. "Seasonal integration and cointegration," Journal of Econometrics, Elsevier, pages 215-238.
    8. Rodrigues, Paulo M. M. & Taylor, A. M. Robert, 2004. "Alternative estimators and unit root tests for seasonal autoregressive processes," Journal of Econometrics, Elsevier, pages 35-73.
    9. 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.
    10. Smith, Richard J. & Taylor, A. M. Robert, 1998. "Additional critical values and asymptotic representations for seasonal unit root tests," Journal of Econometrics, Elsevier, pages 269-288.
    11. Joseph Beaulieu, J. & Miron, Jeffrey A., 1993. "Seasonal unit roots in aggregate U.S. data," Journal of Econometrics, Elsevier, pages 305-328.
    12. Smith, Richard J. & Taylor, A. M. Robert, 1998. "Additional critical values and asymptotic representations for seasonal unit root tests," Journal of Econometrics, Elsevier, pages 269-288.
    13. C-B. An & R. H. Haveman & B. L. Wolfe, "undated". "The "window problem" in studies of children's attainments: A methodological exploration," Institute for Research on Poverty Discussion Papers 977-92, University of Wisconsin Institute for Research on Poverty.
    14. Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2009. "Simple, Robust, And Powerful Tests Of The Breaking Trend Hypothesis," Econometric Theory, Cambridge University Press, pages 995-1029.
    15. Taylor, A. M. Robert, 2005. "Variance ratio tests of the seasonal unit root hypothesis," Journal of Econometrics, Elsevier, pages 33-54.
    16. Castro, Tomás del Barrio & Osborn, Denise R. & Taylor, A.M. Robert, 2012. "On Augmented Hegy Tests For Seasonal Unit Roots," Econometric Theory, Cambridge University Press, pages 1121-1143.
    17. 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, pages 415-442.
    18. Rodrigues, Paulo M.M. & Taylor, A.M. Robert, 2007. "Efficient tests of the seasonal unit root hypothesis," Journal of Econometrics, Elsevier, pages 548-573.
    19. 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(04), pages 645-670, August.
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    21. Hall, Alastair R, 1994. "Testing for a Unit Root in Time Series with Pretest Data-Based Model Selection," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(4), pages 461-470, October.
    22. Castro, Tomás del Barrio & Osborn, Denise R. & Taylor, A.M. Robert, 2012. "On Augmented Hegy Tests For Seasonal Unit Roots," Econometric Theory, Cambridge University Press, pages 1121-1143.
    23. Yoosoon Chang & Joon Park, 2002. "On The Asymptotics Of Adf Tests For Unit Roots," Econometric Reviews, Taylor & Francis Journals, pages 431-447.
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    Cited by:

    1. Tomás del Barrio Castro & Paulo M. M. Rodrigues & A. M. Robert Taylor, 2015. "Semi-Parametric Seasonal Unit Root Tests," DEA Working Papers 72, Universitat de les Illes Balears, Departament d'Economía Aplicada.
    2. Chambers, Marcus J. & Ercolani, Joanne S. & Taylor, A.M. Robert, 2014. "Testing for seasonal unit roots by frequency domain regression," Journal of Econometrics, Elsevier, pages 243-258.
    3. Tomás Barrio Castro & Andrii Bodnar & Andreu Sansó, 2017. "Numerical distribution functions for seasonal unit root tests with OLS and GLS detrending," Computational Statistics, Springer, vol. 32(4), pages 1533-1568, December.
    4. repec:spr:empeco:v:52:y:2017:i:4:d:10.1007_s00181-016-1102-8 is not listed on IDEAS
    5. Castro, Tomás del Barrio & Osborn, Denise R. & Taylor, A.M. Robert, 2012. "On Augmented Hegy Tests For Seasonal Unit Roots," Econometric Theory, Cambridge University Press, pages 1121-1143.
    6. Politis, Dimitris, 2016. "HEGY test under seasonal heterogeneity," University of California at San Diego, Economics Working Paper Series qt2q4054kf, Department of Economics, UC San Diego.
    7. Chambers, Marcus J. & Ercolani, Joanne S. & Taylor, A.M. Robert, 2014. "Testing for seasonal unit roots by frequency domain regression," Journal of Econometrics, Elsevier, pages 243-258.

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