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Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

Author

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  • Pierre Perron

    (Boston University)

  • Mototsugu Shintani

    (University of Tokyo and Vanderbilt University)

  • Tomoyoshi Yabu

    (Keio University)

Abstract

This paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu (2009a) and is based on a Feasible Generalized Least Squares procedure that uses a super-efficient estimator of the sum of the autoregressive coefficients α when α=1. The resulting Wald test statistic asymptotically follows a chi-square limit distribution in both the I(0) and I(1) cases. To improve the finite sample properties of the test, we use a bias corrected version of the OLS estimator of α proposed by Roy and Fuller (2001). We show that our procedure is substantially more powerful than currently available alternatives. We illustrate the usefulness of our method via an application to modeling the trend of global and hemispheric temperatures.

Suggested Citation

  • Pierre Perron & Mototsugu Shintani & Tomoyoshi Yabu, 2015. "Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component," Vanderbilt University Department of Economics Working Papers 15-00001, Vanderbilt University Department of Economics.
  • Handle: RePEc:van:wpaper:vuecon-sub-15-00001
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    References listed on IDEAS

    as
    1. David I. Harvey & Stephen J. Leybourne & Lisa Xiao, 2010. "Testing for nonlinear deterministic components when the order of integration is unknown," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(5), pages 379-391, September.
    2. Pierre Perron & Tomoyoshi Yabu, 2012. "Testing for Trend in the Presence of Autoregressive Error: A Comment," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 844-844, June.
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    5. Perron, Pierre & Yabu, Tomoyoshi, 2009. "Estimating deterministic trends with an integrated or stationary noise component," Journal of Econometrics, Elsevier, vol. 151(1), pages 56-69, July.
    6. Francisco Estrada & Pierre Perron & Benjamin Martinez-Lopez, 2013. "Statistically-derived contributions of diverse human influences to 20th century temperature changes," Boston University - Department of Economics - Working Papers Series 2013-017, Boston University - Department of Economics.
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    8. Perron, Pierre, 1990. "Testing for a Unit Root in a Time Series with a Changing Mean," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(2), pages 153-162, April.
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    10. Pierre Perron & Francisco Estrada & Carlos Gay-García & Benjamín Martínez-López, 2011. "A time-series analysis of the 20th century climate simulations produced for the IPCC’s AR4," Boston University - Department of Economics - Working Papers Series WP2011-051, Boston University - Department of Economics.
    11. Gallant, A. Ronald, 1981. "On the bias in flexible functional forms and an essentially unbiased form : The fourier flexible form," Journal of Econometrics, Elsevier, vol. 15(2), pages 211-245, February.
    12. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    13. Perron, Pierre, 1989. "The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis," Econometrica, Econometric Society, vol. 57(6), pages 1361-1401, November.
    14. Paulo M. M. Rodrigues & A. M. Robert Taylor, 2012. "The Flexible Fourier Form and Local Generalised Least Squares De-trended Unit Root Tests-super-," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 74(5), pages 736-759, October.
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    16. Perron, Pierre & Yabu, Tomoyoshi, 2009. "Testing for Shifts in Trend With an Integrated or Stationary Noise Component," Journal of Business & Economic Statistics, American Statistical Association, vol. 27(3), pages 369-396.
    17. Anindya Roy & Barry Falk & Wayne A. Fuller, 2004. "Testing for Trend in the Presence of Autoregressive Error," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 1082-1091, December.
    18. Serena Ng & Pierre Perron, 2001. "LAG Length Selection and the Construction of Unit Root Tests with Good Size and Power," Econometrica, Econometric Society, vol. 69(6), pages 1519-1554, November.
    19. Ralf Becker & Walter Enders & Junsoo Lee, 2006. "A Stationarity Test in the Presence of an Unknown Number of Smooth Breaks," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(3), pages 381-409, May.
    20. Sam Astill & David I. Harvey & Stephen J. Leybourne & A. M. Robert Taylor, 2015. "Robust and Powerful Tests for Nonlinear Deterministic Components," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 77(6), pages 780-799, December.
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    Cited by:

    1. Claudio, Morana & Giacomo, Sbrana, 2017. "Some Financial Implications of Global Warming: An Empirical Assessment," Working Papers 377, University of Milano-Bicocca, Department of Economics, revised 25 Dec 2017.
    2. Claudio, Morana & Giacomo, Sbrana, 2017. "Temperature anomalies, radiative forcing and ENSO," Working Papers 361, University of Milano-Bicocca, Department of Economics, revised 10 Feb 2017.
    3. Atanu Ghoshray & Madhavi Pundit, 2021. "Economic growth in China and its impact on international commodity prices," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(2), pages 2776-2789, April.
    4. Banerjee, Piyali & Arčabić, Vladimir & Lee, Hyejin, 2017. "Fourier ADL cointegration test to approximate smooth breaks with new evidence from Crude Oil Market," Economic Modelling, Elsevier, vol. 67(C), pages 114-124.
    5. Yeonwoo Rho & Yun Liu & Hie Joo Ahn, 2020. "Revealing Cluster Structures Based on Mixed Sampling Frequencies," Papers 2004.09770, arXiv.org, revised Feb 2021.

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    More about this item

    Keywords

    nonlinear trends; unit root; median-unbiased estimator; GLS procedure; super-efficient estimator;
    All these keywords.

    JEL classification:

    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables

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