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Testing for nonlinear deterministic components when the order of integration is unknown

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  • David I. Harvey
  • Stephen J. Leybourne
  • Lisa Xiao

Abstract

We consider testing for the presence of nonlinearities in the deterministic component of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear deterministic components in a series.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jtsera:v:31:y:2010:i:5:p:379-391
    DOI: 10.1111/j.1467-9892.2010.00671.x
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    References listed on IDEAS

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    1. Timothy J. Vogelsang, 1998. "Trend Function Hypothesis Testing in the Presence of Serial Correlation," Econometrica, Econometric Society, vol. 66(1), pages 123-148, January.
    2. Perron, Pierre & Qu, Zhongjun, 2007. "A simple modification to improve the finite sample properties of Ng and Perron's unit root tests," Economics Letters, Elsevier, vol. 94(1), pages 12-19, January.
    3. 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.
    4. Breitung, Jorg, 2002. "Nonparametric tests for unit roots and cointegration," Journal of Econometrics, Elsevier, vol. 108(2), pages 343-363, June.
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    Cited by:

    1. Bent Jesper Christensen & Robinson Kruse & Philipp Sibbertsen, 2013. "A unified framework for testing in the linear regression model under unknown order of fractional integration," CREATES Research Papers 2013-35, Department of Economics and Business Economics, Aarhus University.
    2. Takamitsu Kurita & Mototsugu Shintani, 2023. "Johansen Test with Fourier-Type Smooth Nonlinear Trends in Cointegrating Relations," CIRJE F-Series CIRJE-F-1216, CIRJE, Faculty of Economics, University of Tokyo.
    3. Pierre Perron & Mototsugu Shintani & Tomoyoshi Yabu, 2017. "Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 79(5), pages 822-850, October.
    4. Pierre Perron & Mototsugu Shintaniz & Tomoyoshi Yabu, 2020. "Trigonometric Trend Regressions of Unknown Frequencies with Stationary or Integrated Noise," Boston University - Department of Economics - Working Papers Series WP2020-012, Boston University - Department of Economics.

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