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

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  • Pierre Perron
  • Mototsugu Shintani
  • Tomoyoshi Yabu
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 superefficient estimator of the sum of the autoregressive coe¢ cients when = 1. The resulting Wald test statistic asymptotically follows chi-square 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 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.
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Suggested Citation

  • 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.
    • Handle: RePEc:bla:obuest:v:79:y:2017:i:5:p:822-850
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    File URL: http://hdl.handle.net/10.1111/obes.12169
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    2. 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.
    3. 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.
    4. 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.
    5. Gallant, A. Ronald & Souza, Geraldo, 1991. "On the asymptotic normality of Fourier flexible form estimates," Journal of Econometrics, Elsevier, vol. 50(3), pages 329-353, December.
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    13. 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.
    14. 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.
    15. 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.
    16. 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.
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    19. 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.
    20. Roy, Anindya & Fuller, Wayne A, 2001. "Estimation for Autoregressive Time Series with a Root Near 1," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(4), pages 482-493, October.
    21. 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.
    22. 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.
    23. Timothy J. Vogelsang, 1998. "Trend Function Hypothesis Testing in the Presence of Serial Correlation," Econometrica, Econometric Society, vol. 66(1), pages 123-148, January.
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    Cited by:

    1. Claudio Morana & Giacomo Sbrana, 2017. "Temperature anomalies, radiative forcing and ENSO," Working Paper series 17-06, Rimini Centre for Economic Analysis.
    2. Jamal G. HUSEIN & S. Murat KARA, 2023. "Are Shocks To Electricity Consumption Permanent Or Transitory? Evidence From A Panel Stationarity Test With Gradual Structural Breaks For 25 Oecd Countries," Applied Econometrics and International Development, Euro-American Association of Economic Development, vol. 23(1), pages 57-76.
    3. 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.
    4. Yeonwoo Rho & Yun Liu & Hie Joo Ahn, 2020. "Revealing Cluster Structures Based on Mixed Sampling Frequencies," Papers 2004.09770, arXiv.org, revised Feb 2021.
    5. 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.
    6. Mübariz Hasanov & Tolga Omay & Vasif Abioglu, 2024. "Re-examining the real interest rate parity hypothesis under temporary gradual breaks and nonlinear convergence," Portuguese Economic Journal, Springer;Instituto Superior de Economia e Gestao, vol. 23(3), pages 355-382, September.
    7. 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.
    8. Razvan Pascalau & Junsoo Lee & Saban Nazlioglu & Yan (Olivia) Lu, 2022. "Johansen‐type cointegration tests with a Fourier function," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(5), pages 828-852, September.
    9. Erhua Zhang & Xiaojun Song & Jilin Wu, 2022. "A non‐parametric test for multi‐variate trend functions," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(6), pages 856-871, November.
    10. Husein, Jamal, 2020. "Current account sustainability for 21 African economies: Evidence based on nonlinear flexible Fourier stationarity and unit-root tests," MPRA Paper 100410, University Library of Munich, Germany.
    11. 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.
    12. Martin B. Schmidt, 2021. "On the evolution of athlete anthropometric measurements: racial integration, expansion, and steroids," Empirical Economics, Springer, vol. 61(6), pages 3419-3443, December.

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    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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