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Assessing the Precision of Turning Point Estimates in Polynomial Regression Functions


  • Florenz Plassmann
  • Neha Khanna


Researchers often report point estimates of turning point(s) obtained in polynomial regression models but rarely assess the precision of these estimates. We discuss three methods to assess the precision of such turning point estimates. The first is the delta method that leads to a normal approximation of the distribution of the turning point estimator. The second method uses the exact distribution of the turning point estimator of quadratic regression functions. The third method relies on Markov chain Monte Carlo methods to provide a finite sample approximation of the exact distribution of the turning point estimator. We argue that the delta method may lead to misleading inference and that the other two methods are more reliable. We compare the three methods using two data sets from the environmental Kuznets curve literature, where the presence and location of a turning point in the income-pollution relationship is the focus of much empirical work.

Suggested Citation

  • Florenz Plassmann & Neha Khanna, 2007. "Assessing the Precision of Turning Point Estimates in Polynomial Regression Functions," Econometric Reviews, Taylor & Francis Journals, vol. 26(5), pages 503-528.
  • Handle: RePEc:taf:emetrv:v:26:y:2007:i:5:p:503-528 DOI: 10.1080/07474930701512105

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    References listed on IDEAS

    1. Bent Nielsen, 1995. "Bartlett correction of the unit root test in autoregressive models," Economics Papers 11 & 98., Economics Group, Nuffield College, University of Oxford.
    2. Nielsen, Bent, 2001. "The Asymptotic Distribution of Unit Root Tests of Unstable Autoregressive Processes," Econometrica, Econometric Society, vol. 69(1), pages 211-219, January.
    3. Bent Nielsen, 2004. "On the Distribution of Likelihood Ratio Test Statistics for Cointegration Rank," Econometric Reviews, Taylor & Francis Journals, vol. 23(1), pages 1-23.
    4. Nielsen, Bent, 2005. "Strong Consistency Results For Least Squares Estimators In General Vector Autoregressions With Deterministic Terms," Econometric Theory, Cambridge University Press, vol. 21(03), pages 534-561, June.
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    Cited by:

    1. Larivière, Jean-Michel & He, Jie, 2012. "L’impact de la taille des firmes industrielles sur la courbe de Kuznets environnementale : le cas des émissions de SO2 en Chine," L'Actualité Economique, Société Canadienne de Science Economique, vol. 88(1), pages 5-36, mars.
    2. Pei-Ing Wu & Je-Liang Liou & Hung-Yi Chang, 2015. "Alternative exploration of EKC for $$\hbox {CO}_{2}$$ CO 2 emissions: inclusion of meta-technical ratio in quantile regression model," Quality & Quantity: International Journal of Methodology, Springer, vol. 49(1), pages 57-73, January.
    3. Onafowora, Olugbenga A. & Owoye, Oluwole, 2014. "Bounds testing approach to analysis of the environment Kuznets curve hypothesis," Energy Economics, Elsevier, vol. 44(C), pages 47-62.
    4. repec:eee:respol:v:46:y:2017:i:10:p:1836-1850 is not listed on IDEAS
    5. Kearsley, Aaron & Riddel, Mary, 2010. "A further inquiry into the Pollution Haven Hypothesis and the Environmental Kuznets Curve," Ecological Economics, Elsevier, vol. 69(4), pages 905-919, February.
    6. J. N. Lye and J. G. Hirschberg, 2012. "Inverse Test Confidence Intervals for Turning points: A," Department of Economics - Working Papers Series 1160, The University of Melbourne.
    7. Siriwardena, Shyamani D. & Boyle, Kevin J. & Holmes, Thomas P. & Wiseman, P. Eric, 2016. "The implicit value of tree cover in the U.S.: A meta-analysis of hedonic property value studies," Ecological Economics, Elsevier, vol. 128(C), pages 68-76.
    8. Khanna, Neha & Plassmann, Florenz, 2004. "The demand for environmental quality and the environmental Kuznets Curve hypothesis," Ecological Economics, Elsevier, vol. 51(3-4), pages 225-236, December.
    9. Brown, Stephen P.A. & McDonough, Ian K., 2016. "Using the Environmental Kuznets Curve to evaluate energy policy: Some practical considerations," Energy Policy, Elsevier, vol. 98(C), pages 453-458.
    10. Barakatou Atte-Oudeyi & Bruno Kestemont & Jean Luc De Meulemeester, 2016. "Road Transport, Economic Growth and Carbon Dioxide Emissions in the BRIICS: Conditions For a Low Carbon Economic Development," Working Papers CEB 16-023, ULB -- Universite Libre de Bruxelles.


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