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Testing linearity using power transforms of regressors

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  • Baek, Yae In
  • Cho, Jin Seo
  • Phillips, Peter C.B.

Abstract

We develop a method of testing linearity using power transforms of regressors, allowing for stationary processes and time trends. The linear model is a simplifying hypothesis that derives from the power transform model in three different ways, each producing its own identification problem. We call this modeling difficulty the trifold identification problem and show that it may be overcome using a test based on the quasi-likelihood ratio (QLR) statistic. More specifically, the QLR statistic may be approximated under each identification problem and the separate null approximations may be combined to produce a composite approximation that embodies the linear model hypothesis. The limit theory for the QLR test statistic depends on a Gaussian stochastic process. In the important special case of a linear time trend regressor and martingale difference errors asymptotic critical values of the test are provided. Test power is analyzed and an empirical application to crop-yield distributions is provided. The paper also considers generalizations of the Box–Cox transformation, which are associated with the QLR test statistic.

Suggested Citation

  • Baek, Yae In & Cho, Jin Seo & Phillips, Peter C.B., 2015. "Testing linearity using power transforms of regressors," Journal of Econometrics, Elsevier, vol. 187(1), pages 376-384.
    • Handle: RePEc:eee:econom:v:187:y:2015:i:1:p:376-384
    DOI: 10.1016/j.jeconom.2015.03.041
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    1. Jin Seo Cho & Isao Ishida & Halbert White, 2013. "Testing for Neglected Nonlinearity Using Twofold Unidentified Models under the Null and Hexic Expansions (published in: Essays in Nonlinear Time Series Econometrics, Festschrift in Honor of Timo Teras," Working papers 2013rwp-55, Yonsei University, Yonsei Economics Research Institute.
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    3. Richard E. Just & Quinn Weninger, 1999. "Are Crop Yields Normally Distributed?," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 81(2), pages 287-304.
    4. Cho, Jin Seo & White, Halbert, 2010. "Testing for unobserved heterogeneity in exponential and Weibull duration models," Journal of Econometrics, Elsevier, vol. 157(2), pages 458-480, August.
    5. Andrews, Donald W.K., 1992. "Generic Uniform Convergence," Econometric Theory, Cambridge University Press, vol. 8(2), pages 241-257, June.
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    7. Jin Seo Cho & Isao Ishida & Halbert White, 2013. "Mathematical Proofs for "Testing for Neglected Nonlinearity Using Twofold Unidentified Models under the Null and Hexic Expansions"," Working papers 2013rwp-55a, Yonsei University, Yonsei Economics Research Institute.
    8. Bierens, Herman J, 1990. "A Consistent Conditional Moment Test of Functional Form," Econometrica, Econometric Society, vol. 58(6), pages 1443-1458, November.
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    10. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(3), pages 468-497, December.
    11. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-430, March.
    12. Bierens, Herman J., 1982. "Consistent model specification tests," Journal of Econometrics, Elsevier, vol. 20(1), pages 105-134, October.
    13. YAE IN BAEK & Jin Seo Cho & PETER C.B. PHILLIPS, 2015. "We provide mathematical proofs for the results in "Testing Linearity Using Power Transforms of Regressors"," Working papers 2015rwp-79a, Yonsei University, Yonsei Economics Research Institute.
    14. Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 15(3), pages 269-298, June.
    15. Cho, Jin Seo & Ishida, Isao, 2012. "Testing for the effects of omitted power transformations," Economics Letters, Elsevier, vol. 117(1), pages 287-290.
    16. Phillips, Peter C.B., 2007. "Regression With Slowly Varying Regressors And Nonlinear Trends," Econometric Theory, Cambridge University Press, vol. 23(4), pages 557-614, August.
    17. Scott M. Swinton & Robert P. King, 1991. "Evaluating Robust Regression Techniques for Detrending Crop Yield Data with Nonnormal Errors," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 73(2), pages 446-451.
    18. Shi, Xiaoxia & Phillips, Peter C.B., 2012. "Nonlinear Cointegrating Regression Under Weak Identification," Econometric Theory, Cambridge University Press, vol. 28(3), pages 509-547, June.
    19. Jarque, Carlos M. & Bera, Anil K., 1980. "Efficient tests for normality, homoscedasticity and serial independence of regression residuals," Economics Letters, Elsevier, vol. 6(3), pages 255-259.
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    As found by EconAcademics.org, the blog aggregator for Economics research:
    1. April Reading
      by Dave Giles in Econometrics Beat: Dave Giles' Blog on 2015-04-02 01:53:00

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    Cited by:

    1. Peter C. B. Phillips & Zhentao Shi, 2021. "Boosting: Why You Can Use The Hp Filter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 62(2), pages 521-570, May.
    2. Jin Seo Cho & Jin Seok Park & Sang Woo Park, 2018. "Testing for the Conditional Geometric Mixture Distribution," Working papers 2018rwp-123, Yonsei University, Yonsei Economics Research Institute.
    3. Jin Seo Cho & Peter C. B. Phillips, 2018. "Sequentially testing polynomial model hypotheses using power transforms of regressors," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 33(1), pages 141-159, January.
    4. Yicong Lin & Hanno Reuvers, 2020. "Cointegrating Polynomial Regressions with Power Law Trends: Environmental Kuznets Curve or Omitted Time Effects?," Papers 2009.02262, arXiv.org, revised Dec 2021.
    5. Cho, Jin Seo & Phillips, Peter C.B., 2018. "Pythagorean generalization of testing the equality of two symmetric positive definite matrices," Journal of Econometrics, Elsevier, vol. 202(1), pages 45-56.
    6. Hu, Zhishui & Phillips, Peter C.B. & Wang, Qiying, 2021. "Nonlinear Cointegrating Power Function Regression With Endogeneity," Econometric Theory, Cambridge University Press, vol. 37(6), pages 1173-1213, December.
    7. Jaedo Choi & Yun Jeong Choi & Minki Kim, 2017. "Vertical Foreclosure with Product Choice and Allocation: Evidence from the Movie Industry," Working papers 2017rwp-107, Yonsei University, Yonsei Economics Research Institute.
    8. Peter C.B. Phillips & Zhentao Shi, 2019. "Boosting the Hodrick-Prescott Filter," Cowles Foundation Discussion Papers 2192, Cowles Foundation for Research in Economics, Yale University.
    9. Jin Seo Cho & Peter C. B. Phillips & Juwon Seo, 2019. "Parametric Inference on the Mean of Functional Data Applied to Lifetime Income Curves," Working papers 2019rwp-153, Yonsei University, Yonsei Economics Research Institute.
    10. Dakyung Seong & Jin Seo Cho & Timo Terasvirta, 2019. "Comprehensive Testing of Linearity against the Smooth Transition Autoregressive Model," Working papers 2019rwp-151, Yonsei University, Yonsei Economics Research Institute.
    11. Kong, Jianning & Phillips, Peter C.B. & Sul, Donggyu, 2019. "Weak σ-convergence: Theory and applications," Journal of Econometrics, Elsevier, vol. 209(2), pages 185-207.
    12. Li, Degui & Phillips, Peter C.B. & Gao, Jiti, 2020. "Kernel-based Inference in Time-Varying Coefficient Cointegrating Regression," Journal of Econometrics, Elsevier, vol. 215(2), pages 607-632.
    13. He, Changli & Kang, Jian & Teräsvirta, Timo & Zhang, Shuhua, 2019. "The shifting seasonal mean autoregressive model and seasonality in the Central England monthly temperature series, 1772–2016," Econometrics and Statistics, Elsevier, vol. 12(C), pages 1-24.
    14. Jin Seo Cho & Peter C. B. Phillips & Juwon Seo, 2022. "Parametric Conditional Mean Inference With Functional Data Applied To Lifetime Income Curves," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 63(1), pages 391-456, February.
    15. Jin Seo Cho & Matthew Greenwood‐Nimmo & Yongcheol Shin, 2023. "Recent developments of the autoregressive distributed lag modelling framework," Journal of Economic Surveys, Wiley Blackwell, vol. 37(1), pages 7-32, February.
    16. Lijuan Huo & Jin Seo Cho, 2020. "Sequentially Estimating Approximate Conditional Mean Using the Extreme Learning Machine," Working papers 2020rwp-180, Yonsei University, Yonsei Economics Research Institute.
    17. Choi, Jaedo & Moon, Hyungsik Roger & Cho, Jin Seo, 2024. "Sequentially Estimating The Structural Equation By Power Transformation," Econometric Theory, Cambridge University Press, vol. 40(1), pages 98-161, February.
    18. Cho, Jin Seo & White, Halbert, 2018. "Directionally Differentiable Econometric Models," Econometric Theory, Cambridge University Press, vol. 34(5), pages 1101-1131, October.
    19. Jin Seo Cho & Meng Huang & Halbert White, 2021. "Testing a Constant Mean Function Using Functional Regression," Working papers 2021rwp-190, Yonsei University, Yonsei Economics Research Institute.

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    More about this item

    Keywords

    Box–Cox transform; Gaussian stochastic process; Neglected nonlinearity; Power transformation; Quasi-likelihood ratio test; Trend exponent; Trifold identification problem;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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