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Testing Linearity Using Power Transforms of Regressors

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. The paper also considers generalizations of the Box-Cox transformation, which are associated with the QLR test statistic.

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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1917.

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Length: 53 pages
Date of creation: Sep 2013
Date of revision:
Publication status: Published in Journal of Econometrics (July 2015), 187(1): 376-384
Handle: RePEc:cwl:cwldpp:1917
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Order Information: Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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  1. Peter C.B. Phillips & Joon Y. Park, 1998. "Asymptotics for Nonlinear Transformations of Integrated Time Series," Cowles Foundation Discussion Papers 1182, Cowles Foundation for Research in Economics, Yale University.
  2. Bierens, Herman J, 1990. "A Consistent Conditional Moment Test of Functional Form," Econometrica, Econometric Society, vol. 58(6), pages 1443-58, November.
  3. Phillips, Peter C.B., 2007. "Regression With Slowly Varying Regressors And Nonlinear Trends," Econometric Theory, Cambridge University Press, vol. 23(04), pages 557-614, August.
  4. Shi, Xiaoxia & Phillips, Peter C.B., 2012. "Nonlinear Cointegrating Regression Under Weak Identification," Econometric Theory, Cambridge University Press, vol. 28(03), pages 509-547, June.
  5. Bierens, Herman J., 1982. "Consistent model specification tests," Journal of Econometrics, Elsevier, vol. 20(1), pages 105-134, October.
  6. 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.
  7. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-30, March.
  8. Weninger, Quinn & Just, Richard E., 1999. "Are Crop Yields Normally Distributed?," Staff General Research Papers Archive 5064, Iowa State University, Department of Economics.
  9. Cho, Jin Seo & Ishida, Isao, 2012. "Testing for the effects of omitted power transformations," Economics Letters, Elsevier, vol. 117(1), pages 287-290.
  10. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 1," Cowles Foundation Discussion Papers 811R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1987.
  11. Peter C.B. Phillips, 1987. "Partially Identified Econometric Models," Cowles Foundation Discussion Papers 845R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1988.
  12. 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.
  13. 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.
  14. Stinchcombe, Maxwell B. & White, Halbert, 1998. "Consistent Specification Testing With Nuisance Parameters Present Only Under The Alternative," Econometric Theory, Cambridge University Press, vol. 14(03), pages 295-325, June.
  15. Octavio A. Ramirez & Sukant Misra & James Field, 2003. "Crop-Yield Distributions Revisited," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 85(1), pages 108-120.
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