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Regression With Slowly Varying Regressors And Nonlinear Trends


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  • Phillips, Peter C.B.


Slowly varying (SV) regressors arise commonly in empirical econometric work, particularly in the form of semilogarithmic regression and log periodogram regression. These regressors are asymptotically collinear. Usual regression formulas for asymptotic standard errors are shown to remain valid, but rates of convergence are affected and the limit distribution of the regression coefficients is shown to be one dimensional. Some asymptotic representations of partial sums of SV functions and central limit theorems with SV weights are given that assist in the development of a regression theory. Multivariate regression and polynomial regression with SV functions are considered and shown to be equivalent, up to standardization, to regression on a polynomial in a logarithmic trend. The theory involves second-, third-, and higher-order forms of slow variation. Some applications to the asymptotic theory of nonlinear trend regression are explored.The author thanks two referees and Pentti Saikkonen for comments and suggestions, Sidney Resnick for references on second-order regular variation, and a Kelly Fellowship and the NSF for partial research support under grants SBR 97-30295 and SES 04-142254. An original draft of the paper was written in June 2000 and circulated under the title Regression with Slowly Varying Regressors.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 23 (2007)
Issue (Month): 04 (August)
Pages: 557-614

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Handle: RePEc:cup:etheor:v:23:y:2007:i:04:p:557-614_07

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Cited by:
  1. Peter C.B. Phillips & Donggyu Sul, 2007. "Transition Modeling and Econometric Convergence Tests," Cowles Foundation Discussion Papers 1595, Cowles Foundation for Research in Economics, Yale University.
  2. Magdalinos, Tassos, 2012. "Mildly explosive autoregression under weak and strong dependence," Journal of Econometrics, Elsevier, vol. 169(2), pages 179-187.
  3. Mynbaev, Kairat, 2009. "Regressions with Asymptotically Collinear Regressor," MPRA Paper 31315, University Library of Munich, Germany.
  4. Yoshimasa Uematsu, 2011. "Regression with a Slowly Varying Regressor in the Presence of a Unit Root," Global COE Hi-Stat Discussion Paper Series gd11-209, Institute of Economic Research, Hitotsubashi University.
  5. Castle, Jennifer L. & Hendry, David F., 2010. "A low-dimension portmanteau test for non-linearity," Journal of Econometrics, Elsevier, vol. 158(2), pages 231-245, October.
  6. Yonghui Zhang & Liangjun Su & Peter C. B. Phillips, 2012. "Testing for common trends in semiā€parametric panel data models with fixed effects," Econometrics Journal, Royal Economic Society, vol. 15(1), pages 56-100, 02.
  7. Tassos Magdalinos, 2008. "Mildly explosive autoregression under weak and strong dependence," Discussion Papers 08/05, University of Nottingham, Granger Centre for Time Series Econometrics.
  8. Mynbaev, Kairat, 2007. "Comment on "Regression with slowly varying regressors and nonlinear trends" by P.C.B. Phillips," MPRA Paper 8838, University Library of Munich, Germany, revised 23 May 2008.
  9. Yoshimasa Uematsu, 2011. "Asymptotic Efficiency of the OLS Estimator with Singular Limiting Sample Moment Matrices," Global COE Hi-Stat Discussion Paper Series gd11-208, Institute of Economic Research, Hitotsubashi University.
  10. Jennifer Castle & David Hendry, 2010. "Automatic Selection for Non-linear Models," Economics Series Working Papers 473, University of Oxford, Department of Economics.
  11. Yae In Baek & Jin Seo Cho & Peter C.B. Phillips, 2013. "Testing Linearity Using Power Transforms of Regressors," Cowles Foundation Discussion Papers 1917, Cowles Foundation for Research in Economics, Yale University.


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