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Regression With Slowly Varying Regressors And Nonlinear Trends

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  • Phillips, Peter C.B.

Abstract

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.â€

Suggested Citation

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

    1. Dong, Chaohua & Linton, Oliver, 2018. "Additive nonparametric models with time variable and both stationary and nonstationary regressors," Journal of Econometrics, Elsevier, vol. 207(1), pages 212-236.
    2. Kairat T. Mynbaev, 2011. "Regressions with asymptotically collinear regressors," Econometrics Journal, Royal Economic Society, vol. 14(2), pages 304-320, July.
    3. Peter C. B. Phillips & Donggyu Sul, 2007. "Transition Modeling and Econometric Convergence Tests," Econometrica, Econometric Society, vol. 75(6), pages 1771-1855, November.
    4. 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.
    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. Gao, Jiti & Robinson, Peter M., 2014. "Inference on nonstationary time series with moving mean," LSE Research Online Documents on Economics 66509, London School of Economics and Political Science, LSE Library.
    7. 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.
    8. 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.
    9. Yonghui Zhang & Liangjun Su & Peter C. B. Phillips, 2012. "Testing for common trends in semi‐parametric panel data models with fixed effects," Econometrics Journal, Royal Economic Society, vol. 15(1), pages 56-100, February.
    10. 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.
    11. Yicong Lin & Hanno Reuvers, 2020. "Cointegrating Polynomial Regressions with Power Law Trends: A New Angle on the Environmental Kuznets Curve," Papers 2009.02262, arXiv.org.
    12. Mynbayev, Kairat & Darkenbayeva, Gulsim, 2017. "Weak convergence of linear and quadratic forms and related statements on Lp-approximability," MPRA Paper 101686, University Library of Munich, Germany, revised Dec 2018.
    13. Norbert Christopeit & Michael Massmann, 2013. "Estimating Structural Parameters in Regression Models with Adaptive Learning," Tinbergen Institute Discussion Papers 13-111/III, Tinbergen Institute.
    14. Mynbayev, Kairat & Darkenbayeva, Gulsim, 2019. "Analyzing variance in central limit theorems," MPRA Paper 101685, University Library of Munich, Germany.
    15. Norbert Christopeit & Michael Massmann, 2017. "Strong consistency of the least squares estimator in regression models with adaptive learning," WHU Working Paper Series - Economics Group 17-07, WHU - Otto Beisheim School of Management.
    16. Uematsu, Yoshimasa, 2016. "Asymptotic efficiency of the OLS estimator with singular limiting sample moment matrices," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 104-110.
    17. Mynbayev, Kairat, 2007. "OLS Asymptotics for Vector Autoregressions with Deterministic Regressors," MPRA Paper 101688, University Library of Munich, Germany, revised 2018.
    18. Tassos Magdalinos, 2008. "Mildly explosive autoregression under weak and strong dependence," Discussion Papers 08/05, University of Nottingham, Granger Centre for Time Series Econometrics.
    19. 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.
    20. Jennifer Castle & David Hendry, 2010. "Automatic Selection for Non-linear Models," Economics Series Working Papers 473, University of Oxford, Department of Economics.
    21. 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.
    22. Norbert Christopeit & Michael Massmann, 2018. "Strong consistency of the least squares estimator in regression models with adaptive learning," Tinbergen Institute Discussion Papers 18-045/III, Tinbergen Institute.
    23. Magdalinos, Tassos, 2012. "Mildly explosive autoregression under weak and strong dependence," Journal of Econometrics, Elsevier, vol. 169(2), pages 179-187.
    24. 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.
    25. Zhishui Hu & Peter C.B. Phillips & Qiying Wang, 2019. "Nonlinear Cointegrating Power Function Regression with Endogeneity," Cowles Foundation Discussion Papers 2211, Cowles Foundation for Research in Economics, Yale University.

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