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Local Limit Theory and Spurious Nonparametric Regression

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Abstract

A local limit theorem is proved for sample covariances of nonstationary time series and integrable functions of such time series that involve a bandwidth sequence. The resulting theory enables an asymptotic development of nonparametric regression with integrated or fractionally integrated processes that includes the important practical case of spurious regressions. Some local regression diagnostics are suggested for forensic analysis of such regresssions, including a local R^2 and a local Durbin Watson (DW) ratio, and their asymptotic behavior is investigated. The most immediate findings extend the earlier work on linear spurious regression (Phillips, 1986), showing that the key behavioral characteristics of statistical significance, low DW ratios and moderate to high R^2 continue to apply locally in nonparametric spurious regression. Some further applications of the limit theory to models of nonlinear functional relations and cointegrating regressions are given. The methods are also shown to be applicable in partial linear semiparametric nonstationary regression.

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  • Peter C.B. Phillips, 2008. "Local Limit Theory and Spurious Nonparametric Regression," Cowles Foundation Discussion Papers 1654, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1654
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    References listed on IDEAS

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    1. Peter C. B. Phillips, 2001. "Descriptive econometrics for non-stationary time series with empirical illustrations," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(3), pages 389-413.
    2. de Jong, Robert M., 2004. "Addendum To," Econometric Theory, Cambridge University Press, vol. 20(03), pages 627-635, June.
    3. Joon Y. Park & Peter C. B. Phillips, 2000. "Nonstationary Binary Choice," Econometrica, Econometric Society, vol. 68(5), pages 1249-1280, September.
    4. P tscher, Benedikt M., 2004. "Nonlinear Functions And Convergence To Brownian Motion: Beyond The Continuous Mapping Theorem," Econometric Theory, Cambridge University Press, vol. 20(01), pages 1-22, February.
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    6. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339 Elsevier.
    7. Phillips, Peter C.B., 2005. "Challenges of trending time series econometrics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(5), pages 401-416.
    8. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
    9. Wang, Qiying & Phillips, Peter C.B., 2009. "Asymptotic Theory For Local Time Density Estimation And Nonparametric Cointegrating Regression," Econometric Theory, Cambridge University Press, vol. 25(03), pages 710-738, June.
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    12. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-161, January.
    13. Yatchew,Adonis, 2003. "Semiparametric Regression for the Applied Econometrician," Cambridge Books, Cambridge University Press, number 9780521812832, December.
    14. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
    15. Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 15(03), pages 269-298, June.
    16. Emmanuel Guerre, 2004. "Design-Adaptive Pointwise Nonparametric Regression Estimation for Recurrent Markov Time Series," Working Papers 2004-22, Center for Research in Economics and Statistics.
    17. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
    18. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, number 8355, June.
    19. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339 Elsevier.
    20. de Jong, Robert & Wang, Chien-Ho, 2005. "Further Results On The Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 21(02), pages 413-430, April.
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    Cited by:

    1. Patrick Saart & Jiti Gao & Nam Hyun Kim, 2014. "Semiparametric methods in nonlinear time series analysis: a selective review," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(1), pages 141-169, March.
    2. Jiti Gao & Peter C.B. Phillips, 2011. "Semiparametric Estimation in Multivariate Nonstationary Time Series Models," Monash Econometrics and Business Statistics Working Papers 17/11, Monash University, Department of Econometrics and Business Statistics.
    3. Chaohua Dong & Jiti Gao, 2012. "Expansion of Lévy Process Functionals and Its Application in Statistical Estimation," Monash Econometrics and Business Statistics Working Papers 2/12, Monash University, Department of Econometrics and Business Statistics.
    4. Yichen Gao & Zheng Li & Zhongjian Lin, 2014. "Semiparametric Estimation of Partially Linear Varying Coefficient Models with Time Trend and Nonstationary Regressors," Emory Economics 1412, Department of Economics, Emory University (Atlanta).
    5. Peter C. B. Phillips & Sainan Jin, 2014. "Testing the Martingale Hypothesis," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(4), pages 537-554, October.
    6. Kasparis, Ioannis & Phillips, Peter C.B., 2012. "Dynamic misspecification in nonparametric cointegrating regression," Journal of Econometrics, Elsevier, vol. 168(2), pages 270-284.
    7. Sun, Yiguo & Hsiao, Cheng & Li, Qi, 2011. "Measuring correlations of integrated but not cointegrated variables: A semiparametric approach," Journal of Econometrics, Elsevier, vol. 164(2), pages 252-267, October.
    8. Gao, Jiti & Phillips, Peter C.B., 2013. "Semiparametric estimation in triangular system equations with nonstationarity," Journal of Econometrics, Elsevier, vol. 176(1), pages 59-79.
    9. Biqing Cai & Jiti Gao, 2013. "Hermite Series Estimation in Nonlinear Cointegrating Models," Monash Econometrics and Business Statistics Working Papers 17/13, Monash University, Department of Econometrics and Business Statistics.
    10. Chaohua Dong & Jiti Gao, 2011. "Expansion of Brownian Motion Functionals and Its Application in Econometric Estimation," Monash Econometrics and Business Statistics Working Papers 19/11, Monash University, Department of Econometrics and Business Statistics.

    More about this item

    Keywords

    Brownian motion; Kernel method; Local R^2; Local Durbin-Watson ratio; Local time; Integrated process; Nonparametric regression; Spurious regression;

    JEL classification:

    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities

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