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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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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1654.

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Length: 31 pages
Date of creation: May 2008
Date of revision:
Publication status: Published in Econometric Theory (2009), 25: 1466-1497
Handle: RePEc:cwl:cwldpp:1654

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

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References

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  1. Guerre, 2004. "Design-Adaptive Pointwise Nonparametric Regression Estimation For Recurrent Markov Time Series," Econometrics, EconWPA 0411007, EconWPA.
  2. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics, Elsevier, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339 Elsevier.
  3. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
  4. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
  5. Peter C.B. Phillips, 2004. "Challenges of Trending Time Series Econometrics," Cowles Foundation Discussion Papers 1472, Cowles Foundation for Research in Economics, Yale University.
  6. Peter C.B. Phillips, 1999. "Descriptive Econometrics for Nonstationary Time Series with Empirical Illustrations," Cowles Foundation Discussion Papers 1219, Cowles Foundation for Research in Economics, Yale University.
  7. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
  8. 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.
  9. Yatchew,Adonis, 2003. "Semiparametric Regression for the Applied Econometrician," Cambridge Books, Cambridge University Press, number 9780521812832, 9.
  10. Qiying Wang & Peter C.B. Phillips, 2006. "Asymptotic Theory for Local Time Density Estimation and Nonparametric Cointegrating Regression," Cowles Foundation Discussion Papers 1594, Cowles Foundation for Research in Economics, Yale University.
  11. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, Econometric Society, vol. 69(1), pages 117-61, January.
  12. Berkes, Istv n & Horv th, Lajos, 2006. "Convergence Of Integral Functionals Of Stochastic Processes," Econometric Theory, Cambridge University Press, vol. 22(02), pages 304-322, April.
  13. 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.
  14. Joon Y. Park & Peter C. B. Phillips, 2000. "Nonstationary Binary Choice," Econometrica, Econometric Society, Econometric Society, vol. 68(5), pages 1249-1280, September.
  15. 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.
  16. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, Econometric Society, vol. 66(6), pages 1299-1326, November.
  17. Granger, C. W. J. & Newbold, P., 1974. "Spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 2(2), pages 111-120, July.
  18. Peter C.B. Phillips & Joon Y. Park, 1998. "Nonstationary Density Estimation and Kernel Autoregression," Cowles Foundation Discussion Papers 1181, Cowles Foundation for Research in Economics, Yale University.
  19. de Jong, Robert M., 2004. "Addendum To," Econometric Theory, Cambridge University Press, vol. 20(03), pages 627-635, June.
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Cited by:
  1. Peter C.B. Phillips & Sainan Jin, 2013. "Testing the Martingale Hypothesis," Cowles Foundation Discussion Papers 1912, Cowles Foundation for Research in Economics, Yale University.
  2. Chaohua Dong & Jiti Gao, 2011. "Expansion of Brownian Motion Functionals and Its Application in Econometric Estimation," Monash Econometrics and Business Statistics Working Papers, Monash University, Department of Econometrics and Business Statistics 19/11, Monash University, Department of Econometrics and Business Statistics.
  3. Yichen Gao & Zheng Li & Zhongjian Lin, 2014. "Semiparametric Estimation of Partially Linear Varying Coefficient Models with Time Trend and Nonstationary Regressors," Emory Economics, Department of Economics, Emory University (Atlanta) 1412, Department of Economics, Emory University (Atlanta).
  4. Kasparis, Ioannis & Phillips, Peter C.B., 2012. "Dynamic misspecification in nonparametric cointegrating regression," Journal of Econometrics, Elsevier, vol. 168(2), pages 270-284.
  5. 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.
  6. Jiti Gao & Peter C.B. Phillips, 2011. "Semiparametric Estimation in Multivariate Nonstationary Time Series Models," Monash Econometrics and Business Statistics Working Papers, Monash University, Department of Econometrics and Business Statistics 17/11, Monash University, Department of Econometrics and Business Statistics.
  7. Patrick Saart & Jiti Gao, 2012. "Semiparametric Methods in Nonlinear Time Series Analysis: A Selective Review," Monash Econometrics and Business Statistics Working Papers, Monash University, Department of Econometrics and Business Statistics 21/12, Monash University, Department of Econometrics and Business Statistics.
  8. Biqing Cai & Jiti Gao, 2013. "Hermite Series Estimation in Nonlinear Cointegrating Models," Monash Econometrics and Business Statistics Working Papers, Monash University, Department of Econometrics and Business Statistics 17/13, Monash University, Department of Econometrics and Business Statistics.
  9. Chaohua Dong & Jiti Gao, 2012. "Expansion of Lévy Process Functionals and Its Application in Statistical Estimation," Monash Econometrics and Business Statistics Working Papers, Monash University, Department of Econometrics and Business Statistics 2/12, Monash University, Department of Econometrics and Business Statistics.
  10. 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.

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