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Regression Asymptotics Using Martingale Convergence Methods

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Abstract

Weak convergence of partial sums and multilinear forms in independent random variables and linear processes to stochastic integrals now plays a major role in nonstationary time series and has been central to the development of unit root econometrics. The present paper develops a new and conceptually simple method for obtaining such forms of convergence. The method relies on the fact that the econometric quantities of interest involve discrete time martingales or semimartingales and shows how in the limit these quantities become continuous martingales and semimartingales. The limit theory itself uses very general convergence results for semimartingales that were obtained in work by Jacod and Shiryaev (2003). The theory that is developed here is applicable in a wide range of econometric models and many examples are given. One notable outcome of the new approach is that it provides a unified treatment of the asymptotics for stationary autoregression and autoregression with roots at or near unity, as both these cases are subsumed within the martingale convergence approach and different rates of convergence are accommodated in a natural way. The approach is also useful in developing asymptotics for certain nonlinear functions of integrated processes, which are now receiving attention in econometric applications, and some new results in this area are presented. The paper is partly of pedagogical interest and the conceptual simplicity of the methods is appealing. Since this is the first time the methods have been used in econometrics, the exposition is presented in some detail with illustrations of new derivations of some well-known existing results, as well as some new asymptotic results and the unification of the limit theory for autoregression.

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File URL: http://cowles.econ.yale.edu/P/cd/d14b/d1473.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1473.

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Length: 42 pages
Date of creation: Jul 2004
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Publication status: Published in Econometric Theory (August 2008), 24(4): 888-947
Handle: RePEc:cwl:cwldpp:1473

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Keywords: Semimartingale; martingale; convergence; stochastic integrals; bilinear forms; multilinear forms; U-statistics; unit root; stationarity; Brownian motion; invariance principle; unification;

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References

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  1. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
  2. Phillips, P.C.B., 1986. "Testing for a Unit Root in Time Series Regression," Cahiers de recherche 8633, Universite de Montreal, Departement de sciences economiques.
  3. Ibragimov, Rustam & Phillips, Peter C.B., 2008. "Regression Asymptotics Using Martingale Convergence Methods," Econometric Theory, Cambridge University Press, vol. 24(04), pages 888-947, August.
  4. Phillips, Peter C B & Ploberger, Werner, 1996. "An Asymptotic Theory of Bayesian Inference for Time Series," Econometrica, Econometric Society, vol. 64(2), pages 381-412, March.
  5. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-61, January.
  6. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  7. 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.
  8. 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.
  9. Peter C.B. Phillips, 1999. "Unit Root Log Periodogram Regression," Cowles Foundation Discussion Papers 1244, Cowles Foundation for Research in Economics, Yale University.
  10. Liudas Giraitis & Peter C.B. Phillips, 2004. "Uniform Limit Theory for Stationary Autoregression," Cowles Foundation Discussion Papers 1475, Cowles Foundation for Research in Economics, Yale University.
  11. Nze, Patrick Ango & Doukhan, Paul, 2004. "Weak Dependence: Models And Applications To Econometrics," Econometric Theory, Cambridge University Press, vol. 20(06), pages 995-1045, December.
  12. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
  13. Saikkonen, Pentti & Choi, In, 2004. "Cointegrating Smooth Transition Regressions," Econometric Theory, Cambridge University Press, vol. 20(02), pages 301-340, April.
  14. In Choi & Pentti Saikkonen, 2004. "Testing linearity in cointegrating smooth transition regressions," Econometrics Journal, Royal Economic Society, vol. 7(2), pages 341-365, December.
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Citations

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Cited by:
  1. Ioannis Kasparis & Peter C.B. Phillips & Tassos Magdalinos, 2012. "Non-linearity Induced Weak Instrumentation," Cowles Foundation Discussion Papers 1872, Cowles Foundation for Research in Economics, Yale University.
  2. Peter C.B. Phillips & Degui Li & Jiti Gao, 2013. "Estimating Smooth Structural Change in Cointegration Models," Cowles Foundation Discussion Papers 1910, Cowles Foundation for Research in Economics, Yale University.
  3. MArcelo Cunha Medeiros & Eduardo Mendes & Les Oxley, 2010. "Nonlinear Cointegration, Misspecification and Bimodality," Textos para discussão 577, Department of Economics PUC-Rio (Brazil).
  4. Pötscher, Benedikt M., 2013. "On The Order Of Magnitude Of Sums Of Negative Powers Of Integrated Processes," Econometric Theory, Cambridge University Press, vol. 29(03), pages 642-658, June.
  5. Chang, Yoosoon, 2004. "Taking a New Contour: A Novel Approach to Panel Unit Root Tests," Working Papers 2004-05, Rice University, Department of Economics.
  6. Hong, Seung Hyun & Phillips, Peter C. B., 2010. "Testing Linearity in Cointegrating Relations With an Application to Purchasing Power Parity," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(1), pages 96-114.
  7. Rustam Ibragimov & Peter C.B. Phillips, 2004. "Regression Asymptotics Using Martingale Convergence Methods," Cowles Foundation Discussion Papers 1473, Cowles Foundation for Research in Economics, Yale University.
  8. M.C. Medeiros & E. Mendes & Les Oxley, 2010. "A Note on Nonlinear Cointegration, Misspecification and Bimodality," Working Papers in Economics 10/01, University of Canterbury, Department of Economics and Finance.
  9. Bent Nielsen & Carlos Caceres, 2007. "Convergence to Stochastic Integrals with Non-linear Integrands," Economics Series Working Papers 2007-W02, University of Oxford, Department of Economics.
  10. Wagner, Martin, 2012. "The Phillips unit root tests for polynomials of integrated processes," Economics Letters, Elsevier, vol. 114(3), pages 299-303.

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