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Regression asymptotics using martingale convergence methods

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  • Ibragimov, Rustam
  • Phillips, Peter C.B.

Abstract

Weak convergence of partial sums and multilinear forms in independent random variables and linear processes and their nonlinear analogues 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 the work of Jacod and Shiryaev (2003, Limit Theorems for Stochastic Processes). 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, explosive, unit root, and local to unity autoregression, and also some general nonlinear time series regressions. All of these cases are subsumed within the martingale convergence approach, and different rates of convergence are accommodated in a natural way. Moreover, the results on multivariate extensions developed in the paper deliver a unification of the asymptotics for, among many others, models with cointegration and also for regressions with regressors that are nonlinear transforms of integrated time series driven by shocks correlated with the equation errors. Because 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, in addition to the provision of new results and the unification of the limit theory for autoregression.

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Bibliographic Info

Paper provided by Harvard University Department of Economics in its series Scholarly Articles with number 2624459.

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Date of creation: 2008
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Publication status: Published in Econometric Theory
Handle: RePEc:hrv:faseco:2624459

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References

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  1. Peter C.B. Phillips, 1999. "Unit Root Log Periodogram Regression," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1244, Cowles Foundation for Research in Economics, Yale University.
  2. Phillips, Peter C B & Ploberger, Werner, 1996. "An Asymptotic Theory of Bayesian Inference for Time Series," Econometrica, Econometric Society, Econometric Society, vol. 64(2), pages 381-412, March.
  3. Peter C.B. Phillips & Joon Y. Park, 1998. "Asymptotics for Nonlinear Transformations of Integrated Time Series," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1182, Cowles Foundation for Research in Economics, Yale University.
  4. Nze, Patrick Ango & Doukhan, Paul, 2004. "Weak Dependence: Models And Applications To Econometrics," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 20(06), pages 995-1045, December.
  5. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, Econometric Society, vol. 55(2), pages 277-301, March.
  6. Saikkonen, Pentti & Choi, In, 2004. "Cointegrating Smooth Transition Regressions," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 20(02), pages 301-340, April.
  7. Liudas Giraitis & Peter C. B. Phillips, 2006. "Uniform Limit Theory for Stationary Autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 51-60, 01.
  8. Joon Y. Park & Peter C.B. Phillips, 1998. "Nonlinear Regressions with Integrated Time Series," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1190, Cowles Foundation for Research in Economics, Yale University.
  9. Peter C.B. Phillips & Tassos Magdalinos, 2004. "Limit Theory for Moderate Deviations from a Unit Root," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1471, Cowles Foundation for Research in Economics, Yale University.
  10. P tscher, Benedikt M., 2004. "Nonlinear Functions And Convergence To Brownian Motion: Beyond The Continuous Mapping Theorem," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 20(01), pages 1-22, February.
  11. Peter C.B. Phillips & Pierre Perron, 1986. "Testing for a Unit Root in Time Series Regression," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 795R, Cowles Foundation for Research in Economics, Yale University, revised Sep 1987.
  12. In Choi & Pentti Saikkonen, 2004. "Testing linearity in cointegrating smooth transition regressions," Econometrics Journal, Royal Economic Society, Royal Economic Society, vol. 7(2), pages 341-365, December.
  13. Ibragimov, Rustam & Phillips, Peter C.B., 2008. "Regression Asymptotics Using Martingale Convergence Methods," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 24(04), pages 888-947, August.
  14. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 932, Cowles Foundation for Research in Economics, Yale University.
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Citations

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Cited by:
  1. Peter C.B. Phillips & Degui Li & Jiti Gao, 2013. "Estimating Smooth Structural Change in Cointegration Models," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1910, Cowles Foundation for Research in Economics, Yale University.
  2. Ioannis Kasparis & Peter C.B. Phillips & Tassos Magdalinos, 2012. "Non-linearity Induced Weak Instrumentation," University of Cyprus Working Papers in Economics, University of Cyprus Department of Economics 02-2012, University of Cyprus Department of Economics.
  3. Pötscher, Benedikt M., 2011. "On the Order of Magnitude of Sums of Negative Powers of Integrated Processes," MPRA Paper 28287, University Library of Munich, Germany.
  4. Yoosoon Chang, 2004. "Taking a New Contour: A Novel Approach to Panel Unit Root Tests," Econometric Society 2004 Far Eastern Meetings, Econometric Society 796, Econometric Society.
  5. Bent Nielsen & Carlos Caceres, 2007. "Convergence to Stochastic Integrals with Non-linear integrands," Economics Papers 2007-W02, Economics Group, Nuffield College, University of Oxford.
  6. M.C. Medeiros & E. Mendes & Les Oxley, 2010. "A Note on Nonlinear Cointegration, Misspecification and Bimodality," Working Papers in Economics, University of Canterbury, Department of Economics and Finance 10/01, University of Canterbury, Department of Economics and Finance.
  7. Ibragimov, Rustam & Phillips, Peter C.B., 2008. "Regression asymptotics using martingale convergence methods," Scholarly Articles 2624459, Harvard University Department of Economics.
  8. 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, American Statistical Association, vol. 28(1), pages 96-114.
  9. MArcelo Cunha Medeiros & Eduardo Mendes & Les Oxley, 2010. "Nonlinear Cointegration, Misspecification and Bimodality," Textos para discussão, Department of Economics PUC-Rio (Brazil) 577, Department of Economics PUC-Rio (Brazil).
  10. Wagner, Martin, 2012. "The Phillips unit root tests for polynomials of integrated processes," Economics Letters, Elsevier, Elsevier, vol. 114(3), pages 299-303.

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