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Nonlinear Functions and Convergence to Brownian Motion: Beyond the Continuous Mapping Theorem

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Abstract

Weak convergence results for sample averages of nonlinear functions of (discrete-time) stochastic processes satisfying a functional central limit theorem (e.g., integrated processes) are given. These results substantially extend recent work by Park and Phillips (1999) and de Jong (2001), in that a much wider class of functions is covered. For example, some of the results hold for the class of all locally integrable functions, thus avoiding any of the various regularity conditions imposed on the functions in Park and Phillips (1999) or de Jong (2001).

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  • Benedikt M. Pötscher, 2001. "Nonlinear Functions and Convergence to Brownian Motion: Beyond the Continuous Mapping Theorem," Vienna Economics Papers vie0203, University of Vienna, Department of Economics.
    • Handle: RePEc:vie:viennp:vie0203
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    Cited by:

    1. Ioannis Kasparis & Peter C. B. Phillips & Tassos Magdalinos, 2014. "Nonlinearity Induced Weak Instrumentation," Econometric Reviews, Taylor & Francis Journals, vol. 33(5-6), pages 676-712, August.
    2. Ibragimov, Rustam & Phillips, Peter C.B., 2008. "Regression Asymptotics Using Martingale Convergence Methods," Econometric Theory, Cambridge University Press, vol. 24(4), pages 888-947, August.
    3. Youngsoo Bae & Robert M. de Jong, 2007. "Money demand function estimation by nonlinear cointegration," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(4), pages 767-793.
    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(3), pages 642-658, June.
    5. Kasparis, Ioannis, 2010. "The Bierens test for certain nonstationary models," Journal of Econometrics, Elsevier, vol. 158(2), pages 221-230, October.
    6. Berenguer Rico, Vanessa, 2013. "Co-summability from linear to non-linear cointegration," UC3M Working papers. Economics we1312, Universidad Carlos III de Madrid. Departamento de Economía.
    7. 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.
    8. Phillips, Peter C.B., 2009. "Local Limit Theory And Spurious Nonparametric Regression," Econometric Theory, Cambridge University Press, vol. 25(6), pages 1466-1497, December.
    9. Vanessa Berenguer-Rico & Bent Nielsen, 2015. "Cumulated sum of squares statistics for non-linear and non-stationary regressions," Economics Papers 2015-W09, Economics Group, Nuffield College, University of Oxford.
    10. Lee, Jungick & de Jong, Robert M., 2008. "Exponential functionals of integrated processes," Economics Letters, Elsevier, vol. 100(2), pages 181-184, August.
    11. Jiti Gao & Peter C. B. Phillips, 2010. "Semiparametric Estimation in Time Series of Simultaneous Equations," Cowles Foundation Discussion Papers 1769, Cowles Foundation for Research in Economics, Yale University.
    12. Kasparis, Ioannis & Andreou, Elena & Phillips, Peter C.B., 2015. "Nonparametric predictive regression," Journal of Econometrics, Elsevier, vol. 185(2), pages 468-494.
    13. Yoon, Gawon, 2005. "Long-memory property of nonlinear transformations of break processes," Economics Letters, Elsevier, vol. 87(3), pages 373-377, June.
    14. 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.
    15. Berenguer-Rico, Vanessa & Gonzalo, Jesús, 2014. "Summability of stochastic processes—A generalization of integration for non-linear processes," Journal of Econometrics, Elsevier, vol. 178(P2), pages 331-341.
    16. Ioannis Kasparis, 2008. "Functional Form Misspecification in Regressions with a Unit Root," University of Cyprus Working Papers in Economics 2-2008, University of Cyprus Department of Economics.

    More about this item

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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