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Nonstationary Index Models


  • Yoosoon Chang
  • Joon Y. Park



This paper considers index models, such as neural network models and smooth transition regressions, with integrated regressors. These are the models that can be ued to analyze various nonlinear relationships among nonstationary economic time series. Asymptotics for the nonlinear least squares (NLS) estimator in such models are fully developed. The estimator is shown to be consistent with a convergence rate that is a mixture of n^(3/4) n^(1/2) and n^(1/4) for neural network models, and of n^(5/4), n, n^(3/4) and n^(1/2) for smooth transition regressions. Its limiting distribution is also obtained. Some of its components are mixed normal, with mixing variates depending upon Brownian local time as well as Brownian motion. However, it also has non-Gaussian components. It is particular shown that applications of usual statistical methods in such models generally yield inefficient estimates and/or invalid tests. We develop a new methodology to efficiently estimate and to correctly test in those models. A simple simulation is conducted to investigate the finite sample properties of the NLS estimators and the newly proposed efficient estimators.

Suggested Citation

  • Yoosoon Chang & Joon Y. Park, 1999. "Nonstationary Index Models," Working Paper Series no7, Institute of Economic Research, Seoul National University.
  • Handle: RePEc:snu:ioerwp:no7

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    References listed on IDEAS

    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Hansen, Bruce E., 1992. "Convergence to Stochastic Integrals for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 8(04), pages 489-500, December.
    3. Donald W. K. Andrews & C. John McDermott, 1995. "Nonlinear Econometric Models with Deterministically Trending Variables," Review of Economic Studies, Oxford University Press, vol. 62(3), pages 343-360.
    4. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
    5. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    6. 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.
    7. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(03), pages 468-497, December.
    8. 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.
    9. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, vol. 59(2), pages 283-306, March.
    10. 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.
    11. repec:cup:etheor:v:8:y:1992:i:4:p:489-500 is not listed on IDEAS
    12. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    13. repec:cup:etheor:v:11:y:1995:i:5:p:888-911 is not listed on IDEAS
    14. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
    15. Saikkonen, Pentti, 1995. "Problems with the Asymptotic Theory of Maximum Likelihood Estimation in Integrated and Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 11(05), pages 888-911, October.
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    Cited by:

    1. Chang, Yoosoon, 2002. "Nonlinear IV unit root tests in panels with cross-sectional dependency," Journal of Econometrics, Elsevier, vol. 110(2), pages 261-292, October.
    2. Yoosoon Chang & Wonho Song, 2002. "Panel Unit Root Tests in the Presence of Cross-Sectional Dependency and Heterogeneity," 10th International Conference on Panel Data, Berlin, July 5-6, 2002 B5-2, International Conferences on Panel Data.

    More about this item


    Index model; integrated time series; nonlinear least squares; neural network model; smooth transition regression; Brownian motion; Brownian local time;

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