Are the Unit-Root Tests Adequate for Nonlinear Models?

  • Gilles Dufrénot
  • Valérie Mignon
Chapter

Abstract

In the previous chapter, we surveyed some basic results achieved in the literature about nonstationary and nonlinear time series models. The motivation of this chapter is to consider these notions together. Indeed, as suggested in chapter 1, many economic time series display both nonlinear and nonstationary behavior and it seems a natural approach to consider them together. As a starting point, we ask the following question. Consider a nonstationary variable x t . What are the properties of the nonlinear model x t = f (x t−1,..., x tp ) + ε t generating x t ? It is impossible to give a general answer without specifying the exact form of the function f. In the sequel, we shall adopt an empirical approach and consider nonlinear processes currently used in the different fields of applied economics. However, this is not enough if we want to answer our question. We also need enquiring about the type of nonstationarity that is produced by nonlinear models. In general terms, the fact that a process has no stationary asymptotic distribution is due to several causes.

Keywords

Random Walk Unit Root Nonlinear Transformation Monte Carlo Experiment Bilinear Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Balke, N.S. and T.B. Fomby (1995). “Threshold Cointegration”, The Review of Economics and Statistics 5, 629–645.Google Scholar
  2. Barro, R. and X. Salai-I-Martin (1995). Economic Growth. McGraw-Hill.Google Scholar
  3. Chan, K.S., Petrucelli, J.D., Tong, H. and S.W. Woolford (1985). “A Multiple Threshold AR(1) Model”, Journal of Applied Probability 22, 267–279.CrossRefGoogle Scholar
  4. Corradi, V. (1994). “Nonlinear Transformations of Integrated Time Series. A Reconsideration”, Journal of Time Series Analysisl6, 539–549.Google Scholar
  5. Dickey, D.A and W.A. Fuller (1979). “Distribution of the Estimators for Autoregressive Time Series with a Unit Root”, Journal of the American Statistical Association 74, 427–431.Google Scholar
  6. Dickey, D.A and W.A. Fuller (1981). “The Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root”, Econometrica 49, 1057–1072.CrossRefGoogle Scholar
  7. Dufrénot, G., J. Drunat and L. Mathieu (1998). “Modeling Burst Phenomena: Bilinear and Autoregressive Exponential Models”, in C. Dunis and B. Zhou, eds, Nonlinear Modeling of High Frequency Time Series. John Wiley and Sons.Google Scholar
  8. Ermini, L. and C.W.J. Granger (1993). “Some Generalization of the Algebra of I(1) Processes”, Journal of Econometrics 58, 369–384.CrossRefGoogle Scholar
  9. Franses, P.H. and M. McAleer (1998). “Testing For Unit-Root and Nonlinear Transformations”, Journal of Time Series Analysis 19, 147–164.CrossRefGoogle Scholar
  10. Godfrey, L.G. and M.R. Wickens (1981). “Testing Linear and Log-linear Regressions for Functional Form”, Review of Economic Studies 48, 487–496.CrossRefGoogle Scholar
  11. Godfrey, L.G., McAleer, M. and C.R. McKenzie (1979). “Variable Addition and Lagrange Multiplier Tests for Linear and Logarithmic Regression Models”, Review of Economics and Statistics 70, 492–503.CrossRefGoogle Scholar
  12. Granger, C.W.J. (1995). “Modeling Nonlinear Relationships between Extended-Memory Variables”, Econometrica 63, 265–279.CrossRefGoogle Scholar
  13. Granger, C.W.J. and J. Hallman (1991). “Nonlinear Transformations in Integrated Series”, Journal of Time Series Analysis 12, 207–224.CrossRefGoogle Scholar
  14. Granger, C.W.J., Inoue, T. and N. Morin (1997). “Nonlinear Stochastic Trends”, Journal of Econometrics 81, 65–92.CrossRefGoogle Scholar
  15. Granger, C.W.J. and P. Newbold (1976). “Forecasting Transformed Series”, Journal of the Royal Statistical Society 38, 189–203.Google Scholar
  16. Igloi, E. and G. Terdik (1999). “Bilinear Stochastic Systems with Fractional Brownian Motion Input”, The Annals of Applied Probability 9, 46–77.CrossRefGoogle Scholar
  17. Keller, G., Kersting E. and U. Roslar (1987). “On the Asymptotic Behavior of Discrete Time Stochastic Growth Processes”, The Annals of Probability 15, 305–343.CrossRefGoogle Scholar
  18. Kersting, E. (1986). “On Recurrence and Transience of Growth Models”, Journal of Applied Probability 22, 614–625.CrossRefGoogle Scholar
  19. Klebaner, F. (1989). “Stochastic Difference Equations and Generalized Gamma Distributions”, The Annals of Probabilityl7(1), 178–188.Google Scholar
  20. Luukkonen, R., Saikkonen, P. and T. Teräsvirta (1988). “Testing Linearity Against Smooth Transition Autoregressive Models”, Biometrika 75, 491–499.CrossRefGoogle Scholar
  21. Lye, J. and V.L. Martin (1995). “Nonlinear Modelling and Distributional Flexibility”, in Creedy J. and V.L. Martin, eds, Chaos and Nonlinear models in Economics. Edward Elgar.Google Scholar
  22. Mignon, V. (1998). Marchés Financiers et Modélisation des Rentabilités Boursières. Economica.Google Scholar
  23. Ozaki, T. (1985). “Nonlinear Time Series Models and Dynamical Systems”, in, Hanan E.J., Krishnaiah P.R. and M.M. Rao, eds, Handbook of Statistics, Volume 5. North Holland.Google Scholar
  24. Page, C.H. (1952). “Instantaneous Power Spectra”, Journal of Applied Physics 23, 103–106.CrossRefGoogle Scholar
  25. Priestley, M.B. and T. Subba Rao (1969). “A Test for Stationarity of Time Series”, Journal of the Royal Statistical Society, Series B, 31, 140–149.Google Scholar
  26. Subba Rao, T. (1968). “A Note on the Asymptotic Relative Efficiency of Cox and Stu-art’s Tests for Testing Trend in Dispersion of p-dependent Time Series”, Biometrika 55, 381–385.Google Scholar
  27. Subba Rao, T. (1979). “On the Theory of Bilinear Time Series Models-2”, Technical Report 121, Department of Mathematics, University of Manchester.Google Scholar
  28. Subba Rao, T. (1992). “Analysis of Nonlinear Time Series and Chaos by BispectralMethods”, in Casdagli, M. and S. Eubank, eds, Nonlinear Modeling and Forecasting, Addison-Wesley, 199–225.Google Scholar
  29. Terdik, G. (2000). Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis. A Frequency Domain Approach. Springer Verlag.Google Scholar
  30. Tjostheim, D. (1976). “Spectral Generating Operators for Non-Stationary Processes”, Advanced Applied Probability 8, 831–846.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Gilles Dufrénot
    • 1
    • 2
  • Valérie Mignon
    • 3
  1. 1.ERUDITEUniversity of Paris 12France
  2. 2.GREQUAM-CNRSUniversity of MarseilleFrance
  3. 3.MODEMUniversity of Paris 10France

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