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Phillips-Perron-type unit root tests in the nonlinear ESTAR framework

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In this paper, we propose Phillips-Perron type, semi-parametric testing procedures to distinguish a unit root process from a mean-reverting exponential smooth transition autoregressive one. The limiting nonstandard distributions are derived under very general conditions and simulation evidence shows that the tests perform better than the standard Phillips-Perron or Dickey-Fuller tests in the region of the null.

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  1. Andrews, D. W. K., Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator.Econometrica 60 953–966.

  2. Andrews, D. W. K., Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative.Econometrica 62 1383–1414.

  3. Baum, C. F., Barkoulas, J. T., Caglayan, M. (2001). Nonlinear adjustment to purchasing power parity in the post-Bretton Woods era.Journal of International Money and Finance 20 379–399.

  4. Davies, R. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative.Biometrika 74 33–43.

  5. Den Haan, W., Levin, A. (1997). A practitioner’s guide to robust covariance matrix estimation. InHandbook of Statistics 15 (G. Maddala and C. Rao, eds.), 299–342. Elsevier, Amsterdam.

  6. Dumas, B. (1992). Dynamic equilibrium and the real exchange rate in a spatially separated world.Review of Financial Studies 5 153–180.

  7. Hansen, B. E. (1992). Convergence to stochastic integrals for dependent heterogeneous processes.Econometric Theory 8 489–500.

  8. Hansen, B. E. (1996). Inference when a nuisance parameter is not identified under the null hypothesis.Econometrica 64 413–430.

  9. Kapetanios, G., Shin, Y. andSnell, A. (2003). Testing for a unit root in the nonlinear star framework.Journal of Econometrics 112 359–379.

  10. Luukkonen, R., Saikkonen, P., Teräsvirta, T. (1988). Testing linearity against smooth transition autoregressive models.Biometrika 75 491–499.

  11. McLeish, D. L. (1975). A maximal inequality and dependent strong laws.Annals of Probability 3 829–839.

  12. Michael, P., Nobay, A. R., Peel, D. A. (1997). Transactions costs and nonlinear adjustment in real exchange rates: An empirical investigation.Journal of Political Economy 105 862–879.

  13. Perron, P. andNg, S. (1996). Useful modifications to some unit root tests with dependent errors and their local asymptotic properties.Review of Economic Studies 63 435–463.

  14. Phillips, P. C. B. (1987). Time series regression with a unit root.Econometrica 55 277–301.

  15. Phillips, P. C. B., Perron, P. (1988). Testing for a unit root in time series regression.Biometrika 75 335–346.

  16. Phillips, P. C. B., Xiao, Z. (1998). A primer on unit root testing.Journal of Economic Surveys 12 423–469.

  17. Said, S., Dickey, D. (1984). Testing for unit roots in autoregressive moving average of unknown order.Biometrika 71 559–607.

  18. Sarno, L. (2000). Real exchange rate behavior in the middle east: A reexamination.Economics Letters 66, 127–136.

  19. Sercu, P., Uppal, R., van Hulle, C. (1995). The exchange rate in the presence of transaction costs: Implications for tests of purchasing power parity.Journal of Finance 50 1309–1319.

  20. Taylor, M. P., Peel, D., Sarno, L. (2001). Nonlinear mean-reversion in real exchange rates: Towards a solution to the purchasing power parity puzzles.International Economic Review 42 1015–1042.

  21. van Dijk, D., Teräsvirta, T., Franses, P. H. (2002). Smooth transition autoregressive models—A survey of recent developments.Econometric Reviews 21 1–47.

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We would like to thank conference participants of the Pfingsttagung 2005 in Münster for their helpful comments.

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Rothe, C., Sibbertsen, P. Phillips-Perron-type unit root tests in the nonlinear ESTAR framework. Allgemeines Statistisches Arch 90, 439–456 (2006). https://doi.org/10.1007/s10182-006-0244-y

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  • Exponential smooth transition autoregressive model
  • unit roots
  • Monte Carlo simulations
  • purchasing power parity


  • C12
  • C32