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Bootstrap Unit Root Tests

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  • Joon Y. Park

    (Seoul National University)

Abstract

We consider the bootstrap unit root tests based on autoregressive integrated models, with or without deterministic time trends. A general methodology is developed to approximate asymptotic distributions for the models driven by integrated time series, and used to obtain asymptotic expansions for the Dickey-Fuller unit root tests. The second-order terms in their expansions are of stochastic orders Op($n^{-1/4}$) and Op($n^{-1/2}$), and involve functionals of Brownian motions and normal random variates. The asymptotic expansions for the bootstrap tests are also derived and compared with those of the Dickey-Fuller tests. We show in particular that the usual nonparametric bootstrap offers asymptotic refinements for the Dickey-Fuller tests, i.e., it corrects their second-order errors. More precisely, it is shown that the critical values obtained by the bootstrap resampling are correct up to the second-order terms, and the errors in rejection probabilities are of order o($n^{-1/2}$) if the tests are based upon the bootstrap critical values. Through simulation, we investigate how effective is the bootstrap correction in small samples.

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Bibliographic Info

Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 1587.

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Date of creation: 01 Aug 2000
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Handle: RePEc:ecm:wc2000:1587

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  1. Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 15(03), pages 269-298, June.
  2. Elliott, Graham & Rothenberg, Thomas J & Stock, James H, 1996. "Efficient Tests for an Autoregressive Unit Root," Econometrica, Econometric Society, Econometric Society, vol. 64(4), pages 813-36, July.
  3. Park, Joon Y., 2002. "An Invariance Principle For Sieve Bootstrap In Time Series," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 18(02), pages 469-490, April.
  4. Courbot, B., 2001. "Rates of convergence in the functional CLT for multidimensional continuous time martingales," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 91(1), pages 57-76, January.
  5. Beveridge, Stephen & Nelson, Charles R., 1981. "A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the `business cycle'," Journal of Monetary Economics, Elsevier, Elsevier, vol. 7(2), pages 151-174.
  6. Evans, G B A & Savin, N E, 1984. "Testing for Unit Roots: 2," Econometrica, Econometric Society, Econometric Society, vol. 52(5), pages 1241-69, September.
  7. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  8. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, Econometric Society, vol. 49(4), pages 1057-72, June.
  9. Nankervis, John C & Savin, N E, 1996. "The Level and Power of the Bootstrap t Test in the AR(1) Model with Trend," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 14(2), pages 161-68, April.
  10. 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.
  11. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, Econometric Society, vol. 49(3), pages 753-79, May.
  12. Yoosoon Chang & Joon Y. Park, 2003. "A Sieve Bootstrap For The Test Of A Unit Root," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(4), pages 379-400, 07.
  13. Donald W.K. Andrews & Moshe Buchinsky, 1997. "On the Number of Bootstrap Repetitions for Bootstrap Standard Errors, Confidence Intervals, and Tests," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1141R, Cowles Foundation for Research in Economics, Yale University.
  14. Horowitz, Joel L., 2001. "The Bootstrap," Handbook of Econometrics, Elsevier, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 52, pages 3159-3228 Elsevier.
  15. Einmahl, Uwe, 1989. "Extensions of results of Komlós, Major, and Tusnády to the multivariate case," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 28(1), pages 20-68, January.
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