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Bootstrap Unit Root Tests

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  • Joon Y. Park

Abstract

We consider the bootstrap unit root tests based on finite order autoregressive integrated models driven by iid innovations, 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 O(p) (n-super- - 1/4) and O(p) (n-super- - 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 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-super- - 1/2) if the tests are based upon the bootstrap critical values. Through simulations, we investigate how effective is the bootstrap correction in small samples. Copyright The Econometric Society 2003.

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Bibliographic Info

Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 71 (2003)
Issue (Month): 6 (November)
Pages: 1845-1895

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Handle: RePEc:ecm:emetrp:v:71:y:2003:i:6:p:1845-1895

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  1. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-79, May.
  2. 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.
  3. Elliott, Graham & Rothenberg, Thomas J & Stock, James H, 1996. "Efficient Tests for an Autoregressive Unit Root," Econometrica, Econometric Society, vol. 64(4), pages 813-36, July.
  4. Park, Joon Y., 2002. "An Invariance Principle For Sieve Bootstrap In Time Series," Econometric Theory, Cambridge University Press, vol. 18(02), pages 469-490, April.
  5. Horowitz, Joel L., 2001. "The Bootstrap," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 52, pages 3159-3228 Elsevier.
  6. Einmahl, Uwe, 1989. "Extensions of results of Komlós, Major, and Tusnády to the multivariate case," Journal of Multivariate Analysis, Elsevier, vol. 28(1), pages 20-68, January.
  7. Courbot, B., 2001. "Rates of convergence in the functional CLT for multidimensional continuous time martingales," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 57-76, January.
  8. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June.
  9. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
  10. Evans, G B A & Savin, N E, 1984. "Testing for Unit Roots: 2," Econometrica, Econometric Society, vol. 52(5), pages 1241-69, September.
  11. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
  12. Peter C.B. Phillips & Joon Y. Park, 1998. "Asymptotics for Nonlinear Transformations of Integrated Time Series," Cowles Foundation Discussion Papers 1182, Cowles Foundation for Research in Economics, Yale University.
  13. 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, vol. 14(2), pages 161-68, April.
  14. 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 1141R, Cowles Foundation for Research in Economics, Yale University.
  15. 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, vol. 7(2), pages 151-174.
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