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Saddlepoint Approximations for Optimal Unit Root Tests

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  • Patrick Marsh
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    Abstract

    This paper provides a (saddlepoint) tail probability approximation for the distribution of an optimal unit root test. Under restrictive assumptions, Gaussianity and known covariance structure, the order of error of the approximation is given. More generally, when innovations are a linear process in martingale differences, the estimated saddlepoint is proven to yield valid asymptotic inference. Numerical evidence demonstrates superiority over approximations for a directly comparable test based on simulation of its limiting stochastic representation. In addition, because the saddlepoint offers an explicit representation P-value sensitivity to model specification is easily analyzed, here in the context of the Nelson and Plosser data.

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    Paper provided by Department of Economics, University of York in its series Discussion Papers with number 09/31.

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    Handle: RePEc:yor:yorken:09/31

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    1. Yoosoon Chang & Joon Park, 2002. "On The Asymptotics Of Adf Tests For Unit Roots," Econometric Reviews, Taylor & Francis Journals, vol. 21(4), pages 431-447.
    2. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2008. "Bootstrap Unit Root Tests For Time Series With Nonstationary Volatility," Econometric Theory, Cambridge University Press, vol. 24(01), pages 43-71, February.
    3. Bühlmann, Peter, 1995. "Moving-average representation of autoregressive approximations," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 331-342, December.
    4. Pantula, Sastry G, 1991. "Asymptotic Distributions of Unit-Root Tests When the Process Is Nearly Stationary," Journal of Business & Economic Statistics, American Statistical Association, vol. 9(1), pages 63-71, January.
    5. Rolf Larsson, 1998. "Distribution approximation of unit root tests in autoregressive models," Econometrics Journal, Royal Economic Society, vol. 1(RegularPa), pages 10-26.
    6. Nabeya, Seiji & Tanaka, Katsuto, 1990. "Limiting power of unit-root tests in time-series regression," Journal of Econometrics, Elsevier, vol. 46(3), pages 247-271, December.
    7. Marsh, Patrick, 2007. "The Available Information For Invariant Tests Of A Unit Root," Econometric Theory, Cambridge University Press, vol. 23(04), pages 686-710, August.
    8. Patrick Richard, 2007. "ARMA Sieve bootstrap unit root tests," Cahiers de recherche 07-05, Departement d'Economique de la Faculte d'administration à l'Universite de Sherbrooke, revised Jul 2009.
    9. Juhl, Ted & Xiao, Zhijie, 2003. "Power Functions And Envelopes For Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 19(02), pages 240-253, April.
    10. Serena Ng & Pierre Perron, 1997. "Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power," Boston College Working Papers in Economics 369, Boston College Department of Economics, revised 01 Sep 2000.
    11. Graham Elliott & Thomas J. Rothenberg & James H. Stock, 1992. "Efficient Tests for an Autoregressive Unit Root," NBER Technical Working Papers 0130, National Bureau of Economic Research, Inc.
    12. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
    13. Abadir, Karim M., 1993. "On the Asymptotic Power of Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 9(02), pages 189-221, April.
    14. Marsh, Patrick W.N., 1998. "Saddlepoint Approximations For Noncentral Quadratic Forms," Econometric Theory, Cambridge University Press, vol. 14(05), pages 539-559, October.
    15. Nabeya, S. & Perron, P., 1991. "Local Asymtotic Distributions Related to the AR(1) MOdel with Dependent Errors," Papers 362, Princeton, Department of Economics - Econometric Research Program.
    16. Marsh, Patrick, 2009. "The Properties Of Kullback–Leibler Divergence For The Unit Root Hypothesis," Econometric Theory, Cambridge University Press, vol. 25(06), pages 1662-1681, December.
    17. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 139-162.
    18. Butler R.W. & Paolella M.S., 2002. "Saddlepoint Approximation and Bootstrap Inference for the Satterthwaite Class of Ratios," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 836-846, September.
    19. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2009. "Heteroskedastic Time Series With A Unit Root," Econometric Theory, Cambridge University Press, vol. 25(05), pages 1228-1276, October.
    20. Francke, Marc K. & de Vos, Aart F., 2007. "Marginal likelihood and unit roots," Journal of Econometrics, Elsevier, vol. 137(2), pages 708-728, April.
    21. Dufour, Jean-Marie & King, Maxwell L., 1991. "Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 115-143, January.
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