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The Calculation of Some Limiting Distributions Arising in Near-Integrated Models with GLS Detrending

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  • Chambers, MJ

Abstract

Many unit root test statistics are nowadays constructed using detrended data, with the method of GLS detrending being popular in the setting of a near-integrated model. This paper determines the properties of some associated limiting distributions when the GLS detrending is based on a linear time trend. A fundamental result for the moment generating function of two key functionals of the relevant stochastic process is provided and used to compute probability density functions and cumulative distribution functions, as well as means and variances, of the limiting distributions of some statistics of interest. Some further applications, including a comparison of limiting power functions and the consideration of a more complicated statistic, are also provided.

Suggested Citation

  • Chambers, MJ, 2013. "The Calculation of Some Limiting Distributions Arising in Near-Integrated Models with GLS Detrending," Economics Discussion Papers 8975, University of Essex, Department of Economics.
  • Handle: RePEc:esx:essedp:8975
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    References listed on IDEAS

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    1. Serena Ng & Pierre Perron, 2001. "LAG Length Selection and the Construction of Unit Root Tests with Good Size and Power," Econometrica, Econometric Society, vol. 69(6), pages 1519-1554, November.
    2. Perron, Pierre, 1989. "The Calculation of the Limiting Distribution of the Least-Squares Estimator in a Near-Integrated Model," Econometric Theory, Cambridge University Press, vol. 5(02), pages 241-255, August.
    3. Elliott, Graham, 1999. "Efficient Tests for a Unit Root When the Initial Observation Is Drawn from Its Unconditional Distribution," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 40(3), pages 767-783, August.
    4. Elliott, Graham & Muller, Ulrich K., 2006. "Minimizing the impact of the initial condition on testing for unit roots," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 285-310.
    5. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
    6. Jan R. Magnus, 1986. "The Exact Moments of a Ratio of Quadratic Forms in Normal Variables," Annals of Economics and Statistics, GENES, issue 4, pages 95-109.
    7. Abadir, Karim M., 1995. "The Limiting Distribution of the t Ratio Under a Unit Root," Econometric Theory, Cambridge University Press, vol. 11(04), pages 775-793, August.
    8. Mehta, Jatinder S. & Swamy, Paravastu A. V. B., 1978. "The existence of moments of some simple Bayes estimators of coefficients in a simultaneous equation model," Journal of Econometrics, Elsevier, vol. 7(1), pages 1-13, February.
    9. Perron, Pierre & Rodriguez, Gabriel, 2003. "GLS detrending, efficient unit root tests and structural change," Journal of Econometrics, Elsevier, vol. 115(1), pages 1-27, July.
    10. Elliott, Graham & Stock, James H., 2001. "Confidence intervals for autoregressive coefficients near one," Journal of Econometrics, Elsevier, vol. 103(1-2), pages 155-181, July.
    11. Karim M. Abadir & Michael Rockinger, 1997. "The 'Devil's Horns' Problem of Inverting Confluent Characteristic Functions," Econometrica, Econometric Society, vol. 65(5), pages 1221-1226, September.
    12. Elliott, Graham & Rothenberg, Thomas J & Stock, James H, 1996. "Efficient Tests for an Autoregressive Unit Root," Econometrica, Econometric Society, vol. 64(4), pages 813-836, July.
    13. repec:adr:anecst:y:1986:i:4:p:05 is not listed on IDEAS
    14. repec:adr:anecst:y:1986:i:4 is not listed on IDEAS
    15. Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Other publications TiSEM c6725407-ac3c-44fd-b6d1-5, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Neil Kellard & Denise Osborn & Jerry Coakley & Marcus J. Chambers, 2015. "Testing for a Unit Root in a Near-Integrated Model with Skip-Sampled Data," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(5), pages 630-649, September.
    2. Chambers, MJ, 2016. "The Effects of Sampling Frequency on Detrending Methods for Unit Root Tests," Economics Discussion Papers 16062, University of Essex, Department of Economics.

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