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Gaussian Semi‐parametric Estimation of Fractional Cointegration


  • Carlos Velasco


. We analyse consistent estimation of the memory parameters of a nonstationary fractionally cointegrated vector time series. Assuming that the cointegrating relationship has substantially less memory than the observed series, we show that a multi‐variate Gaussian semi‐parametric estimate, based on initial consistent estimates and possibly tapered observations, is asymptotically normal. The estimates of the memory parameters can rely either on original (for stationary errors) or on differenced residuals (for nonstationary errors) assuming only a convergence rate for a preliminary slope estimate. If this rate is fast enough, semi‐parametric memory estimates are not affected by the use of residuals and retain the same asymptotic distribution as if the true cointegrating relationship were known. Only local conditions on the spectral densities around zero frequency for linear processes are assumed. We concentrate on a bivariate system but discuss multi‐variate generalizations and show the performance of the estimates with simulated and real data.

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  • Carlos Velasco, 2003. "Gaussian Semi‐parametric Estimation of Fractional Cointegration," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(3), pages 345-378, May.
  • Handle: RePEc:bla:jtsera:v:24:y:2003:i:3:p:345-378
    DOI: 10.1111/1467-9892.00311

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    1. John Geweke & Susan Porter‐Hudak, 1983. "The Estimation And Application Of Long Memory Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(4), pages 221-238, July.
    2. Engle, Robert & Granger, Clive, 2015. "Co-integration and error correction: Representation, estimation, and testing," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 39(3), pages 106-135.
    3. de Jong, Robert M. & Davidson, James, 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals I," Econometric Theory, Cambridge University Press, vol. 16(5), pages 621-642, October.
    4. Michael Dueker & Richard Startz, 1998. "Maximum-Likelihood Estimation Of Fractional Cointegration With An Application To U.S. And Canadian Bond Rates," The Review of Economics and Statistics, MIT Press, vol. 80(3), pages 420-426, August.
    5. Clifford M. Hurvich & Willa W. Chen, 2000. "An Efficient Taper for Potentially Overdifferenced Long‐memory Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 21(2), pages 155-180, March.
    6. Granger, C. W. J., 1981. "Some properties of time series data and their use in econometric model specification," Journal of Econometrics, Elsevier, vol. 16(1), pages 121-130, May.
    7. Liudas Giraitis & Peter M. Robinson & Alexander Samarov, 1997. "Rate Optimal Semiparametric Estimation Of The Memory Parameter Of The Gaussian Time Series With Long‐Range Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(1), pages 49-60, January.
    8. Tanaka, Katsuto, 1999. "The Nonstationary Fractional Unit Root," Econometric Theory, Cambridge University Press, vol. 15(4), pages 549-582, August.
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