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On maximum-likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean

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  • Yin-Wong Cheung
  • Francis X. Diebold

Abstract

There are two approaches to maximum likelihood (ML) estimation of the parameter of fractionally-integrated noise: approximate frequency-domain ML (Fox and Taqqwu, 1986) and exact time-domain ML (Solwell, 1990a). If the mean of the process is known, then a clear finite-sample mean-squared error (MSE) ranking of the estimators emerges: the exact time-domain estimator has smaller MSE. We show in this paper, however, that the finite-sample efficiency of approximate frequency-domain ML relative to exact time-domain ML rises dramatically when the mean result is unknown and instead must be estimated. The intuition for our result is straightforward: The frequency-domain ML estimator is invariant to the true but unknown mean of the process, while the time-domain ML estimator is not. Feasible time-domain estimation must therefore be based upon de-meaned data, but the long memory associated with fractional integration makes precise estimation of the mean difficult. We conclude that the frequency-domain estimator is an attractive and efficient alternative for situations in which large sample sizes render time-domain estimation impractical.

Suggested Citation

  • Yin-Wong Cheung & Francis X. Diebold, 1990. "On maximum-likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean," Discussion Paper / Institute for Empirical Macroeconomics 34, Federal Reserve Bank of Minneapolis.
  • Handle: RePEc:fip:fedmem:34
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    References listed on IDEAS

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    1. Diebold, Francis X & Husted, Steven & Rush, Mark, 1991. "Real Exchange Rates under the Gold Standard," Journal of Political Economy, University of Chicago Press, vol. 99(6), pages 1252-1271, December.
    2. Sowell, Fallaw, 1992. "Maximum likelihood estimation of stationary univariate fractionally integrated time series models," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 165-188.
    3. Diebold, Francis X & Rudebusch, Glenn D, 1991. "Is Consumption Too Smooth? Long Memory and the Deaton Paradox," The Review of Economics and Statistics, MIT Press, vol. 73(1), pages 1-9, February.
    4. Joseph G. Haubrich & Andrew W. Lo, "undated". "The Sources and Nature of Long-Term Memory in the Business Cycle," Rodney L. White Center for Financial Research Working Papers 05-89, Wharton School Rodney L. White Center for Financial Research.
    5. Robinson, P. M., 1991. "Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression," Journal of Econometrics, Elsevier, vol. 47(1), pages 67-84, January.
    6. Diebold, Francis X. & Rudebusch, Glenn D., 1989. "Long memory and persistence in aggregate output," Journal of Monetary Economics, Elsevier, vol. 24(2), pages 189-209, September.
    7. Cheung, Yin-Wong, 1993. "Long Memory in Foreign-Exchange Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(1), pages 93-101, January.
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