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A Bias--Reduced Log--Periodogram Regression Estimator for the Long--Memory Parameter


  • Donald W. K. Andrews

    () (Yale University, New Haven, U.S.A.)

  • Patrik Guggenberger

    () (Yale University, U.S.A.)


In this paper, we propose a simple bias--reduced log--periodogram regression estimator, "ˆd-sub-r", of the long--memory parameter, "d", that eliminates the first-- and higher--order biases of the Geweke and Porter--Hudak (1983) (GPH) estimator. The bias--reduced estimator is the same as the GPH estimator except that one includes frequencies to the power 2"k" for "k"=1,…,"r", for some positive integer "r", as additional regressors in the pseudo--regression model that yields the GPH estimator. The reduction in bias is obtained using assumptions on the spectrum only in a neighborhood of the zero frequency.Following the work of Robinson (1995b) and Hurvich, Deo, and Brodsky (1998), we establish the asymptotic bias, variance, and mean--squared error (MSE) of "ˆd-sub-r", determine the asymptotic MSE optimal choice of the number of frequencies, "m", to include in the regression, and establish the asymptotic normality of "ˆd-sub-r". These results show that the bias of "ˆd-sub-r" goes to zero at a faster rate than that of the GPH estimator when the normalized spectrum at zero is sufficiently smooth, but that its variance only is increased by a multiplicative constant.We show that the bias--reduced estimator "ˆd-sub-r" attains the optimal rate of convergence for a class of spectral densities that includes those that are smooth of order "s"≥1 at zero when "r"≥("s" - 2)/2 and "m" is chosen appropriately. For "s">2, the GPH estimator does not attain this rate. The proof uses results of Giraitis, Robinson, and Samarov (1997).We specify a data--dependent plug--in method for selecting the number of frequencies "m" to minimize asymptotic MSE for a given value of "r".Some Monte Carlo simulation results for stationary Gaussian ARFIMA (1, "d", 1) and (2, "d", 0) models show that the bias--reduced estimators perform well relative to the standard log--periodogram regression estimator. Copyright The Econometric Society 2003.

Suggested Citation

  • Donald W. K. Andrews & Patrik Guggenberger, 2003. "A Bias--Reduced Log--Periodogram Regression Estimator for the Long--Memory Parameter," Econometrica, Econometric Society, vol. 71(2), pages 675-712, March.
  • Handle: RePEc:ecm:emetrp:v:71:y:2003:i:2:p:675-712

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    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes


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