### Abstract

In this paper, we propose a simple bias-reduced log-periodogram regression estimator, d̂_{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 2k 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 d̂_{r}, determine the asymptotic MSE optimal choice of the number of frequencies, m, to include in the regression, and establish the asymptotic normality of d̂_{r}. These results show that the bias of d̂_{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 d̂_{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) model show that the bias-reduced estimators perform well relative to the standard log-periodogram regression estimator.

Original language | English (US) |
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Pages (from-to) | 675-712 |

Number of pages | 38 |

Journal | Econometrica |

Volume | 71 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Economics and Econometrics

### Cite this

*Econometrica*,

*71*(2), 675-712. https://doi.org/10.1111/1468-0262.00420