### Abstract

The bias-reduced log-periodogram estimator d̂ _{LP}(r), r ≥ 1 of Andrews and Guggenberger (2003, Econometrica 71, 675-712) for the long-memory parameter d in a stationary long-memory time series reduces the asymptotic bias of the original log-periodogram estimator d _{GPH} = d̂ _{LP}(0) of Geweke and Porter-Hudak (1983) by an order of magnitude but inflates the asymptotic variance by a multiplicative constant c _{r}, for example, C _{1} = 2.25 and c _{2} = 3.52. In this paper, we introduce a new, computationally attractive estimator d̂ _{WLP}(r) by taking a weighted average of d̂ _{LP}(0) estimators over different bandwidths. We show that, for each fixed r ≥ 0, the new estimator can be designed to have the same asymptotic bias properties as d̂ _{LP}(r) but its asymptotic variance is changed by a constant c* _{r} that can be chosen to be as small as desired, in particular smaller than c _{r}. The same idea is also applied to the local-polynomial Whittle estimator d̂ _{LW}(r) in Andrews and Sun (2004, Econometrica 72, 569-614) leading to the weighted estimator d̂ _{WLW}(r). We establish the asymptotic bias, variance, and mean-squared error of the weighted estimators and show their asymptotic normality. Furthermore, we introduce a data-dependent adaptive procedure for selecting r and the bandwidth m and show that up to a logarithmic factor, the resulting adaptive weighted estimator achieves the optimal rate of convergence. A Monte Carlo study shows that the adaptive weighted estimator compares very favorably to several other adaptive estimators.

Original language | English (US) |
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Pages (from-to) | 863-912 |

Number of pages | 50 |

Journal | Econometric Theory |

Volume | 22 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Social Sciences (miscellaneous)
- Economics and Econometrics

### Cite this

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*Econometric Theory*, vol. 22, no. 5, pp. 863-912. https://doi.org/10.1017/S0266466606060403

**Bias-reduced log-periodogram and whittle estimation of the long-memory parameter without variance inflation.** / Guggenberger, Patrik; Sun, Yixiao.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Bias-reduced log-periodogram and whittle estimation of the long-memory parameter without variance inflation

AU - Guggenberger, Patrik

AU - Sun, Yixiao

PY - 2006/10/1

Y1 - 2006/10/1

N2 - The bias-reduced log-periodogram estimator d̂ LP(r), r ≥ 1 of Andrews and Guggenberger (2003, Econometrica 71, 675-712) for the long-memory parameter d in a stationary long-memory time series reduces the asymptotic bias of the original log-periodogram estimator d GPH = d̂ LP(0) of Geweke and Porter-Hudak (1983) by an order of magnitude but inflates the asymptotic variance by a multiplicative constant c r, for example, C 1 = 2.25 and c 2 = 3.52. In this paper, we introduce a new, computationally attractive estimator d̂ WLP(r) by taking a weighted average of d̂ LP(0) estimators over different bandwidths. We show that, for each fixed r ≥ 0, the new estimator can be designed to have the same asymptotic bias properties as d̂ LP(r) but its asymptotic variance is changed by a constant c* r that can be chosen to be as small as desired, in particular smaller than c r. The same idea is also applied to the local-polynomial Whittle estimator d̂ LW(r) in Andrews and Sun (2004, Econometrica 72, 569-614) leading to the weighted estimator d̂ WLW(r). We establish the asymptotic bias, variance, and mean-squared error of the weighted estimators and show their asymptotic normality. Furthermore, we introduce a data-dependent adaptive procedure for selecting r and the bandwidth m and show that up to a logarithmic factor, the resulting adaptive weighted estimator achieves the optimal rate of convergence. A Monte Carlo study shows that the adaptive weighted estimator compares very favorably to several other adaptive estimators.

AB - The bias-reduced log-periodogram estimator d̂ LP(r), r ≥ 1 of Andrews and Guggenberger (2003, Econometrica 71, 675-712) for the long-memory parameter d in a stationary long-memory time series reduces the asymptotic bias of the original log-periodogram estimator d GPH = d̂ LP(0) of Geweke and Porter-Hudak (1983) by an order of magnitude but inflates the asymptotic variance by a multiplicative constant c r, for example, C 1 = 2.25 and c 2 = 3.52. In this paper, we introduce a new, computationally attractive estimator d̂ WLP(r) by taking a weighted average of d̂ LP(0) estimators over different bandwidths. We show that, for each fixed r ≥ 0, the new estimator can be designed to have the same asymptotic bias properties as d̂ LP(r) but its asymptotic variance is changed by a constant c* r that can be chosen to be as small as desired, in particular smaller than c r. The same idea is also applied to the local-polynomial Whittle estimator d̂ LW(r) in Andrews and Sun (2004, Econometrica 72, 569-614) leading to the weighted estimator d̂ WLW(r). We establish the asymptotic bias, variance, and mean-squared error of the weighted estimators and show their asymptotic normality. Furthermore, we introduce a data-dependent adaptive procedure for selecting r and the bandwidth m and show that up to a logarithmic factor, the resulting adaptive weighted estimator achieves the optimal rate of convergence. A Monte Carlo study shows that the adaptive weighted estimator compares very favorably to several other adaptive estimators.

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U2 - 10.1017/S0266466606060403

DO - 10.1017/S0266466606060403

M3 - Article

AN - SCOPUS:33749352042

VL - 22

SP - 863

EP - 912

JO - Econometric Theory

JF - Econometric Theory

SN - 0266-4666

IS - 5

ER -