Bias-Reduced Log-Periodogram and Whittle Estimation of the Long-Memory Parameter Without Variance Inflation

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Abstract
In Andrews and Guggenberger (2003) a bias-reduced log-periodogram estimator d_{LP}(r) for the long-memory parameter (d) in a stationary long-memory time series has been introduced. Compared to the Geweke and Porter-Hudak (1983) estimator d_{GPH}=d_{LP}(0), the estimator d_{LP}(r) for r larger than 1 generally reduces the asymptotic bias by an order of magnitude but inflates the asymptotic variance by a multiplicative constant c_{r}. In this paper, we introduce a new, computationally attractive estimator d_{WLP}(r) by taking a weighted average of GPH estimators over different bandwidths. We show that, for each fixed r that is larger than zero, the new estimator can be designed to have the same asymptotic bias properties as d_{LP}(r) but its asymptotic variance is changed by a constant 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 d_{LW}(r) in Andrews and Sun (2004) leading to the weighted estimator d_{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.