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

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.