The local Whittle (or Gaussian semiparametric) estimator of long range dependence, proposed by Künsch (1987) and analyzed by Robinson (1995a), has a relatively slow rate of convergence and a ﬁnite sample bias that can be large. In this paper, we generalize the local Whittle estimator to circumvent those problems. Instead of approximating the short-run component of the spectrum, φ(λ), by a constant in a shrinking neighborhood of frequency zero, we approximate its logarithm by a polynomial. This leads to a “local polynomial Whittle” (LPW) estimator. Following the work of Robinson (1995a), we establish the asymptotic bias, variance, mean-squared error (MSE), and normality of the LPW estimator. We determine the asymptotically MSE-optimal bandwidth, and specify a plug-in selection method for its practical implementation. When φ(λ) is smooth enough near the origin, we ﬁnd that the bias of the LPW estimator goes to zero at a faster rate than that of the local Whittle estimator, and its variance is only inflated by a multiplicative constant. In consequence, the rate of convergence of the LPW estimator is faster than that of the local Whittle estimator, given an appropriate choice of the bandwidth m . We show that the LPW estimator attains the optimal rate of convergence for a class of spectra containing those for which φ(λ) is smooth of order s > 1 near zero. When φ(λ) is inﬁnitely smooth near zero, the rate of convergence of the LPW estimator based on a polynomial of high degree is arbitrarily close to n -1/2 .
Andrews, Donald W.K. and Sun, Yixiao, "Local Polynomial Whittle Estimation" (2001). Cowles Foundation Discussion Papers. 1547.