Local Polynomial Whittle Estimation of Long-range Dependence

The local Whittle (or Gaussian semiparametric) estimator of long range dependence, proposed by Kunsch (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, phi(lambda), 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 phi(lambda) 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 varphi(lambda) is smooth of order s > 1 near zero. When phi(lambda) 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}.