A bias-reduced log-periodogram regression estimator for the long-memory parameter

In this paper, we propose a simple bias-reduced log-periodogram regression estimator, d̂_r, of the long-memory parameter, d, that eliminates the first- and higher-order biases of the Geweke and Porter-Hudak (1983) (GPH) estimator. The bias-reduced estimator is the same as the GPH estimator except that one includes frequencies to the power 2k for k = 1, r, for some positive integer r, as additional regressors in the pseudo-regression model that yields the GPH estimator. The reduction in bias is obtained using assumptions on the spectrum only in a neighborhood of the zero frequency. Following the work of Robinson (1995b) and Hurvich, Deo, and Brodsky (1998), we establish the asymptotic bias, variance, and mean-squared error (MSE) of d̂_r, determine the asymptotic MSE optimal choice of the number of frequencies, m, to include in the regression, and establish the asymptotic normality of d̂_r. These results show that the bias of d̂_r goes to zero at a faster rate than that of the GPH estimator when the normalized spectrum at zero is sufficiently smooth, but that its variance only is increased by a multiplicative constant. We show that the bias-reduced estimator d̂_r attains the optimal rate of convergence for a class of spectral densities that includes those that are smooth of order s ≥ 1 at zero when r ≥ (s - 2)/2 and m is chosen appropriately. For s > 2, the GPH estimator does not attain this rate. The proof uses results of Giraitis, Robinson, and Samarov (1997). We specify a data-dependent plug-in method for selecting the number of frequencies m to minimize asymptotic MSE for a given value of r. Some Monte Carlo simulation results for stationary Gaussian ARFIMA(1, d, 1) and (2, d, 0) model show that the bias-reduced estimators perform well relative to the standard log-periodogram regression estimator.