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, "ˆd-sub-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 2"k" 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 "ˆd-sub-r", determine the asymptotic MSE optimal choice of the number of frequencies, "m", to include in the regression, and establish the asymptotic normality of "ˆd-sub-r". These results show that the bias of "ˆd-sub-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 "ˆd-sub-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) models show that the bias--reduced estimators perform well relative to the standard log--periodogram regression estimator. Copyright The Econometric Society 2003.