A Bias–Reduced Log–Periodogram Regression Estimator for the Long–Memory Parameter
Donald W. K. Andrews, Patrik Guggenberger
Econometrica · 2003 · Volume 71 · Issue 2
Abstract
In this paper, we propose a simple bias–reduced log–periodogram regression estimator,^dr, 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 2kfork=1,…,r, for some positive integerr, 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^dr, determine the asymptotic MSE optimal choice of the number of frequencies,m, to include in the regression, and establish the asymptotic normality of^dr. These results show that the bias of^drgoes 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^drattains the optimal rate of convergence for a class of spectral densities that includes those that are smooth of orders≥1 at zero whenr≥(s−2)/2 andmis chosen appropriately. Fors>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 frequenciesmto minimize asymptotic MSE for a given value ofr. 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.
Keywords
asymptotic bias, asymptotic normality, bias reduction, frequency domain, long–range dependence, optimal rate, plug–in estimator, rate of convergence, strongly dependent time series
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