A Bias-Reduced Log-Periodogram Regression Estimator for the Long-Memory Parameter

The widely used log-periodogram regression estimator of the long-memory parameter d proposed by Geweke and Porter-Hudak (1983) (GPH) has been criticized because of its finite-sample bias, see Agiakloglou, Newbold, and Wohar (1993). In this paper, we propose a simple bias-reduced log-periodogram regression estimator, ^dr, that eliminates the first- and higher-order biases of the 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, which is consistent with the semiparametric nature of the long-memory model under consideration.

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 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 ^dr 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. In consequence, the optimal rate of convergence to zero of the MSE of ^dr is faster than that of the GPH estimator.

We establish the optimal rate of convergence of a minimax risk criterion for estimators of d when the normalized spectral density is in a class that includes those that are smooth of order s > 1 at zero. We show that the bias-reduced estimator ^dr attains this rate when r > (s-2)/2 and m is chosen appropriately. For s > 2, the GPH estimator does not attain this rate. The proof of these results uses results of Giraitis, Robinson, and Samarov (1997).

Some Monte Carlo simulation results for stationary Gaussian ARFIMA(1,d,1) models show that the bias-reduced estimators perform well relative to the standard log-periodogram estimator.