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

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Discussion Paper

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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, ^d r , 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 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, 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 ^d r , determine the MSE optimal choice of the number of frequencies, m , to include in the regression, and establish the asymptotic normality of ^d r . These results show that the bias of ^d 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. In consequence, the optimal rate of convergence to zero of the MSE of ^d r 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 ^d r 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.

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