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

*(English)*Zbl 1153.62354Summary: We propose a simple bias-reduced log-periodogram regression estimator, \(\hat d_r\), 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 \(k=1,\dots,r\), 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 P. M. Robinson [Ann. Stat. 23, No. 3, 1048–1072 (1995; Zbl 0838.62085)] and C. M. Hurvich et al. [J. Time Ser. Anal. 19, No. 1, 19–46 (1998; Zbl 0920.62108)], we establish the asymptotic bias, variance, and mean-squared error (MSE) of \(\hat d_r\), determine the asymptotic MSE optimal choice of the number of frequencies, \(m\), to include in the regression, and establish the asymptotic normality of \(\hat d_r\). These results show that the bias of \(\hat d_r\) 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.

We show that the bias-reduced estimator \(\hat d_r\) attains the optimal rate of convergence for a class of spectral densities that includes those that are smooth of order \(s\geq 1\) at zero when \(r\geq (s-2)/2\) and \(m\) is chosen appropriately. For \(s>2\), the GPH estimator does not attain this rate. The proof uses results of L. Giraitis, P. M. Robinson, and A. Samarov [J. Time Ser. Anal. 18, No. 1, 49–60 (1997; Zbl 0870.62073)].

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.

Following the work of P. M. Robinson [Ann. Stat. 23, No. 3, 1048–1072 (1995; Zbl 0838.62085)] and C. M. Hurvich et al. [J. Time Ser. Anal. 19, No. 1, 19–46 (1998; Zbl 0920.62108)], we establish the asymptotic bias, variance, and mean-squared error (MSE) of \(\hat d_r\), determine the asymptotic MSE optimal choice of the number of frequencies, \(m\), to include in the regression, and establish the asymptotic normality of \(\hat d_r\). These results show that the bias of \(\hat d_r\) 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.

We show that the bias-reduced estimator \(\hat d_r\) attains the optimal rate of convergence for a class of spectral densities that includes those that are smooth of order \(s\geq 1\) at zero when \(r\geq (s-2)/2\) and \(m\) is chosen appropriately. For \(s>2\), the GPH estimator does not attain this rate. The proof uses results of L. Giraitis, P. M. Robinson, and A. Samarov [J. Time Ser. Anal. 18, No. 1, 49–60 (1997; Zbl 0870.62073)].

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.