# zbMATH — the first resource for mathematics

A bias-reduced log-periodogram regression estimator for the long-memory parameter. (English) Zbl 1153.62354
Summary: 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.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G08 Nonparametric regression and quantile regression 62M15 Inference from stochastic processes and spectral analysis 62G20 Asymptotic properties of nonparametric inference
Full Text: