COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS
AT YALE UNIVERSITY
Box 208281
New Haven, CT 06520-8281
COWLES FOUNDATION DISCUSSION PAPER NO. 1263
A Bias-Reduced Log-Periodogram Regression Estimator
for the Long-Memory Parameter
Donald W.K. Andrews and Patrik Guggenberger
June 2000
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.
Keywords: Asymptotic bias, asymptotic normality, bias reduction, frequency
domain, long-range dependence, optimal rate, rate of convergence, strongly dependent time
series
JEL Classification: C13, C14, C22 |