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BIAS-REDUCED LOG-PERIODOGRAM AND WHITTLE ESTIMATION OF THE LONG-MEMORY PARAMETER WITHOUT VARIANCE INFLATION
Published online by Cambridge University Press: 30 August 2006
Abstract
The bias-reduced log-periodogram estimator
of Andrews and Guggenberger (2003,
Econometrica 71, 675–712) for the long-memory parameter
d in a stationary long-memory time series reduces the asymptotic
bias of the original log-periodogram estimator
of Geweke and Porter-Hudak (1983) by an order
of magnitude but inflates the asymptotic variance by a multiplicative
constant cr, for example,
c1 = 2.25 and c2 = 3.52. In this
paper, we introduce a new, computationally attractive estimator
by taking a weighted average of
estimators over different bandwidths. We show that, for each fixed
r ≥ 0, the new estimator can be designed to have the same
asymptotic bias properties as
but its asymptotic variance is changed by a constant
cr* that can be chosen to be as small as
desired, in particular smaller than cr. The
same idea is also applied to the local-polynomial Whittle estimator
in Andrews and Sun (2004, Econometrica
72, 569–614) leading to the weighted estimator
.
We establish the asymptotic bias, variance, and mean-squared error of the
weighted estimators and show their asymptotic normality. Furthermore, we
introduce a data-dependent adaptive procedure for selecting r and
the bandwidth m and show that up to a logarithmic factor, the
resulting adaptive weighted estimator achieves the optimal rate of
convergence. A Monte Carlo study shows that the adaptive weighted
estimator compares very favorably to several other adaptive
estimators.We thank a co-editor and three
referees for very helpful suggestions. We are grateful for the
constructive comments offered by Marc Henry, Javier Hidalgo, and
especially Katsumi Shimotsu.
- Type
- Research Article
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- Copyright
- © 2006 Cambridge University Press
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