These estimators are based on the behaviour of the spectrum
of a long-memory time series near the zero frequency, and are
estimated in the frequency band , where
is a bandwidth
parameter less than or equal to
, where
denotes the
integer part operator. The idea is that the range of frequencies
between zero and
captures the long term component, whilst the
remainder of the frequencies capture the local variations which could
be linear or nonlinear. These estimators are denoted
semiparametric in the sense that they depend on a bandwidth parameter
.
Under the assumption of normality, Geweke, and Porter-Hudak (1983)
assumed that the spectrum
near the zero frequency can be
approximated by
![]() |
(14.9) |
![]() |
(14.10) |
where is the sample size.
We consider for this estimator only harmonic frequencies ,
with
, where
is a trimming parameter discarding the
lowest frequencies and
is a bandwidth parameter.
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) d = gph(ar) d
Contents of d [1,] 0.088369
The Robinson (1994b) averaged periodogram estimator is defined by:
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(14.12) |
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(14.13) |
We evaluate the degree of long-memory with this estimator as follows:
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) d = roblm(ar) d
Contents of d [1,] " d Bandwidth q " [2,] "______________________________" [3,] "" [4,] " 0.0927 500 0.5" [5,] " 0.1019 250 0.5" [6,] " 0.1199 125 0.5"
The Robinson (1995a) semiparametric estimator, suggested by
Künsch (1987),
is based on the approximation (14.2) of the spectrum of
a long-memory process in the Whittle approximate maximum likelihood
estimator. An estimator of the fractional degree of integration is
obtained by solving the minimization problem:
![]() |
(14.14) |
![]() |
(14.15) |
![]() |
(14.17) |
The quantlet
robwhittle
computes this local Whittle estimator.
Its syntax is:
d = robwhittle(ary{, m})where
The instructions
library("times") y = read("dmus58.dat") ar = abs(tdiff(y[1:2000])) d = robwhittle(ar) d
Contents of d [1,] " d Bandwidth" [2,] "_____________________" [3,] "" [4,] " 0.0948 500" [5,] " 0.1078 250" [6,] " 0.1188 125"