14.3 Semiparametric Estimators in the Spectral Domain
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
.
14.3.1 Log-periodogram Regression
Under the assumption of normality, Geweke, and Porter-Hudak (1983)
assumed that the spectrum
near the zero frequency can be
approximated by
 |
(14.9) |
and then propose to estimate the long-memory parameter
with the following
spectral regression:
 |
(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
We obtain the following output:
Contents of d
[1,] 0.088369
The Robinson (1994b) averaged periodogram estimator is defined by:
 |
(14.11) |
where
is the average periodogram
![$ \hat{F}(\lambda) = \frac{2 \pi}{n} \sum_{j=1}^{[n \lambda/2\pi]} I(\lambda_j).$](xploreapplichtmlimg1530.gif) |
(14.12) |
By construction, the estimated parameter
is
, i.e.,
is in the stationarity range. This estimator has the following
asymptotic distribution if
 |
(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
We obtain the following output:
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"
14.3.2 Semiparametric Gaussian Estimator
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) |
where
is evaluated for a range of harmonic frequencies
,
bounded by the
bandwidth parameter
, which increases with the sample size
but more slowly: the bandwidth
must satisfy
 |
(14.15) |
If
, this estimator is a Gaussian estimator for the parametric
model
. After eliminating
, the
estimator
is equal to:
 |
(14.16) |
Although this Gaussian estimator has no closed form, it is more
efficient than the averaged periodogram estimator since
 |
(14.17) |
Furthermore, Velasco (1998) has considered the nonstationary case,
i.e., where
, and has shown that, with tapered data, this
estimator is consistent for
and asymptotically normal
for
, i.e., the statistical properties are robust to
nonstationary but nonexplosive alternatives.
The quantlet
robwhittle
computes this local Whittle estimator.
Its syntax is:
d = robwhittle(ary{, m})
where
- ary
- is the series
- m
- is a vector of bandwidth parameters. If this optional
argument is not provided, the default bandwith vector m =
,
,
, where
denotes the sample size.
The results are displayed in the form of a table, the first column
contains the value of the bandwidth parameter, while the second column
contains the estimated value of
.
The instructions
library("times")
y = read("dmus58.dat")
ar = abs(tdiff(y[1:2000]))
d = robwhittle(ar)
d
yield the following table
Contents of d
[1,] " d Bandwidth"
[2,] "_____________________"
[3,] ""
[4,] " 0.0948 500"
[5,] " 0.1078 250"
[6,] " 0.1188 125"