This paper discusses the properties of an estimator of the memory
parameter of a stationary long-memory time-series originally proposed by
Robinson. As opposed to ‘‘narrow-band’’ estimators
of the memory parameter (such as the Geweke and Porter-Hudak or the Gaussian
semiparametric estimators) which use only the periodogram ordinates belonging
to an interval which degenerates to zero as the sample size $n$ increases, this
estimator builds a model of the spectral density of the process over all the
frequency range, hence the name, “broadband.” This is achieved by
estimating the ‘‘short-memory’’ component of the
spectral density, $f*(x) = |1 - e^{ix}|^{2d}f(x)$, where $d \epsilon (-1/2,
1/2)$ is the memory parameter and $f(x)$ is the spectral density, by means of a
truncated Fourier series estimator of log $f*$. Assuming Gaussianity and
additional conditions on the regularity of $f*$ which seem mild, we obtain
expressions for the asymptotic bias and variance of the long-memory parameter
estimator as a function of the truncation order. Under additional assumptions,
we show that this estimator is consistent and asymptotically normal. If the
true spectral density is sufficiently smooth outside the origin, this broadband
estimator outperforms existing semiparametric estimators, attaining an
asymptotic mean-square error $O(\log(n)/n)$ .