Broadband log-periodogram regression of time series with long-range dependence

This paper discusses the properties of an estimator of the memory parameter of a stationary long-memory time-series originally proposed by Robinson. A s opposed to "narrow-band" estimators of the memory parameter (such as the Geweke and Porter-Hudak or the Gaussian semiparametric estimators) which us e only the periodogram ordinates belonging to an interval which degenerates to zero as the sample size n increases, this estimator builds a model of t he spectral density of the process over all the frequency range, hence the name, "broadband." This is achieved by estimating the "short-memory" compon ent of the spectral density, f*(x) = \1 - e(ix)\(2d)f(x), where d is an ele ment of (-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 Gaus sianity and additional conditions on the regularity of fb which seem mild, we obtain expressions for the asymptotic bias and variance of the long-memo ry parameter estimator as a function of the truncation order. Under additio nal assumptions, we show that this estimator is consistent and asymptotical ly normal. If the true spectral density is sufficiently smooth outside the origin, this broadband estimator outperforms existing semiparametric estima tors, attaining an asymptotic mean-square error O(log(n)/n).