## The Annals of Statistics

### Semiparametric Analysis of Long-Memory Time Series

P. M. Robinson

#### Abstract

We study problems of semiparametric statistical inference connected with long-memory covariance stationary time series, having spectrum which varies regularly at the origin: There is an unknown self-similarity parameter, but elsewhere the spectrum satisfies no parametric or smoothness conditions, it need not be in $L_p$, for any $p > 1$, and in some circumstances the slowly varying factor can be of unknown form. The basic statistic of interest is the discretely averaged periodogram, based on a degenerating band of frequencies around the origin. We establish some consistency properties under mild conditions. These are applied to show consistency of new estimates of the self-similarity parameter and scale factor. We also indicate applications of our results to standard errors of least squares estimates of polynomial regression with long-memory errors, to generalized least squares estimates of this model and to estimates of a "cointegrating" relationship between long-memory time series.

#### Article information

Source
Ann. Statist. Volume 22, Number 1 (1994), 515-539.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176325382

Digital Object Identifier
doi:10.1214/aos/1176325382

Mathematical Reviews number (MathSciNet)
MR1272097

Zentralblatt MATH identifier
0795.62082

JSTOR