Semiparametric analysis of long-memory time series. (English) Zbl 0795.62082

Robinson, P. M.

Ann. Stat. 22, No. 1, 515-539 (1994).

Summary: 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.

MSC:

62M15	Inference from stochastic processes and spectral analysis
62G05	Nonparametric estimation
60G18	Self-similar stochastic processes

Keywords:

regular variation; autocorrelation-consistent standard errors; cointegration; semiparametric statistical inference; long-memory covariance stationary time series; unknown self-similarity parameter; discretely averaged periodogram; consistency properties; scale factor; standard errors; least squares estimates of polynomial regression; long- memory errors; generalized least squares estimates