Log-Periodogram Regression of Time Series with Long Range Dependence

Abstract

This paper discusses the estimation of multiple time series models which allow elements of the spectral density matrix to tend to infinity or zero at zero frequency and be unrestricted elsewhere. A form of log-periodogram regression estimate of differencing and scale parameters is proposed, which can provide modest efficiency improvements over a previously proposed method (for which no satisfactory theoretical justification seems previously available) and further improvements in a multivariate context when differencing parameters are a priori equal. Assuming Gaussianity and additional conditions which seem mild, asymptotic normality of the parameter estimates is established.