Log-periodogram Regression of Time Series with Long Range Dependence

This paper discusses the use of fractional exponential models (Robinson (1990), Beran (1994)) to model the spectral density f(x) of a covariance stationary process when f(x) may be decomposed as f(x) = x ?2d f (x), where f (x) is bounded and bounded away from zero. A form of log-periodogram regression technique is presented both in the parametric context (i.e. f (x) is a nite order exponential model in the sense of Bloommeld (1973)) and the semi-parametric context (f (x) is regarded as a nuisance parameter). Assuming gaussianity and additional conditions on the regularity of f (x) which seem mild, asymptotic normality of the parameter estimates in the parametric and the semi-parametric context is established. As a by-product, some improvements over the results presented by Robinson (1994) have been obtained for the large sample distribution of log-periodogram ordinates for Gaussian processes.