Gaussian estimation of parametric spectral density with unknown pole

Abstract

We consider a parametric spectral density with power-law behaviour about a fractional pole at the unknown frequency !. The case of known !, especially ! = 0, is standard in the long memory literature. When ! is unknown, asymptotic distribution theory for estimates of parameters, including the (long) memory parameter, is significantly harder. We study a form of Gaussian estimate. We establish n¡consistency of the estimate of !, and discuss its (non-standard) limiting distributional behaviour. For the remaining parameter estimates, we establish pn - consistency and asymptotic normality