AVERAGED PERIODOGRAM ESTIMATION OF LONG MEMORY

This paper discusses estimates of the parameter H is an element of (1/ 2, 1) which governs the shape of the spectral density near zero freque ncy of a long memory time series. The estimates are semiparametric in the sense that the spectral density is parameterized only within a nei ghborhood of zero frequency. The estimates are based on averages of th e periodogram over a band consisting of m equally-spaced frequencies w hich decays slowly to zero as sample size increases. Robinson (1994a) proposed such an estimate of H which is consistent under very mild con ditions. We describe the limiting distributional behavior of the estim ate and also provide Monte Carlo information on its finite-sample dist ribution. We also give an expression for the asymptotic mean squared e rror of the estimate. In addition to depending on the bandwidth number m, the estimate depends on an additional user-chosen number q, but we show that for H is an element of (1/2, 3/4) there exists an optimal q for each H, and we tabulate this.