Averaged periodogram estimation of long memory

This paper discusses estimates of the parameter H ϵ (12, 1) which governs the shape of the spectral density near zero frequency of a long memory time series. The estimates are semiparametric in the sense that the spectral density is parameterized only within a neighborhood of zero frequency. The estimates are based on averages of the periodogram over a band consisting of m equally-spaced frequencies which decays slowly to zero as sample size increases. Robinson (1994a) proposed such an estimate of H which is consistent under very mild conditions. We describe the limiting distributional behavior of the estimate and also provide Monte Carlo information on its finite-sample distribution. We also give an expression for the asymptotic mean squared error 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 ϵ (12, 34) there exists an optimal q for each H, and we tabulate this.