Higher-order kernel semiparametric M-estimation of long memory

We introduce a broad class of semiparametric estimates of the long memory parameter for stationary time series. A leading \Box-Cox" sub-class, indexed by a single tuning parameter, interpolates between the popular log periodogram and local Whittle estimates, leading to a smooth interpolation of asymptotic variances. The bias of these two estimates also di ers to higher order, and we also show how bias, and asymptotic mean squared error, can be reduced, across the class of estimates studied, by means of a suitable version of higher-order kernels. We thence calculate an optimal bandwidth (the number of low frequency periodogram ordinates employed) which minimizes this mean squared error. Finite sample performance is studied in a small Monte Carlo experiment, and an empirical application to intra-day foreign exchange returns is also included.