Recent work in econometrics has provided large bandwidth asymptotic theory for taper-based studentized estimates of the mean, in the context of nonparametric estimation for serially correlated time series data. These taper-based statistics can be viewed as estimates of the spectral density at frequency zero, and hence it is quite natural to extend the asymptotic theory to non-zero frequencies and thereby obtain a large bandwidth theory for spectarl estimation. This approach was developed by Hashimzade and Vogelsang (2008) for the case of a single frequency. This paper extends their work in several ways: (i) we treat multiple frequencies jointly; (ii) we allow for long-range dependence at differing frequencies; (iii) we allow for piecewise smooth tapers, such as trapezoidal tapers; (iv) we develop a theory of higher order accuracy by a novel expansion of the Laplace Transform of the limit distribution. The theoretical results are complemented by simulations of the limit distributions, an application to confidence band construction, and a discussion of the issue of optimal bandwidth selection.
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