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Automatic Frequency Domain Inference on Semiparametric and Nonparametric Models

P. M. Robinson

Abstract

This paper discusses frequency domain methods of statistical inference for time series, in which the degree of smoothing in nonparametric spectral density estimation is determined from the data. A result on uniform convergence of spectral density estimates is established and applied in developing robust large-sample inference procedures based on instrumental variables estimates of a fairly general linear system with disturbances that have serial correlation of unknown form. Optimal instruments depend on the spectral density of the disturbances and on the frequency response function, and the latter is also nonparametric when the system is incomplete. The system may be parameterized over only a proper subset of the sampling frequencies. We adapt to these nonparametric features, justifying feasible, asymptotically optimal parameter estimates. The above results all assume covariance stationarity of the data, except for the special case of a multivariate linear regression model for which certain trending explanatory variables are permitted. In all these results, the degree of smoothing is allowed to depend on the data in a quite general way. The major technical achievement of the paper is to justify under primitive conditions the consistency of a particular, cross-validation, method of automatically determining a desirable degree of smoothing, which we apply to a multiple regression model. A Monte Carlo study illustrates our methodology and examines its performance in small and moderate samples.