Second order optimality for estimators in time series regression models

We consider the second order asymptotic properties of an efficient frequency domain regression coefficient estimator β ^ proposed by Hannan [Regression for time series, Proc. Sympos. Time Series Analysis (Brown Univ., 1962), Wiley, New York, 1963, pp. 17–37]. This estimator is a semiparametric estimator based on nonparametric spectral estimators. We derive the second order Edgeworth expansion of the distribution of β ^ . Then it is shown that the second order asymptotic properties are independent of the bandwidth choice for residual spectral estimator, which implies that β ^ has the same rate of convergence as in regular parametric estimation. This is a sharp contrast with the general semiparametric estimation theory. We also examine the second order Gaussian efficiency of β ^ . Numerical studies are given to confirm the theoretical results.