Higher Order Asymptotic Theory for Semiparametric Estimation of Spectral Parameters of Stationary Processes

Let g([lambda]) be the spectral density of a stationary process and let f[theta]([lambda]), [theta] [set membership] [Theta], be a fitted spectral model for g([lambda]). A minimum contrast estimator of [theta] is defined that minimizes a distance between , where is a nonparametric spectral density estimator based on n observations. It is known that is asymptotically Gaussian efficient if g([lambda]) = f[theta]([lambda]). Because there are infinitely many candidates for the distance function , this paper discusses higher order asymptotic theory for in relation to the choice of D. First, the second-order Edgeworth expansion for is derived. Then it is shown that the bias-adjusted version of is not second-order asymptotically efficient in general. This is in sharp contrast with regular parametric estimation, where it is known that if an estimator is first-order asymptotically efficient, then it is automatically second-order asymptotically efficient after a suitable bias adjustment (e.g., Ghosh, 1994, Higher Order Asymptotics, p. 57). The paper establishes therefore that for semiparametric estimation it does not hold in general that "first-order efficiency implies second-order efficiency." The paper develops verifiable conditions on D that imply second-order efficiency.