Let g(λ) be the spectral density of a stationary process and let fθ(λ), θ ∈ Θ, be a fitted spectral model for g(λ). A minimum contrast estimator θ̂n of θ is defined that minimizes a distance D(fθ, ĝn) between fθ and ĝn where ĝn is a nonparametric spectral density estimator based on n observations. It is known that θ̂n is asymptotically Gaussian efficient if g(λ) = fθ(λ). Because there are infinitely many candidates for the distance function D(f θ, ĝn), this paper discusses higher order asymptotic theory for θ̂n in relation to the choice of D. First, the second-order Edgeworth expansion for θ̂n is derived. Then it is shown that the bias-adjusted version of θ̂ n 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.
ASJC Scopus subject areas