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Higher Order Asymptotic Theory For Minimum Contrast Estimators Of Spectral Parameters Of Stationary Processes

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  • Taniguchi, Masanobu
  • van Garderen, Kees Jan
  • Puri, Madan L.

Abstract

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 of θ 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(λ) = fθ(λ). 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.This paper was written while the first author was visiting the University of Bristol as a Benjamin Meaker Professor. The second author was previously at Bristol and is now supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences. We are grateful to the co-editor Pentti Saikkonen and two anonymous referees for their valuable comments, which significantly improved the paper.

Suggested Citation

  • Taniguchi, Masanobu & van Garderen, Kees Jan & Puri, Madan L., 2003. "Higher Order Asymptotic Theory For Minimum Contrast Estimators Of Spectral Parameters Of Stationary Processes," Econometric Theory, Cambridge University Press, vol. 19(6), pages 984-1007, December.
  • Handle: RePEc:cup:etheor:v:19:y:2003:i:06:p:984-1007_19
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    Cited by:

    1. Kakizawa, Yoshihide, 2007. "Moderate deviations for quadratic forms in Gaussian stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 992-1017, May.
    2. Tamaki, Kenichiro, 2007. "Second order optimality for estimators in time series regression models," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 638-659, March.

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