### Abstract

Let g(λ) be the spectral density of a stationary process and let f_{θ}(λ), θ ∈ Θ, be a fitted spectral model for g(λ). A minimum contrast estimator θ̂_{n} of θ is defined that minimizes a distance D(f_{θ}, ĝ_{n}) between f_{θ} and ĝ_{n} where ĝ_{n} is a nonparametric spectral density estimator based on n observations. It is known that θ̂_{n} is asymptotically Gaussian efficient if g(λ) = f_{θ}(λ). Because there are infinitely many candidates for the distance function D(f _{θ}, ĝ_{n}), this paper discusses higher order asymptotic theory for θ̂_{n} in relation to the choice of D. First, the second-order Edgeworth expansion for θ̂_{n} 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.

Original language | English |
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Pages (from-to) | 984-1007 |

Number of pages | 24 |

Journal | Econometric Theory |

Volume | 19 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2003 Dec 1 |

### ASJC Scopus subject areas

- Social Sciences (miscellaneous)
- Economics and Econometrics

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## Cite this

*Econometric Theory*,

*19*(6), 984-1007. https://doi.org/10.1017/S0266466603196053