### 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 |
---|---|

Pages (from-to) | 984-1007 |

Number of pages | 24 |

Journal | Econometric Theory |

Volume | 19 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2003 Dec |

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### ASJC Scopus subject areas

- Economics and Econometrics
- Social Sciences (miscellaneous)

### Cite this

*Econometric Theory*,

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

**Higher order asymptotic theory for minimum contrast estimators of spectral parameters of stationary processes.** / Taniguchi, Masanobu; Van Garderen, Kees Jan; Puri, Madan L.

Research output: Contribution to journal › Article

*Econometric Theory*, vol. 19, no. 6, pp. 984-1007. https://doi.org/10.1017/S0266466603196053

}

TY - JOUR

T1 - Higher order asymptotic theory for minimum contrast estimators of spectral parameters of stationary processes

AU - Taniguchi, Masanobu

AU - Van Garderen, Kees Jan

AU - Puri, Madan L.

PY - 2003/12

Y1 - 2003/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0242594559&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242594559&partnerID=8YFLogxK

U2 - 10.1017/S0266466603196053

DO - 10.1017/S0266466603196053

M3 - Article

VL - 19

SP - 984

EP - 1007

JO - Econometric Theory

JF - Econometric Theory

SN - 0266-4666

IS - 6

ER -