### Abstract

Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f_{0}(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator [formula omitted] of θ, we estimate the spectral density matrix f_{0}(ω) by f [formula omitted] (ω). Then we derive asymptotic expansions of the distributions of functions of f [formula omitted] (ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of [formula omitted](ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of [formula omitted], the estimated coherency, and contribution ratio in the principal component analysis based on [formula omitted] in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

Original language | English |
---|---|

Pages (from-to) | 75-96 |

Number of pages | 22 |

Journal | Econometric Theory |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1990 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Economics and Econometrics
- Social Sciences (miscellaneous)

### Cite this

**Asymptotic expansions of the distributions of statistics related to the spectral density matrix in multivariate time series and their applications.** / Taniguchi, Masanobu; Maekawa, Koichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Asymptotic expansions of the distributions of statistics related to the spectral density matrix in multivariate time series and their applications

AU - Taniguchi, Masanobu

AU - Maekawa, Koichi

PY - 1990

Y1 - 1990

N2 - Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f0(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator [formula omitted] of θ, we estimate the spectral density matrix f0(ω) by f [formula omitted] (ω). Then we derive asymptotic expansions of the distributions of functions of f [formula omitted] (ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of [formula omitted](ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of [formula omitted], the estimated coherency, and contribution ratio in the principal component analysis based on [formula omitted] in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

AB - Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f0(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator [formula omitted] of θ, we estimate the spectral density matrix f0(ω) by f [formula omitted] (ω). Then we derive asymptotic expansions of the distributions of functions of f [formula omitted] (ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of [formula omitted](ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of [formula omitted], the estimated coherency, and contribution ratio in the principal component analysis based on [formula omitted] in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

UR - http://www.scopus.com/inward/record.url?scp=84971845389&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84971845389&partnerID=8YFLogxK

U2 - 10.1017/S0266466600004928

DO - 10.1017/S0266466600004928

M3 - Article

AN - SCOPUS:84971845389

VL - 6

SP - 75

EP - 96

JO - Econometric Theory

JF - Econometric Theory

SN - 0266-4666

IS - 1

ER -