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

Masanobu Taniguchi, Koichi Maekawa

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1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)75-96
Number of pages22
JournalEconometric Theory
Volume6
Issue number1
DOIs
Publication statusPublished - 1990
Externally publishedYes

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time series
statistics
time series analysis
multivariate analysis
Multivariate time series
Statistics
Asymptotic expansion
Spectral density
Asymptotic distribution
Cobweb model
Vector autoregressive process
Quasi-maximum likelihood estimator
Reduced form
Time series analysis
Principal component analysis
Stationary process
Eigenvalues

ASJC Scopus subject areas

  • Economics and Econometrics
  • Social Sciences (miscellaneous)

Cite this

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title = "Asymptotic expansions of the distributions of statistics related to the spectral density matrix in multivariate time series and their applications",
abstract = "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.",
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AU - Taniguchi, Masanobu

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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.

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