On a criterion for the selection of models for stationary time series
type: Item
instance of: scholarly article
title: On a criterion for the selection of models for stationary time series
zbMATH Open document ID: 0606.62095
DOI: 10.1007/BF01897812
author: H. Linhart
author: P. Volkers
published in: Metrika
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/176002
review text: Consider a stationary time series \(x_ t\) with a spectral density f. One desires to fit ''the best model'' from a family with spectral densities \(g_{\theta}\), \(\theta \in R_ p\). Denote \(k(\omega,\theta)=\sigma_ g^{-2} 2\pi g_{\theta}(\omega)\), where \(\sigma^ 2_ g\) is the innovation variance. let \[ \phi =\partial k^{-1}/\partial \theta,\quad \psi =\partial^ 2k^{-1}/\partial \theta^ 2,\quad I(\omega)=(2\pi T)^{-1}| \sum^{T}_{s=1}x_ se^{i\omega (s-1)}|^ 2. \] Let \({\hat \theta}_ T\) be a strongly consistent estimator of \(\theta\) based on \(x_ 1,...,x_ T\). The proposed criterion is \[ \Delta_ T({\hat \theta}_ T)+T^{-1} tr({\hat \Omega}^{-1} {\hat \Sigma}), \] where \[ \Delta_ T(\theta)=(2\pi /T)\sum^{T-1}_{t=0}I(\omega_ t)k^{-1}(\omega_ t,\theta),\quad \omega_ t=2\pi t/T, \] \[ {\hat \Omega}=(2\pi /T)\sum^{T-1}_{t=0}\psi (\omega_ t,{\hat \theta}_ T)I(\omega_ t), \] \[ {\hat \Sigma}=(4\pi^ 2/T)\sum^{T-1}_{t=0} \phi(\omega_ t,{\hat \theta}_ T)\phi '(\omega_ t,{\hat \theta}_ T)I^ 2(\omega_ t). \]
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 3980298
zbMATH Keywords: misspecification
zbMATH Keywords: model selection
zbMATH Keywords: model fitting
zbMATH Keywords: ARMA models
zbMATH Keywords: stationary time series
zbMATH Keywords: spectral density
reviewed by: Jiří Anděl
MaRDI profile type: Publication
cites work: Fitting autoregressive models for prediction
cites work: Statistical predictor identification
cites work: Q4769776
cites work: Q5586858
cites work: An exponential model for the spectrum of a scalar time series
cites work: Q5663204
cites work: Q3247497
cites work: The asymptotic theory of linear time-series models
cites work: The asymptotic distribution of serial covariances
cites work: Asymptotic Properties of Non-Linear Least Squares Estimators
cites work: Maximum likelihood estimates of incorrect Markov models for time series and the derivation of AIC
cites work: Some recent advances in time series modeling
cites work: Alternative models for stationary stochastic processes
cites work: On estimation of the integrals of certain functions of spectral density
cites work: On selection of the order of the spectral density model for a stationary process
description: scientific article; zbMATH DE number 3980298