On the residuals of autoregressive processes and polynomial regression
type: Item
instance of: scholarly article
title: On the residuals of autoregressive processes and polynomial regression
zbMATH Open document ID: 0584.62144
DOI: 10.1016/0304-4149(85)90380-1
author: B. George
published in: Stochastic Processes and their Applications
publication date: 1985-01-01T00:00:00Z
review text: The asymptotic behaviour of the residuals obtained from ordinary least squares fitting in autoregression and polynomial regression is investigated. In the autoregressive case the partial sums \[ \hat B_ t^{(n)}=\sigma^{-1}n^{-1/2}\sum^{[nt]}_{i=1}{\hat \epsilon}_{i,p},\quad 0\leq t\leq 1, \] of the residuals \({\hat \epsilon}{}_{i,n}=X_{i,n}-({\hat \beta}_{1,n}X_{i-1}+...+{\hat \beta}_{p,n}X_{i-p})\), \(i=1,...,n,\) converge weakly to the standard Brownian motion \(B_ t\). Similarly in the polynomial case, these residual partial sums converge to generalized Brownian bridges. The author uses the technique of Skorokhod representation. Some applications of this asymptotic representation of the residuals are considered, e.g. parameter estimation based on approximate log likelihood functions of the residuals \(L_ n(\vartheta)=\sum^{n}_{i=1}\log f({\hat \epsilon}_{i,n},\vartheta)\), where f(\(\cdot,\vartheta)\) is the density of \(\epsilon_ i\).
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3936303
zbMATH Keywords: asymptotic behaviour of the residuals
zbMATH Keywords: ordinary least squares fitting
zbMATH Keywords: autoregression
zbMATH Keywords: polynomial regression
zbMATH Keywords: Brownian motion
zbMATH Keywords: residual partial sums
zbMATH Keywords: generalized Brownian bridges
zbMATH Keywords: Skorokhod representation
zbMATH Keywords: asymptotic representation
zbMATH Keywords: parameter estimation
zbMATH Keywords: approximate log likelihood functions
reviewed by: Tomáš Cipra
MaRDI profile type: Publication
full work available at URL: 0304-4149(85)90380-1
OpenAlex ID: W1980948568
cites work: Q5560061
cites work: Q5563135
cites work: Q4082810
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cites work: Q3911791
cites work: Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times
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description: scientific article; zbMATH DE number 3936303