On the residuals of autoregressive processes and polynomial regression
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 \(\epsilon_ i\).
scientific article; zbMATH DE number 3936303