Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process

Let \xₜ\ be a linear stationary process of the form xₜ + ₁ ₈<aᵢxₓ-₈ = eₜ, where \eₜ\ is a sequence of i. i. d. normal random variables with mean 0 and variance ². Given observations x₁, , xₙ, least squares estimates a (k) of a' = (a₁, a₂, ), and ²ₖ of ² are obtained if the kth order autoregressive model is assumed. By using a (k), we can also estimate coefficients of the best predictor based on k successive realizations. An asymptotic lower bound is obtained for the mean squared error of the estimated predictor when k is selected from the data. If k is selected so as to minimize Sₙ (k) = (n + 2k) ²ₖ, then the bound is attained in the limit. The key assumption is that the order of the autoregression of \xₜ\ is infinite.