Linear Prediction by Autoregressive Model Fitting in the Time Domain

Let ₜ\ be a purely nondeterministic stationary process satisfying all the assumptions made by Berk (1974), and ₜ\ be another purely nondeterministic stationary process. Assume that yₜ is independent of xₜ but has exactly the same statistical properties as that of xₜ. Consider the linear prediction of future values of yₜ on the basis of past values, using prediction constants estimated from a realisation of T observations of xₜ by least-squares fitting of an autoregression of order k. By assuming that k → ∞, k³/T → 0 as T → ∞, the effect on the mean square error of prediction of estimating the autoregressive coefficients is determined. This effect is the same as for the case when the prediction constants are estimated by factorising a "windowed" estimate of the spectral density function of xₜ.