Estimation Of The Prediction Error Variance And An R2 Measure By Autoregressive Model Fitting

. For predicting the future values of a stationary process {xt} (t= 0, pL 1, pL 2,…) on the basis of its past, two key parameters are the variance V (h), h≥ 1, of the h‐step prediction error and Z(h) ={R(0) ‐ V(h)}/R(0), the corresponding measure, in an R2 sense, of the predictability of the process from its past, where R(0) denotes the process variance. The estimation of V(h) and Z(h) from a realization of T consecutive observations of {xt} is considered, without requiring that the process follows a finite parameter model. Three different autoregressive estimators are examined and are shown to be asymptotically equivalent in the sense that as T∝ they have the same asymptotic normal distribution. The question of bias in estimating these parameters is also examined and a bias correction is proposed. Finite sample behaviour is investigated by a simulation study.