| In this talk, a rigorous analysis is given to calculate the mean-squared prediction error (MSPE) of the least squares predictor in an unstable autoregressive model, e.g. the random walk model. Our result can precisely measure the contribution due to nonstationarity, and accords with Wei's work (1987) on the accumulated predictive errors. It also complements the work of Fuller and Hasza (1981), only considering the first-order properties without revealing the impact of unstable factors. our analysis can be applied to handle the misspecification case, and may gain certain advantage over the exponential-smoothing method in the ARIMA model. Performances of the out-of-sample and the within-sample predictions are also compared. While most of the model selection and prediction procedures are justified by the out-of-sample prediction, our comparison suggests one should focus on the within-sample one especially when the underlying process is strongly dependent. A new methodology for analyzing the final prediction error in a strongly dependent process is also introduced. |