On Unstable Autoregressive Model Forecasting

Abstract

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. This result can precisely measure the contribution due to non-stationarity, and accords with Wei's work (1987, Ann. Statist.) on the accumulated predictive errors. It also complements the work of Fuller and Hasza (1981, JASA), which only gives the convergent rate of the corresponding MSPE under rather stringent conditions, without revealing the impact of unstability. Finally, performances of the out-of-sample and the within-sample predictions are also compared. While most of the model selection criteria and forecasting procedures are justified by the out-of-sample prediction in the literature, our comparison suggests one should focus on the within-sample one especially when the underlying model is strongly dependent.