Properties of predictors in overdifferenced nearly nonstationary autoregression

Abstract

This paper analyzes the effect of overdifferencing a stationary AR(p + 1) process whose largest root is near unity. It is found that if the largest root is p = exp( -cjT(3), f3 > 1, with T being the sample size and c a fixed constant, the estimators of the overdifferenced model ARIMA (p, 1,0) are root-T consistent. It is also found that this misspecified ARIMA(p, 1,0) has lower predictive mean square error than the properly specified AR(p + 1) model due to its parsimony. The consequences of this result are: (i) for forecasting purposes it is better to overdifferentiate than to underdifferentiate, (ii) the superiority of the overdifferenced predictor is small in the short term forecast but increases with the horizon, (iii) model selection based on predictive performance can lead to the wrong model in nearly nonstationary autoregression.