Forecasting vector autoregressions with mixed roots in the vicinity of unity

This article evaluates the forecast performance of model averaging forecasts in a nonstationary vector autoregression with mixed roots in the vicinity of unity. The deviation from unit root allows for local to unity, moderate deviation from unity and strong unit root, and the direction of such deviation could be from either the stationary or the explosive side. We provide a theoretical foundation for comparison among various forecasts, including the least squares estimator, the constrained estimator imposing the unit root constraint, and the selection or average over these two basic estimators. Furthermore, three new types of estimators are constructed, i.e., the bagging versions of the pretest estimator, the Mallows-pretest estimator that marries the Mallows averaging criterion and the Wald test, and the Mallows-bagging estimator that combines the Mallows averaging criterion and bagging technique. The asymptotic risks are shown to depend on the local parameters, which are not consistently estimable. Via Monte Carlo simulations, graphic comparisons indicate that the Mallows averaging estimator has both robust and outstanding forecasting performance. Model averaging over the vector autoregressive lag order is further considered to address the issue of model uncertainty in the lag specification. Finite sample simulations show that the Mallows averaging estimator performs superior to other frequently used selection and averaging methods. The application to forecasting the financial indices popularly used in the predictive regression further illustrates the practical merit of the proposed estimator.