Impulse response and forecast error variance asymptotics in nonstationary VARs

Estimated impulse responses and forecast error decompositions are shown to be inconsistent at long horizons in unrestricted VARs with some unit roots. Predictions from unrestricted VARs also do not converge to the optimal predictors over long forecast horizons. In contrast, reduced rank regressions produce impulse responses and forecast error variance estimates that are consistent and predictions that are asymptotically optimal, provided the cointegrating rank is correctly specified or consistently estimated by an order selector such as PIC. Some simulations show these findings to be relevant in finite samples in VARs with some unit roots and cointegration.