Impulse response and forecast error variance matrix asymptotics are developed for VAR models with some roots at or near unity and some cointegration. For such models, it is shown that impulse responses that are estimated from an unrestricted VAR are inconsistent at long horizons and tend to random variables rather than the true impulse responses in the limit. The asymmetric distribution of the limit variates helps to explain the asymmetry of the ﬁnite sample distributions of the estimated impulse responses that is often found in simulations. VAR regressions also give inconsistent estimates of the forecast error variance of the optimal predictor at long horizons, and have a tendency to understate this variance. Moreover, predictions from an unrestricted nonstationary VAR are not optimal in the sense that they do not converge to the optimal predictors, at least for long horizons. In these respects, the asymptotic theory of prediction and policy analysis for nonstationary VAR’s is very diﬀerent from that which applies in stationary VAR’s. By contrast, in a reduced rank regression the impulse response and forecast error variance matrix estimates are consistent and predictions from the ﬁtted RRR model are asymptotically optimal, all provided the cointegrating rank is correctly speciﬁed or consistently estimated. Some simulations are reported which show these ﬁndings to be relevant in ﬁnite samples, and which assess the sensitivity of forecasting performance and policy analysis to certain design features of models in the VAR class.
Phillips, Peter C.B., "Impulse Response and Forecast Error Variance Asymptotics in Nonstationary VAR's" (1995). Cowles Foundation Discussion Papers. 1345.