Nonlinear Fore(Back)Casting and Innovation Filtering for Causal–Noncausal VAR Models

We show that the mixed causal–noncausal vector autoregressive (VAR) processes satisfy the Markov property in both calendar and reverse time. Based on that property, we introduce closed-form formulas of forward and backward predictive densities for point and interval forecasting and backcasting out-of-sample. The backcasting formula is used for adjusting the forecast interval to obtain a desired coverage level when the tail quantiles are difficult to estimate. A confidence set for the prediction interval is introduced for assessing the uncertainty due to estimation. We also define new nonlinear past-dependent innovations of mixed causal–noncausal VAR models for impulse response function analysis. Our approach is illustrated by simulations and an application to the joint analysis of oil prices and real gross domestic product (GDP) growth rates.