  We introduce closed-form formulas of out-of-sample predictive densities for
forecasting and backcasting of mixed causal-noncausal (Structural) Vector
Autoregressive VAR models. These nonlinear and time irreversible non-Gaussian
VAR processes are shown to satisfy the Markov property in both calendar and
reverse time. A post-estimation inference method for assessing the forecast
interval uncertainty due to the preliminary estimation step is introduced too.
The nonlinear past-dependent innovations of a mixed causal-noncausal VAR model
are defined and their filtering and identification methods are discussed. Our
approach is illustrated by a simulation study, and an application to
cryptocurrency prices.
