On causal and non-causal cointegrated vector autoregressive time series

Previous-30 treatments of multivariate non-causal time series have assumed stationarity. In this article, we consider integrated processes in a non-causal setting. We generalize the Johansen–Granger representation for causal vector autoregressive (VAR) models to allow for dependence on future errors and discuss how the parameters can be estimated. The asymptotic distribution of the trace statistic is also considered. Some Monte Carlo simulations are presented