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.

中文翻译：

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关于因果和非因果协整向量自回归时间序列

多变量非因果时间序列的先前 30 次处理已假定平稳。在本文中，我们考虑了非因果环境中的集成过程。我们推广因果向量自回归 (VAR) 模型的 Johansen-Granger 表示，以允许依赖未来的错误并讨论如何估计参数。还考虑了迹统计量的渐近分布。介绍了一些蒙特卡罗模拟。