Anticausal time series ($X_t=-{\sum }_{j\ge 1}{\varphi }^{-j}{Z}_{t+j}$)

An anticausal noise representation uses future rather than present and past innovations. For a two-sided AR(1) equation with $|\varphi |>1$, the stationary solution is $X_t=-{\sum }_{j\ge 1}{\varphi }^{-j}{Z}_{t+j}$. Thus stationary existence alone does not require causality.