Consider the unique stationary solution to the following relation $$X_t = \theta X_{t-1} + Z_t$$ where $(Z_t)_{t \in \mathbb{Z}}$ is a white noise series and $\lvert \theta \rvert > 1$. I am trying to find out the projection of $X_t$ onto the closure (in $L^2$) of the linear span of $X_{t-1}, X_{t-2},\ldots$, which I denote by $\Pi_{t-1} X_t$. Clearly,
$$\Pi_{t-1} X_t = \Pi_{t-1} \left(\theta X_{t-1} + Z_t\right) = \theta X_{t-1} + \Pi_{t-1} Z_t$$ Since $\lvert \theta \rvert > 1$, the stationary solution is purely non-causal. So $\Pi_{t-1} Z_t \neq 0$.
I don't think starting with the projection equations is possible as I don't really know what is in the closure, i.e. I cannot say $$\Pi_{t-1} X_t = \sum_{i\geq 1} a_i X_{t-i}$$ since I would be missing the limit points. I feel like I am forgetting a crucial fact that is applicable here. Any help appreciated.