AR(1) — Non-causal Stationary Case

AR(1) — Non-causal Stationary Case ($|\phi| > 1$)

$$X_t = \phi X_{t-1} + Z_t, \qquad |\phi| > 1$$

Rewrite: $X_t = \frac{X_{t+1}}{\phi} - \frac{Z_{t+1}}{\phi}$.

Iterate forward:

$$X_t = -\sum_{j=1}^{\infty} \phi^{-j} Z_{t+j}$$

Since $|\phi^{-1}| < 1$, this sum converges. It is a linear process with $\psi_j = -\phi^{-j}$ for $j \geq 1$ (and $\psi_j = 0$ for $j \leq 0$).

Stationary: yes (the representation is a convergent linear process)
Causal: no — $X_t$ depends on future noise $Z_{t+1}, Z_{t+2}, \ldots$ (future-dependent)

$|\phi| > 1$ gives a stationary but non-causal AR(1). It is not physically realizable.