AR(1) — Causality vs Stationarity Summary
| Condition | Causal? | Stationary? |
|---|---|---|
| $ | \phi | < 1$ |
| $ | \phi | = 1$ |
| $ | \phi | > 1$ |
Key Insight
Causality and stationarity are not the same thing. Causality requires all roots of $\phi(z)$ outside the unit circle; stationarity requires a convergent linear process representation (which can use future noise terms).
In practice, we almost always work with causal models ($|\phi| < 1$), because non-causal models depend on future values.