Causal Linear Process
Definition
A linear process is causal if $\psi_j = 0$ for all $j < 0$:
$$X_t = \sum_{j=0}^{\infty} \psi_j Z_{t-j} = \psi_0 Z_t + \psi_1 Z_{t-1} + \psi_2 Z_{t-2} + \cdots$$with $\sum_{j=0}^{\infty} |\psi_j| < \infty$.
Interpretation
$X_t$ depends only on current and past noise $Z_s, s \leq t$. Equivalently, $X_t$ is uncorrelated with future noise $Z_s, s > t$.
“Causal” = “future-independent.”
Why Causality Matters
- Causal processes are physically realizable (current output depends only on past inputs)
- MA(q) is always causal (finite sum of current + past noise)
- AR(1) with $|\phi| < 1$ is causal; with $|\phi| > 1$ it is stationary but not causal