AR(1) — Causal Case ($|\phi| < 1$)

Model

$$X_t = \phi X_{t-1} + Z_t, \qquad |\phi| < 1, \quad Z_t \sim \text{WN}(0, \sigma^2)$$

Causal MA(∞) Representation

$$X_t = \frac{Z_t}{1 - \phi B} = \sum_{j=0}^{\infty} \phi^j Z_{t-j}$$

This is a causal linear process with $\psi_j = \phi^j$. Since $|\phi| < 1$, $\sum |\psi_j| = 1/(1-|\phi|) < \infty$.

Derivation (Method 1 — geometric series): $X_t = (1-\phi B)^{-1}Z_t = \sum_{j=0}^{\infty}(\phi B)^j Z_t = \sum_{j=0}^{\infty}\phi^j Z_{t-j}$.

Derivation (Method 2 — recursive substitution): $X_t = \phi X_{t-1} + Z_t = \phi(\phi X_{t-2} + Z_{t-1}) + Z_t = Z_t + \phi Z_{t-1} + \phi^2 X_{t-2} = \cdots = \sum_{j=0}^{\infty}\phi^j Z_{t-j}$ (the $\phi^n X_{t-n}$ term vanishes as $n \to \infty$ since $|\phi| < 1$).

Properties

$$E(X_t) = 0$$$$\gamma(0) = \text{Var}(X_t) = \frac{\sigma^2}{1 - \phi^2}$$$$\gamma(h) = \phi^h \gamma(0) = \frac{\phi^h \sigma^2}{1 - \phi^2}, \qquad \forall h > 0$$$$\rho(h) = \phi^h$$

ACF Behavior

$\rho(h) = \phi^h$ decays exponentially:

  • $\phi > 0$: monotone decay toward 0
  • $\phi < 0$: alternating sign, decaying in absolute value (damped oscillation / decayed sine wave)

The ACF never cuts off — it tails off. This is the signature of AR processes in contrast to MA.