MA(q) Deep Dive

Moving-average models with finite memory — ACF cuts off after lag q, PACF decays geometrically. Always stationary.

Invertible

Yes ✓

ρ(1)

0.457

ρ(2)

0.000

Var(y)

1.423

Simulated MA Series

ACF — cuts off after lag q

Impulse Response ψ_k (shock persistence)

θ₁ (shock lag-1)0.65

θ₂ (shock lag-2, 0=MA1)0.00

σ (noise)1.0

Observations600

Seed7

y_{t} = ε_{t} + θ_{1} ε_{t - 1} + θ_{2} ε_{t - 2}, ε_{t} \sim N (0, σ^{2})

ρ (1) = \frac{θ _{1} + θ _{1} θ _{2}}{1 + θ _{1}^{2} + θ _{2}^{2}}, ρ (2) = \frac{θ _{2}}{1 + θ _{1}^{2} + θ _{2}^{2}}, ρ (k) = 0 (k > 2)

The ACF cuts off at lag q — the key identification signature for MA models. Drag θ₂ away from zero to see the ACF extend to lag-2 while still being exactly zero beyond.

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