ARMA(1,1) Deep Dive

Both ACF and PACF tail off — the signature of ARMA. Explore the spectral density and how φ and θ jointly shape dynamics.

Stationary

Yes ✓

ρ(1)

0.000

AR half-life

1.4

σ²

1.00

Simulated ARMA(1,1) Series

ACF — both tail off (vs theoretical)

Spectral Density f(ω) — frequency content

φ (AR coefficient)0.60

θ (MA coefficient)0.40

σ (noise)1.0

Observations600

Seed99

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

ρ (1) = \frac{( ϕ + θ ) ( 1 + ϕθ )}{1 + θ ^{2} + 2 ϕθ}, ρ (k) = ϕ^{k - 1} ρ (1) (k \geq 2)

f (ω) = \frac{σ ^{2}}{2 π} \cdot \frac{1 + θ ^{2} + 2 θ cos ω}{1 + ϕ ^{2} - 2 ϕ cos ω}

When φ and θ have opposite signs, they partially cancel — the ACF peaks later, and the spectral density shifts. Experiment with φ=0.8, θ=−0.6 to see near-cancellation.

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