ARIMA(1,d,1) Deep Dive

Differencing removes stochastic trends — watch the ACF go from persistent (raw) to fast-decaying (differenced).

d (integrations)

Var(yₜ)

0.0

Var(Δᵈyₜ)

0.000

Sample ACF(1) raw

–

Raw Series yₜ (with unit roots)

Differenced Series Δᵈyₜ (d=1)

ACF of Raw yₜ — persistent / slow decay

ACF of Δᵈyₜ — stationary, fast decay

φ (AR)0.50

θ (MA)0.30

d (differencing order)1

σ (noise)1.0

Observations400

Seed55

Δ^{d} y_{t} = ϕ_{1} Δ^{d} y_{t - 1} + θ_{1} ε_{t - 1} + ε_{t}, Δ y_{t} = y_{t} - y_{t - 1}

d = 0 \Rightarrow ARMA; d = 1 \Rightarrow random-walk component; d = 2 \Rightarrow accelerating trend

Set d=0 to confirm the raw ACF decays quickly. Set d=1 or d=2 and watch the raw ACF become near-unit persistent while the differenced ACF recovers fast decay.

← ARMA GARCH →