AR(p) Deep Dive

Explore how autoregressive lag coefficients shape the correlation structure, stationarity region, and impulse response.

Stationary

Yes ✓

ACF(1)

–

Half-life (lags)

∞

Var(y)

0.000

Simulated AR Series (first 300 obs)

Sample ACF vs Theoretical

Sample PACF (cuts off at order p)

AR(2) Stationarity Region — current (φ₁, φ₂) shown as marker

φ₁ (lag-1 coef)0.75

φ₂ (lag-2 coef)0.00

σ (noise)1.0

Observations600

Seed42

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

Stationarity: roots of 1 - ϕ_{1} z - ϕ_{2} z^{2} = 0 outside unit circle

ρ (k) = ϕ_{1} ρ (k - 1) + ϕ_{2} ρ (k - 2), k \geq 1

The PACF cuts off sharply at lag p for an AR(p) process — the key diagnostic for model order selection via Box–Jenkins.