ACF/PACF Order Identification: Reading a Time Series' Autocorrelation Signature

Generating Process

φ₁ 0.70

φ₂ -0.30

θ₁ 0.60

Order Identification

Reading the bars…

Sample size N

300

Variance

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Lag-1 ACF

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95% bound ±

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How it works

The autocorrelation function at lag k measures how correlated the series is with a delayed copy of itself:

r_k = Σ(x_t − x̄)(x_{t−k} − x̄) / Σ(x_t − x̄)²
      t=k+1..N              t=1..N

The partial autocorrelation function at lag k measures the correlation between x_t and x_{t−k} after removing the linear effect of the lags in between. It's computed here with the Durbin–Levinson recursion:

φ_kk = [r_k − Σ φ_{k-1,j} r_{k-j}] / [1 − Σ φ_{k-1,j} r_j]   (j=1..k-1)
φ_kj = φ_{k-1,j} − φ_kk·φ_{k-1,k-j}                          (j=1..k-1)

Box-Jenkins identification reads the two bar charts against the dashed ±1.96/√N significance band:

AR(p) — ACF decays gradually (tails off), PACF cuts off sharply after lag p.
MA(q) — ACF cuts off sharply after lag q, PACF tails off gradually.
Random walk — ACF barely decays at all across lags (non-stationary); differencing once (d=1) turns it back into white noise, which is exactly what the "Difference" toggle demonstrates.

This is the real diagnostic step analysts run before fitting an ARIMA(p,d,q) or SARIMA model — d comes from how many differences it takes to flatten the ACF, p and q come from where PACF and ACF cut off.

ACF r_k PACF φ_kk ±1.96/√N bound

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