ACF/PACF Correlogram: Identifying ARIMA Order

ARMA(1,1) Process

AR coefficient φ 0.60

MA coefficient θ 0.30

Integration order d

Correlogram view

Readouts

ACF(1)

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PACF(1)

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Stationarity check (split-sample heuristic)

Δmean 0% · Δvariance 0%

stationary

How it works

A stationary innovation series is built from an ARMA(1,1) recursion, then integrated d times to model a non-stationary ARIMA(p,d,q) observation — exactly how Box–Jenkins defines the "I":

Wₜ = φ·Wₜ₋₁ + εₜ + θ·εₜ₋₁      (stationary ARMA core)
Xₜ = Δ⁻ᵈ Wₜ                     (integrate d times → observed series)
ΔXₜ = Xₜ − Xₜ₋₁                 (difference back toward stationarity)

The autocorrelation function (ACF) and partial autocorrelation function (PACF) at lag k are:

r_k = Σ(yₜ−ȳ)(yₜ₋ₖ−ȳ) / Σ(yₜ−ȳ)²      (ACF, sample correlation)
φ_kk  from Durbin–Levinson recursion    (PACF, correlation at lag k
                                          with shorter lags' effect removed)

φ / θ sliders — set the AR and MA coefficients of the underlying stationary process.
d buttons — how many times the stationary series is cumulatively summed before you "observe" it. d=0 keeps it stationary; d≥1 injects a trend/random-walk drift.
Raw / Differenced — Raw shows the ACF/PACF of Xₜ as observed. When d≥1 its ACF decays very slowly (a classic non-stationarity signature). Differenced applies Δ d times first, which should recover the sharp ACF/PACF cutoff of the original ARMA(1,1) — this is the core Box–Jenkins identification step.
The dashed band on the bar rows marks the ±1.96/√N approximate 95% confidence bounds for "no autocorrelation at this lag".

Real-world relevance: this same workflow — difference until the ACF collapses, then read AR/MA order off where the ACF or PACF cuts off — is literally how the (p,d,q) order of an ARIMA forecasting model is chosen before fitting.

Time series Xₜ Positive correlation Negative correlation

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