Part IV · Chapter 14
ARIMA Models
ARIMA(p,d,q): take d differences to make the series stationary, then fit ARMA(p,q) to the result. The "I" stands for INTEGRATED — meaning differenced.
\( \phi(L)(1-L)^d y_t = \theta(L) \varepsilon_t \)
Learning objectives
Choose (p,d,q) via Box-Jenkins.
Apply differencing prior to ARMA fitting.
Recognize ARIMA(0,1,0) = random walk.
Use residual diagnostics for adequacy.
ARIMA Builder
Seed:
Box-Jenkins workflow
Identify: plot + ADF + ACF/PACF
Estimate: fit ARIMA(p,d,q)
Diagnose: Ljung-Box on residuals
Forecast: static vs dynamic
📐 ARIMA(p,d,q)
\[ (1-\phi_1 L - \cdots - \phi_p L^p)(1-L)^d y_t = (1+\theta_1 L + \cdots + \theta_q L^q) \varepsilon_t \]
d number of differences
(0,1,0) random walk
(0,1,1) Δy_t follows MA(1)
(1,1,0) Δy_t follows AR(1)
🔍 What to look for
- Original series may trend; differenced series should look stationary.
- Forecast in LEVELS continues the trend (random walk-like for d=1).
- Always validate (p,q) on the differenced (stationary) series.
⚠️ Pro Tip: What to Avoid
Student says
"The I in ARIMA means 'interactive'."
Why this is wrong
The I means INTEGRATED — specifically, the series has been differenced d times to achieve stationarity.
Correct interpretation
ARIMA(p,d,q) = ARMA(p,q) on the d-th difference of y. Always determine d FIRST (ADF + KPSS), then identify p,q.
📝 Mini-quiz
📋 Key Takeaways
| Spec | Meaning |
|---|---|
| ARIMA(0,1,0) | Random walk |
| ARIMA(0,1,1) | Differenced MA(1) — common for ETS-like dynamics |
| ARIMA(1,1,0) | Differenced AR(1) |
| ARIMA(p,0,q) | = ARMA(p,q), already stationary |