Seasonal ARIMA (SARIMA)

Model

A seasonal ARIMA (SARIMA) extends Box–Jenkins ARIMA with seasonal terms, written SARIMA(p, d, q)(P, D, Q)[s]: regular and seasonal AR/MA orders, regular differencing $d$ to remove trend, and seasonal differencing $D$ to remove seasonality, at season length $s$. The canonical case is the "airline model" SARIMA(0,1,1)(0,1,1)[12] — the one Box & Jenkins fit to the airline series — with one regular difference, one seasonal difference, and an MA(1) × seasonal-MA(1), no AR terms.

The airline model is $(1-B)(1-B^{12})\,y_t = (1+\theta B)(1+\Theta B^{12})\,\varepsilon_t$.

Frequentist Box–Jenkins vs Bayesian structural

Two routes are compared. The frequentist Box–Jenkins workflow (identify → estimate by maximum likelihood via the Kalman filter → diagnose residuals) is run in statsmodels and R's forecast, with automatic order selection (auto_arima / auto.arima, the Hyndman–Khandakar stepwise AIC search). The Bayesian route is an interpretable structural regression — a linear trend + monthly seasonal dummies + AR(1) errors — fit by NUTS, giving full posterior-predictive forecast distributions. The two differ in how they handle seasonality: SARIMA uses seasonal differencing ($1-B^s$, stochastic seasonality that can drift), the structural model uses fixed monthly effects (deterministic seasonality) — both legitimate, and they forecast similarly when the seasonal shape is stable.

Notebooks

Five examples span the range of ARIMA behaviour — fixed-period seasonality, variable-period cyclicality, a near-deterministic trend-plus-season, a barely-forecastable business cycle, and a structural break — building the ARIMA → changepoint → regime-switching ladder. Example 1 — AirPassengers (multiplicative seasonality). Monthly airline passengers, 1949–1960 (Box & Jenkins): a clear upward trend and multiplicative seasonality (swings grow with the level), the textbook case for SARIMA on the log scale. Stationarity tests (ADF / KPSS, opposite nulls) call for a regular and a seasonal difference; both selectors pick the airline model SARIMA(0,1,1)(0,1,1)[12], residuals pass Ljung–Box, and the held-out 2 years forecast to MAPE ≈ 8–9%. A judgement caveat surfaces: left to choose $d$ itself, auto_arima picks $d=0$, leaving the trend in and degrading the forecast — automated selectors still need the differencing pinned by the unit-root tests. Example 2 — Sunspots (cyclicality, not seasonality). Annual sunspot counts, 1700–2008: the famous ~11-year solar cycle is cyclicality — a variable, unknown period (~9–13 yr, wandering), not a fixed seasonal one. There is no seasonal differencing; the series is stationary ($d=0$) and a plain AR(p) generates the pseudo-cycle through complex-conjugate roots whose angle sets the period. The ACF is a damped sine wave (vs. sharp seasonal spikes), all three engines select ~AR(9) and recover a ~10–11-year cycle (Yule's 1927 AR(2) on this very series was the original demonstration), and the forecast damps toward the mean within a cycle or two — a stochastic cycle is only predictable a short way out. The Bayesian AR adds a genuine posterior distribution over the cycle length, read from the complex roots of each draw. Example 3 — Mauna Loa CO₂ (near-deterministic trend + season). Monthly atmospheric CO₂, 1958–2001 (the Keeling curve): an accelerating upward trend plus a very regular additive annual cycle (~7 ppm, constant amplitude — so no log transform). STL splits it into a smooth trend, an almost perfectly repeating cycle, and a tiny remainder; auto_arima picks a parsimonious SARIMA, and the held-out 5 years forecast to MAPE ≈ 0.2% — among the most forecastable series anywhere, the opposite extreme from the damping sunspot cycle. Example 4 — US GNP growth (the business cycle). Quarterly real GNP growth, 1951–1984 (Hamilton 1989): the macroeconomic business cycle is cyclicality with no fixed period — recessions arrive irregularly. Growth is already stationary (a difference of log GNP, so $d=0$) with only weak autocorrelation, so it is close to unforecastable: the ARMA's out-of-sample RMSE essentially ties predicting the mean and the forecast reverts within ~2 quarters — the honest empirical basis for "nobody forecasts GDP growth," the opposite extreme from CO₂. The Bayesian AR(1) puts $\phi \approx 0.3$ (a growth-shock half-life under a quarter). Crucially, a linear ARMA is symmetric, but real recessions are sharper than expansions — which is exactly what the Markov-switching AR adds on this very series; ARMA is the linear benchmark it improves on. Example 5 — Nile river flow (structural break / intervention). Annual Nile flow at Aswan, 1871–1970, which drops abruptly around 1899 (the old Aswan dam). The lesson: an ignored level shift masquerades as non-stationarity — differencing "fixes" it mechanically, but the right model adds a step-dummy regressor (Box–Tiao intervention analysis: regression with ARIMA errors), giving a parsimonious stationary AR(1) with a clean estimate of the break ($\approx -250$ units) and a lower AIC than a differenced ARIMA. The Bayesian version goes further — a changepoint model that estimates the break year itself as a parameter $\tau$, whose posterior concentrates tightly on 1899 without being told. A changepoint is "Markov-switching with a single, absorbing switch" — completing the ARIMA → changepoint → Markov-switching ladder. Together they bridge from seasonality/cyclicality through structural breaks toward the regime-switching and GARCH / stochastic-volatility work, where the variance (or the regime) itself becomes the moving part. Each example is cross-checked across statsmodels / pmdarima, PyMC, and R forecast.

References

• Box, G. E. P. & Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control. Holden-Day. — the Box–Jenkins methodology, the AirPassengers series, and the airline model
• Hyndman, R. J. & Khandakar, Y. (2008). Automatic time series forecasting: the forecast package for R. Journal of Statistical Software 27(3), 1–22. — the auto.arima / auto_arima stepwise order selection
• Yule, G. U. (1927). On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers. Philosophical Transactions of the Royal Society A 226, 267–298. — the AR model of the sunspot cycle
• Box, G. E. P. & Tiao, G. C. (1975). Intervention analysis with applications to economic and environmental problems. Journal of the American Statistical Association 70(349), 70–79. — the step-dummy / intervention model behind the Nile analysis
• Cobb, G. W. (1978). The problem of the Nile: conditional solution to a changepoint problem. Biometrika 65(2), 243–251. — the Nile flow changepoint at 1899
• Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2), 357–384. — the US GNP-growth data and the business-cycle motivation for regime switching

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