No; it depends on the specification.

Is SARIMA the same as seasonal adjustment?

No; they serve different modeling purposes.

AI Glossary / Time series and signal processing / Advanced

Seasonal Autoregressive Integrated Moving-Average

Definition

Seasonal Autoregressive Integrated Moving-Average, commonly abbreviated SARIMA, is a time-series model that combines ordinary ARIMA terms with terms operating at seasonal lags. It can represent short-term dependence, seasonal dependence and differencing in one model. It is commonly written ARIMA(p,d,q)(P,D,Q)_m, where m is the seasonal period.

What it means

A series can depend on both recent observations and observations from the corresponding season in earlier cycles. For quarterly data with an annual cycle, that seasonal lag is four observations; for monthly data with an annual cycle, it is twelve.

Lowercase orders describe the nonseasonal part, and uppercase orders describe the seasonal part. The orders may be zero. A seasonal ARIMA model therefore need not contain a nonzero autoregressive coefficient, moving-average coefficient and differencing operation in every part simply because all three concepts appear in its name.

How it works

The orders p and q specify ordinary autoregressive and moving-average terms, while d specifies ordinary differencing. P, Q and D have corresponding seasonal roles. Seasonal operators use lags that are multiples of m.

In a multiplicative SARIMA specification, the seasonal and nonseasonal operators multiply. Expanding them can create cross-lag terms whose coefficients are constrained by that multiplication. This is different from fitting an unrestricted coefficient for every lag that appears.

Choose an appropriate sampling interval and seasonal period, estimate a defensible model and examine its innovations. Check forecasts against simpler alternatives. More terms or more differencing do not automatically improve a model, especially when only a few seasonal cycles are available.

Three practical examples

1. Quarterly operations: model an annual workload pattern alongside short-term persistence in changes from the same quarter last year.

2. Monthly demand: assess whether seasonal and ordinary lag relationships help predict future activity.

3. AI forecasting benchmarks: compare a complex forecasting system with a transparent statistical model using the same information and forecast dates.

Think of it this way

Think of a planner who asks two questions: what happened recently, and what happened at this point in the seasonal cycle? SARIMA provides a structured mathematical way to combine those kinds of information, while allowing differencing when the model requires it.

Worked example

Consider an invented quarterly index with eight observations:

Year 1: [100,120,130,110].

Year 2: [108,128,138,118].

Suppose the teaching model is ARIMA(1,0,0)(0,1,0)_4, without a constant. Define the seasonal difference w_t = y_t - y_(t-4), and assume w_t = 0.5w_(t-1) + e_t. The innovations are independent Gaussian variables with mean zero and variance 4. The coefficient and variance are stipulated, not estimated from these eight numbers.

The notation means one ordinary autoregressive term, no ordinary differencing, no moving-average terms and one seasonal difference at lag four. The seasonal period is four quarters, not four years.

For quarters 5 through 8, the seasonal differences are 108 - 100 = 8, 128 - 120 = 8, 138 - 130 = 8 and 118 - 110 = 8. The latest observed difference is therefore w_8 = 8.

At quarter 8, forecast the next four differences by repeatedly multiplying by 0.5. They are 4, 2, 1 and 0.5. These are forecasts of differences, not forecasts of the original index levels.

Recover the levels by adding the relevant observations from four quarters earlier:

Quarter 9: 108 + 4 = 112.

Quarter 10: 128 + 2 = 130.

Quarter 11: 138 + 1 = 139.

Quarter 12: 118 + 0.5 = 118.5.

A seasonal-naive forecast would instead repeat [108,128,138,118]. Our stipulated model adds a decaying positive change because the latest seasonal difference is positive. This arithmetic does not establish that its forecasts would outperform the seasonal-naive alternative on real data.

An expansion checks the model's lag structure. Substituting the definition of w into its equation gives y_t - y_(t-4) = 0.5[y_(t-1) - y_(t-5)] + e_t. Rearranging gives y_t = 0.5y_(t-1) + y_(t-4) - 0.5y_(t-5) + e_t.

The lag-five term is essential. For quarter 9, the conditional mean is 0.5 × 118 + 108 - 0.5 × 110 = 112. Omitting the lag-five subtraction would give 167 and would describe a different model. The three displayed lag coefficients are not three independently estimated autoregressive parameters.

Now consider uncertainty. For the first four forecast horizons, the seasonal base observations are already known. The level forecast errors therefore equal the corresponding forecast errors in w. Their variances are 4, 4(1 + 0.5^2) = 5, 4(1 + 0.5^2 + 0.5^4) = 5.25 and 4(1 + 0.5^2 + 0.5^4 + 0.5^6) = 5.3125.

The one-quarter standard deviation is two, giving an approximate 95% Gaussian prediction interval of 112 ± 1.96 × 2, or [108.08,115.92]. This calculation conditions on the known parameters and observed history. Estimation uncertainty and model misspecification are not included.

At horizon five, the seasonal base is itself a forecast. The mean for quarter 13 is the quarter-9 forecast, 112, plus the forecast difference 0.5^5 × 8 = 0.25, giving 112.25. Uncertainty must now account for shared future innovations in both pieces; simply reusing the first-four-horizon variance calculation would be wrong.

Illustrative case study

Fictional Cedar Forecasting receives a report that treats a seasonal period of four as four years and drops the lag-five term during implementation. Its analyst reconstructs the equations and identifies both mistakes.

Cedar then compares historical forecasts from the corrected specification with seasonal-naive forecasts. This invented case illustrates verification of a proposed model, not a measured accuracy improvement or a recommendation to use these particular orders.

Common misconceptions and mistakes

"The moving-average part is a rolling average of observations." It represents lagged innovations.

"Seasonal and ordinary orders are interchangeable." They operate at different lags.

"Differencing produces adjusted levels." Differences must be converted back for level forecasts.

"More parameters guarantee better forecasts." They can instead overfit.

Business use

Record the complete model orders, seasonal period, transformation and constant convention. Evaluate forecasts at operationally relevant horizons and keep simple benchmarks. Review large errors for structural changes rather than assuming a historical seasonal relationship will continue indefinitely.

Check your understanding

What does m = 4 mean here?

Answer: Four quarterly observations per annual cycle.

What is the quarter-9 level forecast?

Answer: 112.

Why is lag five present?

Answer: Multiplying the specified operators creates it.

Related terms

Seasonal difference
Prediction interval
ARIMA
autoregression
moving-average process
backshift operator
innovation
seasonal-naive forecast

Learn next

Replace the autoregressive coefficient 0.5 with 0.25, keeping the observed history unchanged. Calculate the first four level forecasts and explain how the smaller coefficient changes the persistence of seasonal differences.

Sources

Rob J Hyndman and George Athanasopoulos, Forecasting: Principles and Practice, ARIMA models, seasonal specification and forecasting: https://otexts.com/fpp3/arima.html

Statsmodels developers, SARIMAX model documentation, ordinary and seasonal orders: https://www.statsmodels.org/stable/generated/statsmodels.tsa.statespace.sarimax.SARIMAX.html

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