SARIMA Models: Seasonal ARIMA

1. Motivation: Why SARIMA Models Are Needed

Many real-world time series exhibit seasonality, meaning that patterns repeat every fixed number of periods $S$ (e.g., monthly data with yearly cycles, $S=12$).
Standard ARIMA models handle nonstationarity caused by trends through differencing, but they do not explicitly model seasonal structure.

SARIMA models extend ARIMA by incorporating seasonal differencing and seasonal AR/MA components, allowing dependencies across timesteps that are $S$ periods apart.

2. Definition of a SARIMA Process

A process $(X_t)_{t=-\infty}^{\infty}$​ is said to follow a

$$\text{SARIMA}(p,d,q) \times (P,D,Q)_S$$

model if

$$Y_t = (1-B)^d (1-B^S)^D X_t$$

is a causal stationary ARMA process satisfying

$$\phi(B)\,\Phi(B^S)\,Y_t = \theta(B)\,\Theta(B^S)\,W_t,$$

where:

• $B$ is the backshift operator,
• $\phi(B)$: nonseasonal AR polynomial of degree $p$,
• $\theta(B)$: nonseasonal MA polynomial of degree $q$,
• $\Phi(B^S)$: seasonal AR polynomial of degree $P$,
• $\Theta(B^S)$: seasonal MA polynomial of degree $Q$,
• $S$: seasonal period,
• $W_t$​: white noise.

Equivalently, the model for the original series is:

$$\phi(B)\Phi(B^S)(1-B)^d(1-B^S)^D X_t = \theta(B)\Theta(B^S)W_t.$$

3. Interpreting Seasonal vs. Nonseasonal Components

Although the combined AR polynomial has degree $p + PS$ and the MA polynomial has degree $q + QS$, it is far more informative to represent the model using seasonal and nonseasonal components separately.
This preserves the interpretation that some dynamics occur within seasons and others across seasons.

Example

$$(1-0.5B)(1-B^4)X_t = (1 + 0.3B^4 + 0.8B^8)W_t$$

• Seasonal period: $S=4$
• Seasonal differencing: $D=1$
• Nonseasonal AR: $p=1$
• Seasonal MA: $Q=2$

Natural classification:

$$\text{SARIMA}(1,0,0) \times (0,1,2)_4$$

Although it could technically be written as a high-order nonseasonal MA model, that representation obscures the seasonal structure and is therefore less appropriate.

4. ACF Behavior of Seasonal ARMA Processes

After removing nonstationary components, the ACF of a seasonal ARMA process exhibits:

• Strong correlations at multiples of S (e.g., $S, 2S, 3S$), often decaying geometrically if seasonal AR terms are present.
• Spillover correlations at nearby lags (e.g., $S \pm 1$) when nonseasonal MA terms exist.

Example:

$$(1-0.6B^8)Y_t = (1-0.3B)W_t$$

This corresponds to an ARMA(0,1) × (1,0)$_8$​ process.
The ACF shows prominent spikes at lags 8, 16, 24, etc., with additional influence at neighboring lags due to the MA(1) term.

5. Example: Fitting a SARIMA Model to the AirPassengers Data

Step 1: Variance Stabilization

The variance of the AirPassengers series increases with its level, so a logarithmic transformation is applied:

$$Z_t = \log(X_t).$$

Step 2: Seasonal Differencing

Monthly data exhibit yearly seasonality, so $S=12$:

$$(1-B^{12})Z_t.$$

Step 3: Testing for Additional Nonseasonal Differencing

The ACF of the seasonally differenced series decays slowly.
An augmented Dickey–Fuller test yields a large p-value, indicating that a nonseasonal unit root remains.
Therefore, apply an additional first difference:

$$Y_t = (1-B)(1-B^{12})Z_t,$$

so $d=1$, $D=1$.

Step 4: Identifying ARMA Orders

The ACF of $Y_t$​ shows:

• a significant spike at lag 1 → nonseasonal MA(1),
• a significant spike at lag 12 → seasonal MA(1).

This suggests:

$$\text{ARMA}(0,1) \times (0,1)_{12}.$$

Hence, the original log-series is modeled as:

$$\text{SARIMA}(0,1,1) \times (0,1,1)_{12}.$$

Step 5: Model Estimation

Using R:

sarima(log(AirPassengers), 0,1,1, 0,1,1, 12)

The fitted model is:

$$(1-B)(1-B^{12})Z_t = (1-0.4018B)(1-0.5569B^{12})W_t.$$

Step 6: Model Diagnostics

All standard diagnostics are satisfactory:

• standardized residuals appear stationary,
• residual ACF shows no systematic autocorrelation,
• Q–Q plot shows near-normality,
• Ljung–Box p-values are consistently large.

Thus, the model is accepted.

6. Practical SARIMA Identification Strategy

From this lesson, a practical workflow is:

1. Identify the seasonal period $S$.
2. Apply a log transform if variance increases with level.
3. Apply seasonal differencing ($D$) first.
4. Use ACF and the ADF test to decide whether nonseasonal differencing ($d$) is needed.
5. Use ACF/PACF of the differenced series to propose $(p,q)$ and $(P,Q)$.
6. Fit candidate models and validate them using residual diagnostics.