Seasonality in a time series is a regular pattern of changes that repeats over S time periods, where S defines the number of time periods until the pattern repeats again.

For example, there is seasonality in monthly data for which high values tend always to occur in some particular months and low values tend always to occur in other particular months. In this case, S = 12 (months per year) is the span of the periodic seasonal behavior. For quarterly data, S = 4 time periods per year.

In a seasonal ARIMA model, seasonal AR and MA terms predict \(x_{t}\) using data values and errors at times with lags that are multiples of S (the span of the seasonality).

• With monthly data (and S = 12), a seasonal first order autoregressive model would use \(x_{t-12}\) to predict \(x_{t}\). For instance, if we were selling cooling fans we might predict this August’s sales using last August’s sales. (This relationship of predicting using last year’s data would hold for any month of the year.)
• A seasonal second order autoregressive model would use \(x_{t-12}\) and \(x_{t-24}\) to predict \(x_{t}\). Here we would predict this August’s values from the past two Augusts.
• A seasonal first order MA(1) model (with S = 12) would use \(w_{t-12}\) as a predictor. A seasonal second order MA(2) model would use \(w_{t-12}\) and \(w_{t-24}\)

Differencing

Almost by definition, it may be necessary to examine differenced data when we have seasonality. Seasonality usually causes the series to be nonstationary because the average values at some particular times within the seasonal span (months, for example) may be different than the average values at other times. For instance, our sales of cooling fans will always be higher in the summer months.

Seasonal differencing

Seasonal differencing is defined as a difference between a value and a value with lag that is a multiple of S.

• With S = 12, which may occur with monthly data, a seasonal difference is \(\left( 1 - B ^ { 12 } \right) x _ { t } = x _ { t } - x _ { t - 12 }\).

The differences (from the previous year) may be about the same for each month of the year giving us a stationary series.

• With S = 4, which may occur with quarterly data, a seasonal difference is \(\left( 1 - B ^ { 4 } \right) x _ { t } = x _ { t } - x _ { t - 4 }\).

Seasonal differencing removes seasonal trend and can also get rid of a seasonal random walk type of nonstationarity.

Non-seasonal differencing

If trend is present in the data, we may also need non-seasonal differencing. Often (not always) a first difference (non-seasonal) will “detrend” the data. That is, we use \(( 1 - B ) x _ { t } = x _ { t } - x _ { t - 1 }\) in the presence of trend.

Differencing for Trend and Seasonality

When both trend and seasonality are present, we may need to apply both a non-seasonal first difference and a seasonal difference.

That is, we may need to examine the ACF and PACF of \(\left( 1 - B ^ { 12 } \right) ( 1 - B ) x _ { t } = \left( x _ { t } - x _ { t - 1 } \right) - \left( x _ { t - 12 }  - x _ { t - 13 } \right)\).

Removing trend doesn't mean that we have removed the dependency. We may have removed the mean, \(\mu_t\), part of which may include a periodic component. In some ways we are breaking the dependency down into recent things that have happened and long-range things that have happened.

Non-seasonal behavior will still matter...

With seasonal data, it is likely that short run non-seasonal components will still contribute to the model. In the monthly sales of cooling fans mentioned above, for instance, sales in the previous month or two, along with the sales from the same month a year ago, may help predict this month’s sales.

We’ll have to look at the ACF and PACF behavior over the first few lags (less than S) to assess what non-seasonal terms might work in the model.

Seasonal ARIMA Model

The seasonal ARIMA model incorporates both non-seasonal and seasonal factors in a multiplicative model. One shorthand notation for the model is

ARIMA \((p, d, q) \times (P, D, Q)S\)

with p = non-seasonal AR order, d = non-seasonal differencing, q = non-seasonal MA order, P = seasonal AR order, D = seasonal differencing, Q = seasonal MA order, and S = time span of repeating seasonal pattern.

Without differencing operations, the model could be written more formally as

(1) \(\Phi(B^S)\phi(B)(x_t-\mu)=\Theta(B^S)\theta(B)w_t\) 

1. Step 1: Do a time series plot of the data.

   Examine it for features such as trend and seasonality. You’ll know that you’ve gathered seasonal data (months, quarters, etc.,) so look at the pattern across those time units (months, etc.) to see if there is indeed a seasonal pattern.

2. Step 2: Do any necessary differencing.
3. If there is seasonality and no trend, then take a difference of lag S. For instance, take a 12th difference for monthly data with seasonality. Seasonality will appear in the ACF by tapering slowly at multiples of S. (View theTS plot and ACF/PACF plots
   TS plot and ACF/PACF plots

   ×

   for an example of data that requires a seasonal difference. Note that the TS plot shows a clear seasonal pattern that repeats over 12 time points. Seasonal differences are supported in the ACF/PACF of the original data because the first seasonal lag in the ACF is close to 1 and decays slowly over multiples of S=12. Once seasonal differences are taken, the ACF/PACF plots of twelfth differences support a seasonal AR(1) pattern.)
   ACF/PACF plots

   ×

1. If there is linear trend and no obvious seasonality, then take a first difference. If there is a curved trend, consider a transformation of the data before differencing.
2. If there is both trend and seasonality, apply a seasonal difference to the data and then re-evaluate the trend. If a trend remains, then take first differences. For instance, if the series is called x, the commands in R would be:

   diff12=diff(x, 12)
   plot(diff12)
   acf2(diff12)
   diff1and12 = diff(diff12, 1) 

3. If there is neither obvious trend nor seasonality, don’t take any differences.
4. Step 3: Examine the ACF and PACF of the differenced data (if differencing is necessary).

   We’re using this information to determine possible models. This can be tricky going involving some (educated) guessing. Some basic guidance:

   • Non-seasonal terms: Examine the early lags (1, 2, 3, …) to judge non-seasonal terms. Spikes in the ACF (at low lags) with a tapering PACF indicate non-seasonal MA terms. Spikes in the PACF (at low lags) with a tapering ACF indicate possible non-seasonal AR terms.

   • Seasonal terms: Examine the patterns across lags that are multiples of S. For example, for monthly data, look at lags 12, 24, 36, and so on (probably won’t need to look at much more than the first two or three seasonal multiples). Judge the ACF and PACF at the seasonal lags in the same way you do for the earlier lags.

5. Step 4: Estimate the model(s) that might be reasonable on the basis of step 3.

   Don’t forget to include any differencing that you did before looking at the ACF and PACF. In the software, specify the original series as the data and then indicate the desired differencing when specifying parameters in the arima command that you’re using.

6. Step 5: Examine the residuals (with ACF, Box-Pierce, and any other means) to see if the model seems good.

   Compare AIC or BIC values to choose among several models.

   If things don’t look good here, it’s back to Step 3 (or maybe even Step 2).