Seasonal ARIMA - Understanding the S term

I am reviewing the use and creation of seasonal ARIMA (SARIMA (P,D,Q)(p,d,q)[S]) models to use on some financial time-series data. However, as soon as I started, I hit a snag. More than one source is saying that if you have "quarterly seasonality" then the S term = 4, and if you have yearly (which they call "monthly") seasonality, then S = 12. I'm confused:

In their examples
- "monthly seasonality" meant things like peaks every December, and troughs every June.
- "quarterly seasonality" meant something that repeated every quarter.

It therefore makes no sense for S to be 12 for one, and 4 for the other. Logically, S=12 implies "length of the season". The length of a quarter is 3 months. If both data are monthly, there is 1 season in 1 year, but we don't set S=1. With the length of a quarter being 3 months, shouldn't S be 3?

Can someone tell me if this is an error in the text(s), or otherwise explain why quarterly seasonality would use S=4?

Semi-annual would likely use S=6, (not S=2) correct? Yearly is always shown as 12.

• It's about frequency in an interval, it's not about length. Aug 28 '19 at 7:29

The $$S$$ parameter indicates that each season is $$S$$ underlying time periods long, or equivalently, that seasonal patterns recur every $$S$$ underlying time periods. For instance:
1. Monthly data, yearly seasonality: $$S=12$$ (12 months in a year)
2. Quarterly data, yearly seasonality: $$S=4$$ (4 quarters in a year)
3. Monthly data, quarterly seasonality (e.g., if sales people try to hit their quarterly quota and start selling more aggressively as the quarter closes): $$S=3$$ (3 months in a quarter)
4. Daily data, weekly seasonality: $$S=7$$