# 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

## 1 Answer

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: $$S=7$$

Somewhat confusingly, "quarterly seasonality" is sometimes used to refer to example 2 and sometimes to example 3 above. It's better practice to always explicitly note the time granularity (e.g., monthly or quarterly) and when seasonality is expected to recur (every year, or every quarter).

This holds especially if you have . For instance, with daily data, you could have intra-weekly seasonality, but also intra-yearly (and possibly also intra-monthly, for paycheck effects in countries where salaries are paid monthly).

• Thank you. That makes more sense with pairing of granularity. Maybe I've missed this in some examples. Otherwise it does follow what I was imagining with Example 3. Aug 28 '19 at 9:45