1. Pure seasonal models
In this chapter, you will learn about fitting and forecasting seasonal ARIMA models
2. Pure Seasonal Models
We often collect data with known seasonal components such as the monthly Air Passengers time series and the quarterly Johnson & Johnson earnings time series. Both have an annual cycle, 1 cycle every 12 months for the Air Passengers and 1 cycle every 4 quarters for Johnson and Johnson.
The letter S denotes the seasonal period.
3. Pure Seasonal Models
Although not realistic, it is instructive to first consider pure seasonal models such as the seasonal AR shown here it is an AR(1) model, but only at the seasonal lag of 12.
The model, which might be appropriate for average monthly temperatures, states that what happens this month depends on what happened in the same month last year plus some noise.
You see a simulation of 3 years from such a model. Notice, for example, that January is typically small whereas June and July are typically large. The behavior of the ACF and PACF for seasonal models
is similar to the nonseasonal model, but things happen only at the seasons, 1S, 2S, 3S, and so on.
4. ACF and PACF of Pure Seasonal Models
Here's an example of the ACF and PACF of a seasonal AR(1) with S equals 12 and of a seasonal MA(1) with S equals 12. Note the similarity with the nonseasonal models, but in these cases, everything happens at the seasonal lags, 1S, 2S, and so on. The Seasonal ARMA case is not shown, but the ACF and the PACF would tail off at the seasons.
5. Let's practice!
Now it's your turn!