Mixed seasonal models

1. Mixed seasonal models

Purely seasonal time series are rare, and typically we have to mix the seasonal part with the non-seasonal part.

2. Mixed Seasonal Model

These mixed models are called SARIMA(p,d,q)x(P,D, Q)_S, the lower case letters denote the orders of the nonseasonal components and the capital letters refer to the seasonal components. Here is an example of a mixed model that could be used for monthly data. You see that this month's value is related to last year using a seasonal AR(1). In addition, this month's value is related to last month's shock (or error).

3. ACF and PACF of SARIMA(0,0,1) x (1,0,0) s=12

Here we see the ACF and PACF of the model with a seasonal AR parameter of point-8 and a nonseasonal MA parameter of -.5. These type of correlation relationships, although idealized here, are typically seen with seasonal data.

4. ACF and PACF of SARIMA(0,0,1) x (1,0,0) s=12

First, focus on the seasonal lags, 12, 24,36,48. Note that the ACF tails and the PACF cuts off at lag 12. This indicates the seasonal AR component is needed.

5. ACF and PACF of SARIMA(0,0,1) x (1,0,0) s=12

Then, focus on the nonseasonal lags, 1,2,3,4,5. The ACF cuts off at lag 1 and the PACF tails. This indicates the nonseasonal MA component is needed.

6. Seasonal Persistence

The data shown are the quarterly occpancy rate for Hawaii and are typical of seasonal data. Notice that quarters 1 and 3 are always up, while quarters 2 and 4 are always down.

7. Seasonal Persistence

Next, look at the extracted seasonal component, it barely changes from year to year. This is called seasonal persistence,

8. Seasonal Persistence

and it can be taken care of by seasonally differencing. In this example, the seasonal difference is the difference between successive quarters. Note how this removes the seasonal persistence in the data.

9. Air Passengers

Let's analyze the Air Passengers series. We'll fit the best model and then forecast the series by the end of the chapter. The data are put into x for convenience. First note that the variance is increasing, so logging the data is appropriate. Notice how the variance stabilizes after the log transformation. There is obvious trend, so differencing the data is needed. Looking at the differenced data, notice the seasonal persistence where there are three cycles per year. This can be removed by taking a seasonal difference. Now the data look stationary and we can model the differenced data. At this point, we have little d equals 1 and capital D equals 1 with S equals 12

10. Air Passengers: ACF and PACF of ddlx

First inspect the ACF and PACF at the seasonal components.

11. Air Passengers: ACF and PACF of ddlx

It appears that the ACF is cutting off at lag 1s (s equals 12) and

12. Air Passengers: ACF and PACF of ddlx

the PACF tailing off at lags 1s, 2s, 3s, 4s. This suggests a seasonal MA(1), so P equals 0, Q equals 1 .

13. Air Passengers: ACF and PACF of ddlx

Next look at the nonseasonal lags. It seems that both the ACF and PACF are tailing off and this suggests an ARMA(1,1), so little p equals little q equals 1.

14. Air Passengers

Finally, we conclude that an SARIMA(1,1,1)x(0,1,1)_12 model is appropriate on the logged data. When we fit the model, we see that the non-seasonal AR component is not significant. Thus we take it out and fit an SARIMA(0,1,1)x(0,1,1)_12 model.

15. Air Passengers

The residual diagnostics are shown for this model, and everything looks good, so this is our final model.