1. 1 Chapter 3:Box-Jenkins Seasonal Modelling 3.1Stationarity Transformation “Pre-differencing transformation” is often used to stablize the seasonal variation of the time series. A common transformation is of the form:

2. 2 “Differencing transformation”: 1) (first non-seasonal difference) 2) (first seasonal difference, where L is the number of seasons in a year) 3) (first seasonal and first non- seasonal difference) Of course, one can also obtain second and higher order differences by simply applying the same rule.

3. 3 3.2 Autocorrelation and Partial Autocorrelation To determine if the data are stationary, we examine the behaviour of the autocorrelation and partial autocorrelation of the series at both the seasonal and non-seasonal level. The behaviour of the SAC and SPAC functions at lags 1 to L-3 is often considered as the behaviour of these functions at the non-seasonal level. A spike (significant memory) is said to exist if the corresponding SAC or SPAC are greater than twice their respective standard deviations. The time series is considered to be stationary if the SAC of the series cuts off or dies down reasonably quickly at both the seasonal & non-seasonal levels.

4. 4 Example 3.1 Figure 3.1 shows the monthly passenger totals (y t ) in thousands of passengers from 1949-59. The plot levels patterns of increasing seasonal variations. Figure 3.2 shows, which seems to have equalized the seasonal variations.

5. 5 Figure 3.1 Figure 3.1

6. 6 Figure 3.2 Figure 3.2

7. 7 The following SAS output shows the SAC’s of, its first difference at the non-seasonal level, at the seasonal level and at both the non-seasonal and seasonal levels. On the basis of the SAC’s, it appears that first difference at either seasonal level, or at both seasonal and non- seasonal levels are necessary to ensure the stationarity of the data.

8. 8

9. 9

10. 10

11. 11

12. 12 Example 3.2 Figure 3.3 shows the monthly values of the number of people (X t ) in Wisconsin employed in trade from 1961 to 1975. No predifferencing transformation appears to be necessary.

13. 13 Figure 3.3 Figure 3.3

14. 14 Next, let’s examine the SAC’s of X t, its first difference at the non-seasonal level, at the seasonal level and at both the seasonal and non-seasonal levels.

15. 15

16. 16

17. 17

18. 18

19. 19 Notations Now, suppose that is a pre-differencing transformed series, the general stationarity transformation is: where B is the lag (backward shift) operator, D is the degree of seasonal differencing and d is the degree of non-seasonal differencing.

20. 20 3.3Estimation and Diagnostic Checking The general seasonal Box-Jenkins model can be written in the form,  p (B)  p (B L )Z t = δ+θ q (B)  Q (B L )  t where  p (B) = (1   1 B   2 B 2  …   p B p ) is the non-seasonal autoregressive operator of order p,  p (B L ) = (1   1,L B L   2,L B 2L  …   p,L B pL ) is the seasonal autoregressive operator of order P,  q (B) = (1   1 B   2 B 2  …   p B q ) is the non-seasonal moving average operator of order q,  Q (B L ) = (1   1,L B L   2,L B 2L  …   Q,L B QL ) is the seasonal moving average operator of order Q,  =  p (B)  P (B L ) The ARIMA notation is usually written as ARIMA (p, d, q) (P, D, Q) L.

21. 21 Identification of the order p, q, P and Q are basically the same as in non- seasonal Box-Jenkins models. The following table provides some guidelines for choosing non-seasonal and seasonal operators

22. 22

23. 23 Estimation is usually carried out using maximum likelihood, as in the case of non-seasonal Box-Jenkins analysis. As an example, consider the SPAC of the time series of example 3.2, after first difference at both seasonal and non-seasonal levels.

24. 24

25. 25 At the non-seasonal level, both the SAC and SPAC appear to have a significant spike at lag 1 and cuts off after lag 1. One can tentatively identify an AR(1), MA(1) or ARMA(1, 1) models for the non-seasonal part of the series. At the seasonal level, the SPAL appears to be dying down, while the SAC cuts off after lag 12. Hence a seasonal MA(1) model is identified. Combining both the seasonal & non-seasonal levels, we have the following tentative models: ARIMA(1, 1, 0) (0, 1, 1) 12, ARIMA(0, 1, 1) (0, 1, 1) 12, ARIMA(1, 1, 1) (0, 1, 1) 12

26. 26 The SAS program for estimating these models is as follows: data employ; input @7 x; cards; 239.6 236.4 236.8 241.5 etc. ; proc arima data=employ; identify var=x(1,12); estimate p=1 q=(12) printall plot method=ml; estimate q=(1) (12) printall plot method=ml; estimate p=1 q=(1) (12) printall plot method=ml; run;

27. 27 Estimation results of an model

28. 28 Estimation results of an model

29. 29 Estimation results of an model

30. 30 Diagnostic checking is conducted using the Ljung-Box-Pierce Statistic where n is number of observations available after differencing.