Given

• Orders: ,
• Parameters: , ,
• Initial conditions: ,
• Data:

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

Step 1 — formula

Substitute , into the ARMA recursion:

Forecast (set ):

Innovation:

Step 2 — apply at

Plug in , , , , :

Step 3 — iterate

Each row uses the previous observation and the previous innovation . Values rounded to 4 decimal places.

1	12
2	10
3	8
4	11
5	14
6	12
7	9
8	13
9	16
10	14
11	11
12	15
13	18
14	16
15	13
16	17
17	—		—