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 ARIMA recursion:

Difference — apply to convert to a stationary series:

In the differenced space, the model is ARMA:

Forecast the difference (set ):

Undifference — convert the difference forecast back to a forecast for :

Innovation:

Step 2 — apply at

First difference: .

Plug in , , , :

Step 3 — iterate

Two-stage pipeline at each : difference ARMA-forecast the difference undifference. Values rounded to 4 decimal places.

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	—	—			—