Given

• Orders: , , , , , ,
• Parameters: , ,
• 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 orders into the SARIMA recursion. With and :

Expand the operator product:

Forecast (set ):

Innovation:

Pre-compute the cross term: .

Step 2 — apply at (first usable step: needs )

Plug in , , , , :

Step 3 — iterate

Each row uses lag-1 (), lag- (), and lag- ().

		, ,
6	12
7	9
8	13
9	16
10	14
11	11
12	15
13	18
14	16
15	13
16	17
17	—				—

Forecasts now sit much closer to the data than AR(1) — the seasonal lag tracks the within-cycle pattern, and lag-1 picks up the trend.