Given

• Order:
• 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 into the AR recursion:

Forecast (set ):

Innovation:

Step 2 — apply at

Plug in , , and :

Step 3 — iterate

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	—		—

Residuals are large because AR(1) cannot capture the trend or seasonality in — that is the role of ARIMA / SARIMA / ETS, applied to the same data in their own examples.