Given

• Orders: , , , , , , ,
• Parameters: , , ,
• Endogenous 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

• Exogenous regressor :

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

Step 1 — formula

Substitute orders into the SARIMAX recursion. With and :

Expand the operator product (same as SARIMA, plus the exogenous term):

Forecast (set ):

Innovation:

Pre-compute the cross term: .

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

Plug in , , , , , , :

Step 3 — iterate

Each row adds the exogenous contribution on top of the SARIMA forecast.

			SARIMA part:
6	12
7	9
8	13
9	16
10	14
11	11
12	15
13	18
14	16
15	13
16	17
17	—				—

Here is collinear with ‘s linear trend, so adds an extra drift on top of the AR/seasonal structure — forecasts now overshoot. With a less collinear regressor, would correct part of that AR/seasonal lags miss.