Prediction intervals for ensemble time series forecasts

M1

	Period	DEMOGR	INDUST	INDUSTRIAL	MACRO1	MACRO2	MICRO1	MICRO2	MICRO3
1	MONTHLY	75	183	0	64	92	10	89	104
2	QUARTERLY	39	17	1	45	59	5	21	16
3	YEARLY	30	35	0	30	29	16	29	12

Makridakis et al, 1982

M3

	Period	DEMOGRAPHI-	DEMOGRAPHIC	FINANCE	INDUSTRY	MACRO	MICRO	OTHER
1	MONTHLY	0	111	145	334	312	474	52
2	OTHER	0	0	29	0	0	4	141
3	QUARTERLY	57	0	76	83	336	204	0
4	YEARLY	0	245	58	102	83	146	11

Makridakis et al, 2000

Tourism

	Period	TOURISM
1	MONTHLY	366
2	QUARTERLY	427
3	YEARLY	518

Athanasopoulos et al, 2011

• Bates and Granger (1969) The Combination of Forecasts
• Many confirmations since.

For example:

model	two	four	six	eight
Theta	0.77	1.06	1.35	1.62
ARIMA-ETS average	0.72	1.07	1.38	1.75
ARIMA	0.75	1.12	1.43	1.79
ETS	0.75	1.11	1.44	1.82
Naive	1.08	1.11	1.74	1.87

Mean absolute scaled error of forecasts for 756 quarterly series from the M3 competition, forecast horizon ranging from two to eight quarters.

• Usually presumed some kind of weighted average of the components
• Weights might be estimated based on in-sample errors