Time series and forecasting · insight-1231
Decide the ARIMA constant deliberately, because with differencing it sets the long-run trend
- Evidence type
- documented fact
- Difficulty
- hard
- Roles
- Data scientist, ML engineer
- Industries
- energy, finance, cross-industry
Summary: situation, action, reason, boundary
Situation
An automatic ARIMA run on quarterly fuel-cost data returns d = 1 with a non-zero constant, and the five-year forecast climbs in a straight line.
Action
Write down the long-run shape you believe in (flat mean, wandering level, steady drift), choose d and the constant to match, and compare the model with and without the constant on rolling origins and on interval width at the longest horizon.
Reason
Per FPP3, d = 0 with a constant converges to the series mean, d = 1 with a constant follows a straight line, and d = 2 with a constant follows a quadratic that the book advises against; a larger d also makes intervals grow faster.
Boundary
ARIMA and seasonal ARIMA fitted automatically across many series.
Symptom or risk
The constant is not a nuisance parameter: combined with differencing it fixes the long-run shape, so a default can inject a trend, or with d = 2 a curve, that nobody intended.
Exceptions
At short horizons the long-run behaviour has little effect on the forecast.
Trade-offs
Dropping the constant removes drift that the data may really contain.
Pitfalls
Reading a significant constant in a differenced model as evidence that the series has a stable mean.
Lessons and concepts
Claims and their sources
FPP3 lists the long-term forecast shape by differencing order and constant: mean for d = 0 with a constant, straight-line trend for d = 1 with a constant, quadratic trend for d = 2 with a constant (advised against), and faster-growing intervals as d increases.